ab-angle->ABCF B

Percentage Accurate: 54.5% → 67.8%

Time: 10.2s

Alternatives: 26

Speedup: 13.7×

• 33.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	return ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	return ((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	return ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	return ((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 67.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(-0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := \frac{t\_0}{2}\\ t_2 := \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\\ t_3 := \left(angle \cdot \pi\right) \cdot 0.005555555555555556\\ t_4 := \cos \left(\frac{t\_3 + t\_3}{2}\right)\\ t_5 := \frac{t\_3}{2}\\ \mathbf{if}\;b\_m \leq 6.4 \cdot 10^{-97}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{t\_3 + t\_0}{2}\right) \cdot t\_4\right)\right) \cdot t\_2\\ \mathbf{elif}\;b\_m \leq 1.1 \cdot 10^{+58}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right) \cdot \mathsf{fma}\left(\cos t\_5, \cos t\_1, \sin t\_5 \cdot \sin t\_1\right)\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, t\_3\right)}{2}\right) \cdot t\_4\right)\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (* -0.005555555555555556 angle) PI))
        (t_1 (/ t_0 2.0))
        (t_2
         (*
          (* (sin (* (* PI angle) 0.005555555555555556)) (+ a b_m))
          (- b_m a)))
        (t_3 (* (* angle PI) 0.005555555555555556))
        (t_4 (cos (/ (+ t_3 t_3) 2.0)))
        (t_5 (/ t_3 2.0)))
   (if (<= b_m 6.4e-97)
     (* (* 2.0 (* (cos (/ (+ t_3 t_0) 2.0)) t_4)) t_2)
     (if (<= b_m 1.1e+58)
       (*
        (*
         2.0
         (*
          (cos
           (/
            (fma
             (* 0.005555555555555556 angle)
             PI
             (* (* angle PI) (- 0.005555555555555556)))
            2.0))
          (fma (cos t_5) (cos t_1) (* (sin t_5) (sin t_1)))))
        t_2)
       (*
        (*
         2.0
         (* (cos (/ (fma (* angle PI) -0.005555555555555556 t_3) 2.0)) t_4))
        t_2)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (-0.005555555555555556 * angle) * ((double) M_PI);
	double t_1 = t_0 / 2.0;
	double t_2 = (sin(((((double) M_PI) * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a);
	double t_3 = (angle * ((double) M_PI)) * 0.005555555555555556;
	double t_4 = cos(((t_3 + t_3) / 2.0));
	double t_5 = t_3 / 2.0;
	double tmp;
	if (b_m <= 6.4e-97) {
		tmp = (2.0 * (cos(((t_3 + t_0) / 2.0)) * t_4)) * t_2;
	} else if (b_m <= 1.1e+58) {
		tmp = (2.0 * (cos((fma((0.005555555555555556 * angle), ((double) M_PI), ((angle * ((double) M_PI)) * -0.005555555555555556)) / 2.0)) * fma(cos(t_5), cos(t_1), (sin(t_5) * sin(t_1))))) * t_2;
	} else {
		tmp = (2.0 * (cos((fma((angle * ((double) M_PI)), -0.005555555555555556, t_3) / 2.0)) * t_4)) * t_2;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(-0.005555555555555556 * angle) * pi)
	t_1 = Float64(t_0 / 2.0)
	t_2 = Float64(Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(a + b_m)) * Float64(b_m - a))
	t_3 = Float64(Float64(angle * pi) * 0.005555555555555556)
	t_4 = cos(Float64(Float64(t_3 + t_3) / 2.0))
	t_5 = Float64(t_3 / 2.0)
	tmp = 0.0
	if (b_m <= 6.4e-97)
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(Float64(t_3 + t_0) / 2.0)) * t_4)) * t_2);
	elseif (b_m <= 1.1e+58)
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(fma(Float64(0.005555555555555556 * angle), pi, Float64(Float64(angle * pi) * Float64(-0.005555555555555556))) / 2.0)) * fma(cos(t_5), cos(t_1), Float64(sin(t_5) * sin(t_1))))) * t_2);
	else
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(fma(Float64(angle * pi), -0.005555555555555556, t_3) / 2.0)) * t_4)) * t_2);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(-0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(N[(t$95$3 + t$95$3), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 / 2.0), $MachinePrecision]}, If[LessEqual[b$95$m, 6.4e-97], N[(N[(2.0 * N[(N[Cos[N[(N[(t$95$3 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[b$95$m, 1.1e+58], N[(N[(2.0 * N[(N[Cos[N[(N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(N[(angle * Pi), $MachinePrecision] * (-0.005555555555555556)), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t$95$5], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision] + N[(N[Sin[t$95$5], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[N[(N[(N[(angle * Pi), $MachinePrecision] * -0.005555555555555556 + t$95$3), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 6.39999999999999961e-97

   1. Initial program 56.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 62.4%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
   10. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \mathsf{PI}\left(\right), -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       5. lower-+.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       9. lift-*.f64 62.6

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       10. lift-neg.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       11. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       12. lift-PI.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       14. distribute-rgt-neg-in N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       17. associate-*r* N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       18. lower-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       19. lower-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       20. lift-PI.f64 65.4

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   11. Applied rewrites 65.4%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   if 6.39999999999999961e-97 < b < 1.1e58

   1. Initial program 41.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 41.2%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 54.0%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       3. lift--.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       5. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       7. lift-neg.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} - \left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} - \left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       9. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} - \left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} - \left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   11. Applied rewrites 54.6%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \mathsf{fma}\left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556}{2}\right), \cos \left(\frac{\left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2}\right), \sin \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556}{2}\right) \cdot \sin \left(\frac{\left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2}\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]

   if 1.1e58 < b

   1. Initial program 37.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 74.5%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
   10. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \mathsf{PI}\left(\right), -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       6. lift-neg.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       10. distribute-rgt-neg-in N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180} + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       12. lower-fma.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{-1}{180}, \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{-1}{180}, \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       14. lift-PI.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       17. lift-PI.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       18. lift-*.f64 80.6

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   11. Applied rewrites 80.6%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{-97}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(angle \cdot \pi\right) \cdot 0.005555555555555556}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+58}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right) \cdot \mathsf{fma}\left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556}{2}\right), \cos \left(\frac{\left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2}\right), \sin \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556}{2}\right) \cdot \sin \left(\frac{\left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2}\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(angle \cdot \pi\right) \cdot 0.005555555555555556}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ t_1 := \left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+260}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(a + b\_m\right)\right) \cdot \sin \left(2 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\left(\left(\left(a + b\_m\right) \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(b\_m - a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0)))
        (t_1
         (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))))
   (if (<= t_1 -1e+260)
     (* (* (* (* PI angle) (+ a b_m)) (- b_m a)) 0.011111111111111112)
     (if (<= t_1 5e+286)
       (* (* (- b_m a) (+ a b_m)) (sin (* 2.0 t_0)))
       (*
        (* 2.0 (cos (* (* PI angle) 0.005555555555555556)))
        (* (* (* (* (+ a b_m) PI) angle) 0.005555555555555556) (- b_m a)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double t_1 = ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0);
	double tmp;
	if (t_1 <= -1e+260) {
		tmp = (((((double) M_PI) * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	} else if (t_1 <= 5e+286) {
		tmp = ((b_m - a) * (a + b_m)) * sin((2.0 * t_0));
	} else {
		tmp = (2.0 * cos(((((double) M_PI) * angle) * 0.005555555555555556))) * (((((a + b_m) * ((double) M_PI)) * angle) * 0.005555555555555556) * (b_m - a));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double t_1 = ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0);
	double tmp;
	if (t_1 <= -1e+260) {
		tmp = (((Math.PI * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	} else if (t_1 <= 5e+286) {
		tmp = ((b_m - a) * (a + b_m)) * Math.sin((2.0 * t_0));
	} else {
		tmp = (2.0 * Math.cos(((Math.PI * angle) * 0.005555555555555556))) * (((((a + b_m) * Math.PI) * angle) * 0.005555555555555556) * (b_m - a));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = math.pi * (angle / 180.0)
	t_1 = ((2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)
	tmp = 0
	if t_1 <= -1e+260:
		tmp = (((math.pi * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112
	elif t_1 <= 5e+286:
		tmp = ((b_m - a) * (a + b_m)) * math.sin((2.0 * t_0))
	else:
		tmp = (2.0 * math.cos(((math.pi * angle) * 0.005555555555555556))) * (((((a + b_m) * math.pi) * angle) * 0.005555555555555556) * (b_m - a))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	t_1 = Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0))
	tmp = 0.0
	if (t_1 <= -1e+260)
		tmp = Float64(Float64(Float64(Float64(pi * angle) * Float64(a + b_m)) * Float64(b_m - a)) * 0.011111111111111112);
	elseif (t_1 <= 5e+286)
		tmp = Float64(Float64(Float64(b_m - a) * Float64(a + b_m)) * sin(Float64(2.0 * t_0)));
	else
		tmp = Float64(Float64(2.0 * cos(Float64(Float64(pi * angle) * 0.005555555555555556))) * Float64(Float64(Float64(Float64(Float64(a + b_m) * pi) * angle) * 0.005555555555555556) * Float64(b_m - a)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = pi * (angle / 180.0);
	t_1 = ((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0);
	tmp = 0.0;
	if (t_1 <= -1e+260)
		tmp = (((pi * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	elseif (t_1 <= 5e+286)
		tmp = ((b_m - a) * (a + b_m)) * sin((2.0 * t_0));
	else
		tmp = (2.0 * cos(((pi * angle) * 0.005555555555555556))) * (((((a + b_m) * pi) * angle) * 0.005555555555555556) * (b_m - a));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+260], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[t$95$1, 5e+286], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(a + b$95$m), $MachinePrecision] * Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < -1.00000000000000007e260

   1. Initial program 50.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 54.6

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      16. lift--.f64 68.4

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112 \]

   7. Applied rewrites 68.4%

      \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112 \]

   if -1.00000000000000007e260 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 5.0000000000000004e286

   1. Initial program 59.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \]

   6. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   if 5.0000000000000004e286 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 36.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 73.8%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\frac{1}{180} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)\right)\right) \cdot \left(\color{blue}{b} - a\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      8. lift-PI.f64 70.4

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(b - a\right)\right) \]

   10. Applied rewrites 70.4%

       \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\color{blue}{b} - a\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq -1 \cdot 10^{+260}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot \sin t\_1\right) \cdot \cos t\_1 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right) \cdot 1\right)\right) \cdot \left(\left(\sin t\_0 \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos t\_0\right) \cdot \left(\left(\left(\left(\left(a + b\_m\right) \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(b\_m - a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (* PI (/ angle 180.0))))
   (if (<=
        (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (sin t_1)) (cos t_1))
        5e+286)
     (*
      (*
       2.0
       (*
        (cos
         (/
          (fma
           (* 0.005555555555555556 angle)
           PI
           (* (* angle PI) (- 0.005555555555555556)))
          2.0))
        1.0))
      (* (* (sin t_0) (+ a b_m)) (- b_m a)))
     (*
      (* 2.0 (cos t_0))
      (* (* (* (* (+ a b_m) PI) angle) 0.005555555555555556) (- b_m a))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * sin(t_1)) * cos(t_1)) <= 5e+286) {
		tmp = (2.0 * (cos((fma((0.005555555555555556 * angle), ((double) M_PI), ((angle * ((double) M_PI)) * -0.005555555555555556)) / 2.0)) * 1.0)) * ((sin(t_0) * (a + b_m)) * (b_m - a));
	} else {
		tmp = (2.0 * cos(t_0)) * (((((a + b_m) * ((double) M_PI)) * angle) * 0.005555555555555556) * (b_m - a));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 5e+286)
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(fma(Float64(0.005555555555555556 * angle), pi, Float64(Float64(angle * pi) * Float64(-0.005555555555555556))) / 2.0)) * 1.0)) * Float64(Float64(sin(t_0) * Float64(a + b_m)) * Float64(b_m - a)));
	else
		tmp = Float64(Float64(2.0 * cos(t_0)) * Float64(Float64(Float64(Float64(Float64(a + b_m) * pi) * angle) * 0.005555555555555556) * Float64(b_m - a)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 5e+286], N[(N[(2.0 * N[(N[Cos[N[(N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(N[(angle * Pi), $MachinePrecision] * (-0.005555555555555556)), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(a + b$95$m), $MachinePrecision] * Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 5.0000000000000004e286

   1. Initial program 56.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 61.3%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

   10. Taylor expanded in angle around 0

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot 1\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 60.2%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot 1\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]

   if 5.0000000000000004e286 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 36.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 73.8%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\frac{1}{180} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)\right)\right) \cdot \left(\color{blue}{b} - a\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      8. lift-PI.f64 70.4

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(b - a\right)\right) \]

   10. Applied rewrites 70.4%

       \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\color{blue}{b} - a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right) \cdot 1\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot \sin t\_1\right) \cdot \cos t\_1 \leq 10^{+306}:\\ \;\;\;\;\left(2 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \left(\left(\sin t\_0 \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos t\_0\right) \cdot \left(\left(\left(\left(\left(a + b\_m\right) \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(b\_m - a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (* PI (/ angle 180.0))))
   (if (<=
        (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (sin t_1)) (cos t_1))
        1e+306)
     (*
      (* 2.0 (sin (fma (* 0.005555555555555556 angle) PI (/ PI 2.0))))
      (* (* (sin t_0) (+ a b_m)) (- b_m a)))
     (*
      (* 2.0 (cos t_0))
      (* (* (* (* (+ a b_m) PI) angle) 0.005555555555555556) (- b_m a))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * sin(t_1)) * cos(t_1)) <= 1e+306) {
		tmp = (2.0 * sin(fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) / 2.0)))) * ((sin(t_0) * (a + b_m)) * (b_m - a));
	} else {
		tmp = (2.0 * cos(t_0)) * (((((a + b_m) * ((double) M_PI)) * angle) * 0.005555555555555556) * (b_m - a));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 1e+306)
		tmp = Float64(Float64(2.0 * sin(fma(Float64(0.005555555555555556 * angle), pi, Float64(pi / 2.0)))) * Float64(Float64(sin(t_0) * Float64(a + b_m)) * Float64(b_m - a)));
	else
		tmp = Float64(Float64(2.0 * cos(t_0)) * Float64(Float64(Float64(Float64(Float64(a + b_m) * pi) * angle) * 0.005555555555555556) * Float64(b_m - a)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 1e+306], N[(N[(2.0 * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(a + b$95$m), $MachinePrecision] * Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 1.00000000000000002e306

   1. Initial program 55.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 57.2%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. sin-+PI/2-rev N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(2 \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      11. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \frac{\pi}{2}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left(2 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \mathsf{PI}\left(\right), \frac{\pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \mathsf{PI}\left(\right), \frac{\pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      14. lift-PI.f64 60.7

          \[\leadsto \left(2 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 60.7%

      \[\leadsto \left(2 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   if 1.00000000000000002e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 38.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 76.3%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\frac{1}{180} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)\right)\right) \cdot \left(\color{blue}{b} - a\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(b - a\right)\right) \]

      8. lift-PI.f64 73.4

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(b - a\right)\right) \]

   10. Applied rewrites 73.4%

       \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\color{blue}{b} - a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq 10^{+306}:\\ \;\;\;\;\left(2 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\left(\left(\left(a + b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -1 \cdot 10^{+270}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot 1\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<=
        (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
        -1e+270)
     (* (* (* (* PI angle) (+ a b_m)) (- b_m a)) 0.011111111111111112)
     (*
      (* 2.0 1.0)
      (*
       (* (sin (* (* 0.005555555555555556 angle) PI)) (+ a b_m))
       (- b_m a))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= -1e+270) {
		tmp = (((((double) M_PI) * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	} else {
		tmp = (2.0 * 1.0) * ((sin(((0.005555555555555556 * angle) * ((double) M_PI))) * (a + b_m)) * (b_m - a));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0)) <= -1e+270) {
		tmp = (((Math.PI * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	} else {
		tmp = (2.0 * 1.0) * ((Math.sin(((0.005555555555555556 * angle) * Math.PI)) * (a + b_m)) * (b_m - a));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = math.pi * (angle / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)) <= -1e+270:
		tmp = (((math.pi * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112
	else:
		tmp = (2.0 * 1.0) * ((math.sin(((0.005555555555555556 * angle) * math.pi)) * (a + b_m)) * (b_m - a))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= -1e+270)
		tmp = Float64(Float64(Float64(Float64(pi * angle) * Float64(a + b_m)) * Float64(b_m - a)) * 0.011111111111111112);
	else
		tmp = Float64(Float64(2.0 * 1.0) * Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle) * pi)) * Float64(a + b_m)) * Float64(b_m - a)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = pi * (angle / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= -1e+270)
		tmp = (((pi * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	else
		tmp = (2.0 * 1.0) * ((sin(((0.005555555555555556 * angle) * pi)) * (a + b_m)) * (b_m - a));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], -1e+270], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(N[(2.0 * 1.0), $MachinePrecision] * N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < -1e270

   1. Initial program 51.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 55.4

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      16. lift--.f64 69.4

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112 \]

   7. Applied rewrites 69.4%

      \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112 \]

   if -1e270 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 50.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      2. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      9. lift-PI.f64 63.5

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 63.5%

      \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   10. Taylor expanded in angle around 0

       \[\leadsto \left(2 \cdot 1\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 61.7%

       \[\leadsto \left(2 \cdot 1\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq -1 \cdot 10^{+270}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot 1\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -1 \cdot 10^{+260}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<=
        (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
        -1e+260)
     (* (* (* (* PI angle) (+ a b_m)) (- b_m a)) 0.011111111111111112)
     (*
      2.0
      (*
       (* (sin (* (* PI angle) 0.005555555555555556)) (+ a b_m))
       (- b_m a))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= -1e+260) {
		tmp = (((((double) M_PI) * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * ((sin(((((double) M_PI) * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0)) <= -1e+260) {
		tmp = (((Math.PI * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * ((Math.sin(((Math.PI * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = math.pi * (angle / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)) <= -1e+260:
		tmp = (((math.pi * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112
	else:
		tmp = 2.0 * ((math.sin(((math.pi * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= -1e+260)
		tmp = Float64(Float64(Float64(Float64(pi * angle) * Float64(a + b_m)) * Float64(b_m - a)) * 0.011111111111111112);
	else
		tmp = Float64(2.0 * Float64(Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(a + b_m)) * Float64(b_m - a)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = pi * (angle / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= -1e+260)
		tmp = (((pi * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	else
		tmp = 2.0 * ((sin(((pi * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], -1e+260], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(2.0 * N[(N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < -1.00000000000000007e260

   1. Initial program 50.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 54.6

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      16. lift--.f64 68.4

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112 \]

   7. Applied rewrites 68.4%

      \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112 \]

   if -1.00000000000000007e260 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 51.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]

   8. Taylor expanded in angle around 0

      \[\leadsto 2 \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 62.1%

       \[\leadsto 2 \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq -1 \cdot 10^{+260}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-209}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\right) \cdot \left(a \cdot \left(b\_m - a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b_m 2.0) (pow a 2.0)))))
   (if (<= t_0 -4e-209)
     (* (* -0.011111111111111112 a) (* (* angle PI) a))
     (if (<= t_0 INFINITY)
       (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112)
       (* (* (* PI angle) (* a (- b_m a))) 0.011111111111111112)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = 2.0 * (pow(b_m, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_0 <= -4e-209) {
		tmp = (-0.011111111111111112 * a) * ((angle * ((double) M_PI)) * a);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112;
	} else {
		tmp = ((((double) M_PI) * angle) * (a * (b_m - a))) * 0.011111111111111112;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = 2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0));
	double tmp;
	if (t_0 <= -4e-209) {
		tmp = (-0.011111111111111112 * a) * ((angle * Math.PI) * a);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = ((Math.PI * (b_m * b_m)) * angle) * 0.011111111111111112;
	} else {
		tmp = ((Math.PI * angle) * (a * (b_m - a))) * 0.011111111111111112;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = 2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))
	tmp = 0
	if t_0 <= -4e-209:
		tmp = (-0.011111111111111112 * a) * ((angle * math.pi) * a)
	elif t_0 <= math.inf:
		tmp = ((math.pi * (b_m * b_m)) * angle) * 0.011111111111111112
	else:
		tmp = ((math.pi * angle) * (a * (b_m - a))) * 0.011111111111111112
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_0 <= -4e-209)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle * pi) * a));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112);
	else
		tmp = Float64(Float64(Float64(pi * angle) * Float64(a * Float64(b_m - a))) * 0.011111111111111112);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = 2.0 * ((b_m ^ 2.0) - (a ^ 2.0));
	tmp = 0.0;
	if (t_0 <= -4e-209)
		tmp = (-0.011111111111111112 * a) * ((angle * pi) * a);
	elseif (t_0 <= Inf)
		tmp = ((pi * (b_m * b_m)) * angle) * 0.011111111111111112;
	else
		tmp = ((pi * angle) * (a * (b_m - a))) * 0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-209], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(N[(N[(Pi * angle), $MachinePrecision] * N[(a * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.0000000000000002e-209

   1. Initial program 51.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 45.4

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 45.4

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   8. Applied rewrites 45.4%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 45.4

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   10. Applied rewrites 45.4%

       \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 54.3

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   12. Applied rewrites 54.3%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]

   if -4.0000000000000002e-209 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < +inf.0

   1. Initial program 57.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 58.8

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{90} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      6. pow2 N/A

         \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      7. lift-*.f64 58.8

         \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112 \]

   8. Applied rewrites 58.8%

      \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112 \]

   if +inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 0.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 62.3

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   6. Taylor expanded in a around inf

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(a \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.1%

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(a \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(-0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\\ t_2 := \left(angle \cdot \pi\right) \cdot 0.005555555555555556\\ t_3 := \cos \left(\frac{t\_2 + t\_2}{2}\right)\\ \mathbf{if}\;b\_m \leq 1.15 \cdot 10^{-96}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{t\_2 + t\_0}{2}\right) \cdot t\_3\right)\right) \cdot t\_1\\ \mathbf{elif}\;b\_m \leq 1.15 \cdot 10^{+58}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right) \cdot \sin \left(\frac{t\_2 - t\_0}{2} + \frac{\pi}{2}\right)\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, t\_2\right)}{2}\right) \cdot t\_3\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (* -0.005555555555555556 angle) PI))
        (t_1
         (*
          (* (sin (* (* PI angle) 0.005555555555555556)) (+ a b_m))
          (- b_m a)))
        (t_2 (* (* angle PI) 0.005555555555555556))
        (t_3 (cos (/ (+ t_2 t_2) 2.0))))
   (if (<= b_m 1.15e-96)
     (* (* 2.0 (* (cos (/ (+ t_2 t_0) 2.0)) t_3)) t_1)
     (if (<= b_m 1.15e+58)
       (*
        (*
         2.0
         (*
          (cos
           (/
            (fma
             (* 0.005555555555555556 angle)
             PI
             (* (* angle PI) (- 0.005555555555555556)))
            2.0))
          (sin (+ (/ (- t_2 t_0) 2.0) (/ PI 2.0)))))
        t_1)
       (*
        (*
         2.0
         (* (cos (/ (fma (* angle PI) -0.005555555555555556 t_2) 2.0)) t_3))
        t_1)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (-0.005555555555555556 * angle) * ((double) M_PI);
	double t_1 = (sin(((((double) M_PI) * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a);
	double t_2 = (angle * ((double) M_PI)) * 0.005555555555555556;
	double t_3 = cos(((t_2 + t_2) / 2.0));
	double tmp;
	if (b_m <= 1.15e-96) {
		tmp = (2.0 * (cos(((t_2 + t_0) / 2.0)) * t_3)) * t_1;
	} else if (b_m <= 1.15e+58) {
		tmp = (2.0 * (cos((fma((0.005555555555555556 * angle), ((double) M_PI), ((angle * ((double) M_PI)) * -0.005555555555555556)) / 2.0)) * sin((((t_2 - t_0) / 2.0) + (((double) M_PI) / 2.0))))) * t_1;
	} else {
		tmp = (2.0 * (cos((fma((angle * ((double) M_PI)), -0.005555555555555556, t_2) / 2.0)) * t_3)) * t_1;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(-0.005555555555555556 * angle) * pi)
	t_1 = Float64(Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(a + b_m)) * Float64(b_m - a))
	t_2 = Float64(Float64(angle * pi) * 0.005555555555555556)
	t_3 = cos(Float64(Float64(t_2 + t_2) / 2.0))
	tmp = 0.0
	if (b_m <= 1.15e-96)
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(Float64(t_2 + t_0) / 2.0)) * t_3)) * t_1);
	elseif (b_m <= 1.15e+58)
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(fma(Float64(0.005555555555555556 * angle), pi, Float64(Float64(angle * pi) * Float64(-0.005555555555555556))) / 2.0)) * sin(Float64(Float64(Float64(t_2 - t_0) / 2.0) + Float64(pi / 2.0))))) * t_1);
	else
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(fma(Float64(angle * pi), -0.005555555555555556, t_2) / 2.0)) * t_3)) * t_1);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(-0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(N[(t$95$2 + t$95$2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 1.15e-96], N[(N[(2.0 * N[(N[Cos[N[(N[(t$95$2 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[b$95$m, 1.15e+58], N[(N[(2.0 * N[(N[Cos[N[(N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(N[(angle * Pi), $MachinePrecision] * (-0.005555555555555556)), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(N[(t$95$2 - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[N[(N[(N[(angle * Pi), $MachinePrecision] * -0.005555555555555556 + t$95$2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.15e-96

   1. Initial program 56.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 62.4%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
   10. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \mathsf{PI}\left(\right), -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       5. lower-+.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       9. lift-*.f64 62.6

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       10. lift-neg.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       11. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       12. lift-PI.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       14. distribute-rgt-neg-in N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       17. associate-*r* N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       18. lower-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       19. lower-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       20. lift-PI.f64 65.4

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   11. Applied rewrites 65.4%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   if 1.15e-96 < b < 1.15000000000000001e58

   1. Initial program 41.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 41.2%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 54.0%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]

       2. sin-+PI/2-rev N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \sin \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]

       3. lower-sin.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \sin \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]

       4. lower-+.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \sin \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   11. Applied rewrites 46.3%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \sin \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2} + \frac{\pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]

   if 1.15000000000000001e58 < b

   1. Initial program 37.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 74.5%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
   10. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \mathsf{PI}\left(\right), -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       6. lift-neg.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       10. distribute-rgt-neg-in N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180} + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       12. lower-fma.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{-1}{180}, \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{-1}{180}, \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       14. lift-PI.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       17. lift-PI.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       18. lift-*.f64 80.6

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   11. Applied rewrites 80.6%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-96}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(angle \cdot \pi\right) \cdot 0.005555555555555556}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+58}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right) \cdot \sin \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2} + \frac{\pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(angle \cdot \pi\right) \cdot 0.005555555555555556}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ t_1 := \left(angle \cdot \pi\right) \cdot 0.005555555555555556\\ t_2 := \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\\ \mathbf{if}\;b\_m \leq 1.6 \cdot 10^{-200}:\\ \;\;\;\;\left(\left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right) \cdot 2\right) \cdot \frac{\sin \left(t\_0 + t\_0\right) + \sin \left(\mathsf{fma}\left(\pi, \frac{angle}{180}, \left(-\pi\right) \cdot \frac{angle}{180}\right)\right)}{2}\\ \mathbf{elif}\;b\_m \leq 1.12 \cdot 10^{+56}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right) \cdot 1\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, t\_1\right)}{2}\right) \cdot \cos \left(\frac{t\_1 + t\_1}{2}\right)\right)\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0)))
        (t_1 (* (* angle PI) 0.005555555555555556))
        (t_2
         (*
          (* (sin (* (* PI angle) 0.005555555555555556)) (+ a b_m))
          (- b_m a))))
   (if (<= b_m 1.6e-200)
     (*
      (* (* (+ b_m a) (- b_m a)) 2.0)
      (/
       (+
        (sin (+ t_0 t_0))
        (sin (fma PI (/ angle 180.0) (* (- PI) (/ angle 180.0)))))
       2.0))
     (if (<= b_m 1.12e+56)
       (*
        (*
         2.0
         (*
          (cos
           (/
            (fma
             (* 0.005555555555555556 angle)
             PI
             (* (* angle PI) (- 0.005555555555555556)))
            2.0))
          1.0))
        t_2)
       (*
        (*
         2.0
         (*
          (cos (/ (fma (* angle PI) -0.005555555555555556 t_1) 2.0))
          (cos (/ (+ t_1 t_1) 2.0))))
        t_2)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double t_1 = (angle * ((double) M_PI)) * 0.005555555555555556;
	double t_2 = (sin(((((double) M_PI) * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a);
	double tmp;
	if (b_m <= 1.6e-200) {
		tmp = (((b_m + a) * (b_m - a)) * 2.0) * ((sin((t_0 + t_0)) + sin(fma(((double) M_PI), (angle / 180.0), (-((double) M_PI) * (angle / 180.0))))) / 2.0);
	} else if (b_m <= 1.12e+56) {
		tmp = (2.0 * (cos((fma((0.005555555555555556 * angle), ((double) M_PI), ((angle * ((double) M_PI)) * -0.005555555555555556)) / 2.0)) * 1.0)) * t_2;
	} else {
		tmp = (2.0 * (cos((fma((angle * ((double) M_PI)), -0.005555555555555556, t_1) / 2.0)) * cos(((t_1 + t_1) / 2.0)))) * t_2;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	t_1 = Float64(Float64(angle * pi) * 0.005555555555555556)
	t_2 = Float64(Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(a + b_m)) * Float64(b_m - a))
	tmp = 0.0
	if (b_m <= 1.6e-200)
		tmp = Float64(Float64(Float64(Float64(b_m + a) * Float64(b_m - a)) * 2.0) * Float64(Float64(sin(Float64(t_0 + t_0)) + sin(fma(pi, Float64(angle / 180.0), Float64(Float64(-pi) * Float64(angle / 180.0))))) / 2.0));
	elseif (b_m <= 1.12e+56)
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(fma(Float64(0.005555555555555556 * angle), pi, Float64(Float64(angle * pi) * Float64(-0.005555555555555556))) / 2.0)) * 1.0)) * t_2);
	else
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(fma(Float64(angle * pi), -0.005555555555555556, t_1) / 2.0)) * cos(Float64(Float64(t_1 + t_1) / 2.0)))) * t_2);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.6e-200], N[(N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[Sin[N[(t$95$0 + t$95$0), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision] + N[((-Pi) * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.12e+56], N[(N[(2.0 * N[(N[Cos[N[(N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(N[(angle * Pi), $MachinePrecision] * (-0.005555555555555556)), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[N[(N[(N[(angle * Pi), $MachinePrecision] * -0.005555555555555556 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$1 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.59999999999999991e-200

   1. Initial program 55.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \]

      4. cos-neg-rev N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right)}\right) \]

      5. sin-cos-mult N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \frac{angle}{180} - \left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right)\right) + \sin \left(\pi \cdot \frac{angle}{180} + \left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right)\right)}{2}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \frac{angle}{180} - \left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right)\right) + \sin \left(\pi \cdot \frac{angle}{180} + \left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right)\right)}{2}} \]

   6. Applied rewrites 60.1%

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \frac{angle}{180} - \left(-\pi \cdot \frac{angle}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\pi, \frac{angle}{180}, -\pi \cdot \frac{angle}{180}\right)\right)}{2}} \]

   if 1.59999999999999991e-200 < b < 1.12e56

   1. Initial program 49.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 56.6%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 55.7%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

   10. Taylor expanded in angle around 0

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot 1\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 56.7%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot 1\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]

   if 1.12e56 < b

   1. Initial program 37.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 74.5%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
   10. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \mathsf{PI}\left(\right), -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       6. lift-neg.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       10. distribute-rgt-neg-in N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180} + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       12. lower-fma.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{-1}{180}, \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{-1}{180}, \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       14. lift-PI.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       17. lift-PI.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       18. lift-*.f64 80.6

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   11. Applied rewrites 80.6%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-200}:\\ \;\;\;\;\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \frac{\sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right) + \sin \left(\mathsf{fma}\left(\pi, \frac{angle}{180}, \left(-\pi\right) \cdot \frac{angle}{180}\right)\right)}{2}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{+56}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right) \cdot 1\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(angle \cdot \pi\right) \cdot 0.005555555555555556}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\\ t_1 := \left(angle \cdot \pi\right) \cdot 0.005555555555555556\\ t_2 := \cos \left(\frac{t\_1 + t\_1}{2}\right)\\ \mathbf{if}\;b\_m \leq 1.2 \cdot 10^{+56}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{t\_1 + \left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2}\right) \cdot t\_2\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, t\_1\right)}{2}\right) \cdot t\_2\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0
         (*
          (* (sin (* (* PI angle) 0.005555555555555556)) (+ a b_m))
          (- b_m a)))
        (t_1 (* (* angle PI) 0.005555555555555556))
        (t_2 (cos (/ (+ t_1 t_1) 2.0))))
   (if (<= b_m 1.2e+56)
     (*
      (*
       2.0
       (* (cos (/ (+ t_1 (* (* -0.005555555555555556 angle) PI)) 2.0)) t_2))
      t_0)
     (*
      (*
       2.0
       (* (cos (/ (fma (* angle PI) -0.005555555555555556 t_1) 2.0)) t_2))
      t_0))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (sin(((((double) M_PI) * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a);
	double t_1 = (angle * ((double) M_PI)) * 0.005555555555555556;
	double t_2 = cos(((t_1 + t_1) / 2.0));
	double tmp;
	if (b_m <= 1.2e+56) {
		tmp = (2.0 * (cos(((t_1 + ((-0.005555555555555556 * angle) * ((double) M_PI))) / 2.0)) * t_2)) * t_0;
	} else {
		tmp = (2.0 * (cos((fma((angle * ((double) M_PI)), -0.005555555555555556, t_1) / 2.0)) * t_2)) * t_0;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(a + b_m)) * Float64(b_m - a))
	t_1 = Float64(Float64(angle * pi) * 0.005555555555555556)
	t_2 = cos(Float64(Float64(t_1 + t_1) / 2.0))
	tmp = 0.0
	if (b_m <= 1.2e+56)
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(Float64(t_1 + Float64(Float64(-0.005555555555555556 * angle) * pi)) / 2.0)) * t_2)) * t_0);
	else
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(fma(Float64(angle * pi), -0.005555555555555556, t_1) / 2.0)) * t_2)) * t_0);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(t$95$1 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 1.2e+56], N[(N[(2.0 * N[(N[Cos[N[(N[(t$95$1 + N[(N[(-0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[N[(N[(N[(angle * Pi), $MachinePrecision] * -0.005555555555555556 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.20000000000000007e56

   1. Initial program 54.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 61.4%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
   10. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \mathsf{PI}\left(\right), -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       5. lower-+.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       9. lift-*.f64 60.7

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       10. lift-neg.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       11. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       12. lift-PI.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       14. distribute-rgt-neg-in N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       17. associate-*r* N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       18. lower-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       19. lower-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       20. lift-PI.f64 63.0

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   11. Applied rewrites 63.0%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   if 1.20000000000000007e56 < b

   1. Initial program 37.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 74.5%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
   10. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \mathsf{PI}\left(\right), -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       6. lift-neg.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       10. distribute-rgt-neg-in N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180} + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       12. lower-fma.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{-1}{180}, \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{-1}{180}, \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       14. lift-PI.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       17. lift-PI.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{-1}{180}, \left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       18. lift-*.f64 80.6

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   11. Applied rewrites 80.6%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{+56}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(-0.005555555555555556 \cdot angle\right) \cdot \pi}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(angle \cdot \pi\right) \cdot 0.005555555555555556}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle \cdot \pi, -0.005555555555555556, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(angle \cdot \pi\right) \cdot 0.005555555555555556}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(angle \cdot \pi\right) \cdot 0.005555555555555556\\ \mathbf{if}\;angle \leq 3.2 \cdot 10^{+168}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, 0.005555555555555556 \cdot \pi, \left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)}{2}\right) \cdot \cos \left(\frac{t\_0 + t\_0}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(\frac{b\_m}{a} \cdot \frac{b\_m}{a} - 1\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (* angle PI) 0.005555555555555556)))
   (if (<= angle 3.2e+168)
     (*
      (*
       2.0
       (*
        (cos
         (/
          (fma
           angle
           (* 0.005555555555555556 PI)
           (* (* -0.005555555555555556 angle) PI))
          2.0))
        (cos (/ (+ t_0 t_0) 2.0))))
      (* (* (sin (* (* PI angle) 0.005555555555555556)) (+ a b_m)) (- b_m a)))
     (*
      (*
       (* 2.0 (* (- (* (/ b_m a) (/ b_m a)) 1.0) (* a a)))
       (sin (* PI (/ angle 180.0))))
      (cos (* PI (* 0.005555555555555556 angle)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (angle * ((double) M_PI)) * 0.005555555555555556;
	double tmp;
	if (angle <= 3.2e+168) {
		tmp = (2.0 * (cos((fma(angle, (0.005555555555555556 * ((double) M_PI)), ((-0.005555555555555556 * angle) * ((double) M_PI))) / 2.0)) * cos(((t_0 + t_0) / 2.0)))) * ((sin(((((double) M_PI) * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a));
	} else {
		tmp = ((2.0 * ((((b_m / a) * (b_m / a)) - 1.0) * (a * a))) * sin((((double) M_PI) * (angle / 180.0)))) * cos((((double) M_PI) * (0.005555555555555556 * angle)));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(angle * pi) * 0.005555555555555556)
	tmp = 0.0
	if (angle <= 3.2e+168)
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(fma(angle, Float64(0.005555555555555556 * pi), Float64(Float64(-0.005555555555555556 * angle) * pi)) / 2.0)) * cos(Float64(Float64(t_0 + t_0) / 2.0)))) * Float64(Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(a + b_m)) * Float64(b_m - a)));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(Float64(Float64(b_m / a) * Float64(b_m / a)) - 1.0) * Float64(a * a))) * sin(Float64(pi * Float64(angle / 180.0)))) * cos(Float64(pi * Float64(0.005555555555555556 * angle))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[angle, 3.2e+168], N[(N[(2.0 * N[(N[Cos[N[(N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision] + N[(N[(-0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(N[(N[(b$95$m / a), $MachinePrecision] * N[(b$95$m / a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.2000000000000001e168

   1. Initial program 53.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 70.3%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
   10. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \mathsf{PI}\left(\right), -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, \frac{1}{180} \cdot \mathsf{PI}\left(\right), -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, \frac{1}{180} \cdot \mathsf{PI}\left(\right), -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       10. lift-PI.f64 71.0

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, 0.005555555555555556 \cdot \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       11. lift-neg.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       13. lift-PI.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       14. lift-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       15. distribute-rgt-neg-in N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       17. *-commutative N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       18. associate-*r* N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       19. lower-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       20. lower-*.f64 N/A

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       21. lift-PI.f64 71.6

           \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, 0.005555555555555556 \cdot \pi, \left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   11. Applied rewrites 71.6%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, 0.005555555555555556 \cdot \pi, \left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   if 3.2000000000000001e168 < angle

   1. Initial program 36.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left({a}^{2} \cdot \left(\frac{{b}^{2}}{{a}^{2}} - 1\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{{b}^{2}}{{a}^{2}} - 1\right) \cdot \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{{b}^{2}}{{a}^{2}} - 1\right) \cdot \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{{b}^{2}}{{a}^{2}} - 1\right) \cdot {\color{blue}{a}}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b \cdot b}{{a}^{2}} - 1\right) \cdot {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b \cdot b}{a \cdot a} - 1\right) \cdot {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. times-frac N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot \left(a \cdot \color{blue}{a}\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lower-*.f64 35.0

          \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot \left(a \cdot \color{blue}{a}\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Applied rewrites 35.0%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot \left(a \cdot a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 41.5

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right) \]

   8. Applied rewrites 41.5%

      \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3.2 \cdot 10^{+168}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(angle, 0.005555555555555556 \cdot \pi, \left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(angle \cdot \pi\right) \cdot 0.005555555555555556}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\\ t_1 := \cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right)\\ \mathbf{if}\;a \leq 10^{+235}:\\ \;\;\;\;\left(2 \cdot \left(t\_1 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(t\_1 \cdot 1\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0
         (*
          (* (sin (* (* PI angle) 0.005555555555555556)) (+ a b_m))
          (- b_m a)))
        (t_1
         (cos
          (/
           (fma
            (* 0.005555555555555556 angle)
            PI
            (* (* angle PI) (- 0.005555555555555556)))
           2.0))))
   (if (<= a 1e+235)
     (* (* 2.0 (* t_1 (cos (* (* angle PI) 0.005555555555555556)))) t_0)
     (* (* 2.0 (* t_1 1.0)) t_0))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (sin(((((double) M_PI) * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a);
	double t_1 = cos((fma((0.005555555555555556 * angle), ((double) M_PI), ((angle * ((double) M_PI)) * -0.005555555555555556)) / 2.0));
	double tmp;
	if (a <= 1e+235) {
		tmp = (2.0 * (t_1 * cos(((angle * ((double) M_PI)) * 0.005555555555555556)))) * t_0;
	} else {
		tmp = (2.0 * (t_1 * 1.0)) * t_0;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(a + b_m)) * Float64(b_m - a))
	t_1 = cos(Float64(fma(Float64(0.005555555555555556 * angle), pi, Float64(Float64(angle * pi) * Float64(-0.005555555555555556))) / 2.0))
	tmp = 0.0
	if (a <= 1e+235)
		tmp = Float64(Float64(2.0 * Float64(t_1 * cos(Float64(Float64(angle * pi) * 0.005555555555555556)))) * t_0);
	else
		tmp = Float64(Float64(2.0 * Float64(t_1 * 1.0)) * t_0);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(N[(angle * Pi), $MachinePrecision] * (-0.005555555555555556)), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, 1e+235], N[(N[(2.0 * N[(t$95$1 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(2.0 * N[(t$95$1 * 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.0000000000000001e235

   1. Initial program 51.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 62.4%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 64.2%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

   10. Taylor expanded in angle around 0

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       3. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       4. lift-*.f64 64.2

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   12. Applied rewrites 64.2%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   if 1.0000000000000001e235 < a

   1. Initial program 45.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 61.0%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 61.0%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

   10. Taylor expanded in angle around 0

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot 1\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 83.2%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot 1\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 10^{+235}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right) \cdot 1\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 9 \cdot 10^{+168}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right) \cdot 1\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(\frac{b\_m}{a} \cdot \frac{b\_m}{a} - 1\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 9e+168)
   (*
    (*
     2.0
     (*
      (cos
       (/
        (fma
         (* 0.005555555555555556 angle)
         PI
         (* (* angle PI) (- 0.005555555555555556)))
        2.0))
      1.0))
    (* (* (sin (* (* PI angle) 0.005555555555555556)) (+ a b_m)) (- b_m a)))
   (*
    (*
     (* 2.0 (* (- (* (/ b_m a) (/ b_m a)) 1.0) (* a a)))
     (sin (* PI (/ angle 180.0))))
    (cos (* PI (* 0.005555555555555556 angle))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 9e+168) {
		tmp = (2.0 * (cos((fma((0.005555555555555556 * angle), ((double) M_PI), ((angle * ((double) M_PI)) * -0.005555555555555556)) / 2.0)) * 1.0)) * ((sin(((((double) M_PI) * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a));
	} else {
		tmp = ((2.0 * ((((b_m / a) * (b_m / a)) - 1.0) * (a * a))) * sin((((double) M_PI) * (angle / 180.0)))) * cos((((double) M_PI) * (0.005555555555555556 * angle)));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 9e+168)
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(fma(Float64(0.005555555555555556 * angle), pi, Float64(Float64(angle * pi) * Float64(-0.005555555555555556))) / 2.0)) * 1.0)) * Float64(Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(a + b_m)) * Float64(b_m - a)));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(Float64(Float64(b_m / a) * Float64(b_m / a)) - 1.0) * Float64(a * a))) * sin(Float64(pi * Float64(angle / 180.0)))) * cos(Float64(pi * Float64(0.005555555555555556 * angle))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 9e+168], N[(N[(2.0 * N[(N[Cos[N[(N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(N[(angle * Pi), $MachinePrecision] * (-0.005555555555555556)), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(N[(N[(b$95$m / a), $MachinePrecision] * N[(b$95$m / a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 9.00000000000000024e168

   1. Initial program 53.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      8. count-2-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      9. cos-neg-rev N/A

         \[\leadsto \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      10. sum-cos N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \cdot \cos \left(\frac{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 70.3%

      \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right)\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

   10. Taylor expanded in angle around 0

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}{2}\right) \cdot 1\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 71.3%

       \[\leadsto \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}{2}\right) \cdot 1\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]

   if 9.00000000000000024e168 < angle

   1. Initial program 36.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left({a}^{2} \cdot \left(\frac{{b}^{2}}{{a}^{2}} - 1\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{{b}^{2}}{{a}^{2}} - 1\right) \cdot \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{{b}^{2}}{{a}^{2}} - 1\right) \cdot \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{{b}^{2}}{{a}^{2}} - 1\right) \cdot {\color{blue}{a}}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b \cdot b}{{a}^{2}} - 1\right) \cdot {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b \cdot b}{a \cdot a} - 1\right) \cdot {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. times-frac N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot \left(a \cdot \color{blue}{a}\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lower-*.f64 35.0

          \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot \left(a \cdot \color{blue}{a}\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Applied rewrites 35.0%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot \left(a \cdot a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 41.5

         \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right) \]

   8. Applied rewrites 41.5%

      \[\leadsto \left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 9 \cdot 10^{+168}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right) \cdot 1\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(\frac{b}{a} \cdot \frac{b}{a} - 1\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 3.6 \cdot 10^{+102}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \mathbf{elif}\;angle \leq 3.6 \cdot 10^{+164}:\\ \;\;\;\;\left(\left(\left(\sin \left(\mathsf{fma}\left(-0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(a + b\_m\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 3.6e+102)
   (*
    (* 2.0 (cos (* (* PI angle) 0.005555555555555556)))
    (* (* (sin (* (* 0.005555555555555556 angle) PI)) (+ a b_m)) (- b_m a)))
   (if (<= angle 3.6e+164)
     (*
      (*
       (*
        (* (sin (fma -0.005555555555555556 (* angle PI) (* 0.5 PI))) (+ a b_m))
        (- b_m a))
       (sin (* (* angle PI) 0.005555555555555556)))
      2.0)
     (* (* (- b_m a) (+ a b_m)) (sin (* 2.0 (* PI (/ angle 180.0))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 3.6e+102) {
		tmp = (2.0 * cos(((((double) M_PI) * angle) * 0.005555555555555556))) * ((sin(((0.005555555555555556 * angle) * ((double) M_PI))) * (a + b_m)) * (b_m - a));
	} else if (angle <= 3.6e+164) {
		tmp = (((sin(fma(-0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI)))) * (a + b_m)) * (b_m - a)) * sin(((angle * ((double) M_PI)) * 0.005555555555555556))) * 2.0;
	} else {
		tmp = ((b_m - a) * (a + b_m)) * sin((2.0 * (((double) M_PI) * (angle / 180.0))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 3.6e+102)
		tmp = Float64(Float64(2.0 * cos(Float64(Float64(pi * angle) * 0.005555555555555556))) * Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle) * pi)) * Float64(a + b_m)) * Float64(b_m - a)));
	elseif (angle <= 3.6e+164)
		tmp = Float64(Float64(Float64(Float64(sin(fma(-0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi))) * Float64(a + b_m)) * Float64(b_m - a)) * sin(Float64(Float64(angle * pi) * 0.005555555555555556))) * 2.0);
	else
		tmp = Float64(Float64(Float64(b_m - a) * Float64(a + b_m)) * sin(Float64(2.0 * Float64(pi * Float64(angle / 180.0)))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 3.6e+102], N[(N[(2.0 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 3.6e+164], N[(N[(N[(N[(N[Sin[N[(-0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 3.6000000000000002e102

   1. Initial program 56.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      2. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      9. lift-PI.f64 73.4

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 73.4%

      \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   if 3.6000000000000002e102 < angle < 3.5999999999999999e164

   1. Initial program 18.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\color{blue}{\left(-\pi \cdot \frac{angle}{180}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

      8. lift-PI.f64 32.6

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \frac{\color{blue}{\pi}}{2}\right)\right) \]

   6. Applied rewrites 32.6%

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \frac{\pi}{2}\right)}\right) \]

   7. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{2} \]

   9. Applied rewrites 32.4%

      \[\leadsto \color{blue}{\left(\left(\left(\sin \left(\mathsf{fma}\left(-0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot 2} \]

   if 3.5999999999999999e164 < angle

   1. Initial program 35.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \]

   6. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3.6 \cdot 10^{+102}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{elif}\;angle \leq 3.6 \cdot 10^{+164}:\\ \;\;\;\;\left(\left(\left(\sin \left(\mathsf{fma}\left(-0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 4.5 \cdot 10^{+53}:\\ \;\;\;\;\left(2 \cdot \sin \left(\mathsf{fma}\left(-0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(a + b\_m\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 4.5e+53)
   (*
    (* 2.0 (sin (fma (* -0.005555555555555556 angle) PI (/ PI 2.0))))
    (* (* (sin (* (* 0.005555555555555556 angle) PI)) (+ a b_m)) (- b_m a)))
   (* (* (- b_m a) (+ a b_m)) (sin (* 2.0 (* PI (/ angle 180.0)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 4.5e+53) {
		tmp = (2.0 * sin(fma((-0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) / 2.0)))) * ((sin(((0.005555555555555556 * angle) * ((double) M_PI))) * (a + b_m)) * (b_m - a));
	} else {
		tmp = ((b_m - a) * (a + b_m)) * sin((2.0 * (((double) M_PI) * (angle / 180.0))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 4.5e+53)
		tmp = Float64(Float64(2.0 * sin(fma(Float64(-0.005555555555555556 * angle), pi, Float64(pi / 2.0)))) * Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle) * pi)) * Float64(a + b_m)) * Float64(b_m - a)));
	else
		tmp = Float64(Float64(Float64(b_m - a) * Float64(a + b_m)) * sin(Float64(2.0 * Float64(pi * Float64(angle / 180.0)))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 4.5e+53], N[(N[(2.0 * N[Sin[N[(N[(-0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.5000000000000002e53

   1. Initial program 57.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      2. lift-PI.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      9. lift-PI.f64 74.2

         \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 74.2%

      \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

       2. cos-neg-rev N/A

          \[\leadsto \left(2 \cdot \cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \cos \left(\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \cos \left(\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \left(2 \cdot \cos \left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \cos \left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \left(2 \cdot \cos \left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \cos \left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       10. lift-neg.f64 N/A

           \[\leadsto \left(2 \cdot \cos \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

       11. sin-+PI/2-rev N/A

           \[\leadsto \left(2 \cdot \sin \left(\left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

       12. lower-sin.f64 N/A

           \[\leadsto \left(2 \cdot \sin \left(\left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   11. Applied rewrites 74.9%

       \[\leadsto \left(2 \cdot \sin \left(\mathsf{fma}\left(-0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

   if 4.5000000000000002e53 < angle

   1. Initial program 32.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 36.9%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \]

   6. Applied rewrites 36.9%

      \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 4.5 \cdot 10^{+53}:\\ \;\;\;\;\left(2 \cdot \sin \left(\mathsf{fma}\left(-0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;angle \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\left(2 \cdot \cos t\_0\right) \cdot \left(\left(\sin t\_0 \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(a + b\_m\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (if (<= angle 5e+57)
     (* (* 2.0 (cos t_0)) (* (* (sin t_0) (+ a b_m)) (- b_m a)))
     (* (* (- b_m a) (+ a b_m)) (sin (* 2.0 (* PI (/ angle 180.0))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (angle <= 5e+57) {
		tmp = (2.0 * cos(t_0)) * ((sin(t_0) * (a + b_m)) * (b_m - a));
	} else {
		tmp = ((b_m - a) * (a + b_m)) * sin((2.0 * (((double) M_PI) * (angle / 180.0))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if (angle <= 5e+57) {
		tmp = (2.0 * Math.cos(t_0)) * ((Math.sin(t_0) * (a + b_m)) * (b_m - a));
	} else {
		tmp = ((b_m - a) * (a + b_m)) * Math.sin((2.0 * (Math.PI * (angle / 180.0))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = (math.pi * angle) * 0.005555555555555556
	tmp = 0
	if angle <= 5e+57:
		tmp = (2.0 * math.cos(t_0)) * ((math.sin(t_0) * (a + b_m)) * (b_m - a))
	else:
		tmp = ((b_m - a) * (a + b_m)) * math.sin((2.0 * (math.pi * (angle / 180.0))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (angle <= 5e+57)
		tmp = Float64(Float64(2.0 * cos(t_0)) * Float64(Float64(sin(t_0) * Float64(a + b_m)) * Float64(b_m - a)));
	else
		tmp = Float64(Float64(Float64(b_m - a) * Float64(a + b_m)) * sin(Float64(2.0 * Float64(pi * Float64(angle / 180.0)))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = (pi * angle) * 0.005555555555555556;
	tmp = 0.0;
	if (angle <= 5e+57)
		tmp = (2.0 * cos(t_0)) * ((sin(t_0) * (a + b_m)) * (b_m - a));
	else
		tmp = ((b_m - a) * (a + b_m)) * sin((2.0 * (pi * (angle / 180.0))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[angle, 5e+57], N[(N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.99999999999999972e57

   1. Initial program 57.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]

   if 4.99999999999999972e57 < angle

   1. Initial program 33.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \]

   6. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq -\infty:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(angle \cdot \left(\left(b\_m - a\right) \cdot \left(a + b\_m\right)\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (- INFINITY))
   (* (* -0.011111111111111112 a) (* (* angle PI) a))
   (* (* PI (* angle (* (- b_m a) (+ a b_m)))) 0.011111111111111112)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= -((double) INFINITY)) {
		tmp = (-0.011111111111111112 * a) * ((angle * ((double) M_PI)) * a);
	} else {
		tmp = (((double) M_PI) * (angle * ((b_m - a) * (a + b_m)))) * 0.011111111111111112;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) <= -Double.POSITIVE_INFINITY) {
		tmp = (-0.011111111111111112 * a) * ((angle * Math.PI) * a);
	} else {
		tmp = (Math.PI * (angle * ((b_m - a) * (a + b_m)))) * 0.011111111111111112;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) <= -math.inf:
		tmp = (-0.011111111111111112 * a) * ((angle * math.pi) * a)
	else:
		tmp = (math.pi * (angle * ((b_m - a) * (a + b_m)))) * 0.011111111111111112
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= Float64(-Inf))
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle * pi) * a));
	else
		tmp = Float64(Float64(pi * Float64(angle * Float64(Float64(b_m - a) * Float64(a + b_m)))) * 0.011111111111111112);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) <= -Inf)
		tmp = (-0.011111111111111112 * a) * ((angle * pi) * a);
	else
		tmp = (pi * (angle * ((b_m - a) * (a + b_m)))) * 0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(angle * N[(N[(b$95$m - a), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -inf.0

   1. Initial program 55.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 49.5

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 49.5

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   8. Applied rewrites 49.5%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 49.5

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   10. Applied rewrites 49.5%

       \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 68.6

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   12. Applied rewrites 68.6%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]

   if -inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 49.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 54.6

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      2. lift-PI.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      7. associate-*l* N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90} \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\pi \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\pi \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90} \]

      11. *-commutative N/A

          \[\leadsto \left(\pi \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\pi \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \frac{1}{90} \]

      13. lift--.f64 N/A

          \[\leadsto \left(\pi \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \frac{1}{90} \]

      14. +-commutative N/A

          \[\leadsto \left(\pi \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right)\right) \cdot \frac{1}{90} \]

      15. lower-+.f64 54.6

          \[\leadsto \left(\pi \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right)\right) \cdot 0.011111111111111112 \]

   7. Applied rewrites 54.6%

      \[\leadsto \left(\pi \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right)\right) \cdot 0.011111111111111112 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 59.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq -5 \cdot 10^{+124}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(\left(\left(b\_m - a\right) \cdot \left(a + b\_m\right)\right) \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) -5e+124)
   (* (* -0.011111111111111112 a) (* (* angle PI) a))
   (* (* PI angle) (* (* (- b_m a) (+ a b_m)) 0.011111111111111112))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= -5e+124) {
		tmp = (-0.011111111111111112 * a) * ((angle * ((double) M_PI)) * a);
	} else {
		tmp = (((double) M_PI) * angle) * (((b_m - a) * (a + b_m)) * 0.011111111111111112);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) <= -5e+124) {
		tmp = (-0.011111111111111112 * a) * ((angle * Math.PI) * a);
	} else {
		tmp = (Math.PI * angle) * (((b_m - a) * (a + b_m)) * 0.011111111111111112);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) <= -5e+124:
		tmp = (-0.011111111111111112 * a) * ((angle * math.pi) * a)
	else:
		tmp = (math.pi * angle) * (((b_m - a) * (a + b_m)) * 0.011111111111111112)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= -5e+124)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle * pi) * a));
	else
		tmp = Float64(Float64(pi * angle) * Float64(Float64(Float64(b_m - a) * Float64(a + b_m)) * 0.011111111111111112));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) <= -5e+124)
		tmp = (-0.011111111111111112 * a) * ((angle * pi) * a);
	else
		tmp = (pi * angle) * (((b_m - a) * (a + b_m)) * 0.011111111111111112);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+124], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * angle), $MachinePrecision] * N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.9999999999999996e124

   1. Initial program 49.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 43.8

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 43.8%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 43.8

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   8. Applied rewrites 43.8%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 43.8

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   10. Applied rewrites 43.8%

       \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 56.2

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   12. Applied rewrites 56.2%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]

   if -4.9999999999999996e124 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 51.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 57.6

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      6. associate-*l* N/A

         \[\leadsto \left(\pi \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90}\right)} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot \frac{1}{90}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \frac{1}{90}\right) \]

      9. *-commutative N/A

         \[\leadsto \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \frac{1}{90}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90}\right)} \]

      11. *-commutative N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \frac{1}{90}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \frac{1}{90}\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot \frac{1}{90}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\frac{1}{90}}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \frac{1}{90}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \frac{1}{90}\right) \]

      17. lift--.f64 N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \frac{1}{90}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \frac{1}{90}\right) \]

      19. lower-+.f64 57.6

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot 0.011111111111111112\right) \]

   7. Applied rewrites 57.6%

      \[\leadsto \left(\pi \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot 0.011111111111111112\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 58.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq -4 \cdot 10^{-209}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\right) \cdot \left(b\_m \cdot \left(b\_m - a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) -4e-209)
   (* (* -0.011111111111111112 a) (* (* angle PI) a))
   (* (* (* PI angle) (* b_m (- b_m a))) 0.011111111111111112)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= -4e-209) {
		tmp = (-0.011111111111111112 * a) * ((angle * ((double) M_PI)) * a);
	} else {
		tmp = ((((double) M_PI) * angle) * (b_m * (b_m - a))) * 0.011111111111111112;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) <= -4e-209) {
		tmp = (-0.011111111111111112 * a) * ((angle * Math.PI) * a);
	} else {
		tmp = ((Math.PI * angle) * (b_m * (b_m - a))) * 0.011111111111111112;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) <= -4e-209:
		tmp = (-0.011111111111111112 * a) * ((angle * math.pi) * a)
	else:
		tmp = ((math.pi * angle) * (b_m * (b_m - a))) * 0.011111111111111112
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= -4e-209)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle * pi) * a));
	else
		tmp = Float64(Float64(Float64(pi * angle) * Float64(b_m * Float64(b_m - a))) * 0.011111111111111112);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) <= -4e-209)
		tmp = (-0.011111111111111112 * a) * ((angle * pi) * a);
	else
		tmp = ((pi * angle) * (b_m * (b_m - a))) * 0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-209], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * angle), $MachinePrecision] * N[(b$95$m * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.0000000000000002e-209

   1. Initial program 51.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 45.4

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 45.4

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   8. Applied rewrites 45.4%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 45.4

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   10. Applied rewrites 45.4%

       \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 54.3

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   12. Applied rewrites 54.3%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]

   if -4.0000000000000002e-209 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 50.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 59.2

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.9%

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 57.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq -4 \cdot 10^{-209}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) -4e-209)
   (* (* -0.011111111111111112 a) (* (* angle PI) a))
   (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= -4e-209) {
		tmp = (-0.011111111111111112 * a) * ((angle * ((double) M_PI)) * a);
	} else {
		tmp = ((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) <= -4e-209) {
		tmp = (-0.011111111111111112 * a) * ((angle * Math.PI) * a);
	} else {
		tmp = ((Math.PI * (b_m * b_m)) * angle) * 0.011111111111111112;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) <= -4e-209:
		tmp = (-0.011111111111111112 * a) * ((angle * math.pi) * a)
	else:
		tmp = ((math.pi * (b_m * b_m)) * angle) * 0.011111111111111112
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= -4e-209)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle * pi) * a));
	else
		tmp = Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) <= -4e-209)
		tmp = (-0.011111111111111112 * a) * ((angle * pi) * a);
	else
		tmp = ((pi * (b_m * b_m)) * angle) * 0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-209], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.0000000000000002e-209

   1. Initial program 51.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 45.4

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 45.4

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   8. Applied rewrites 45.4%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 45.4

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   10. Applied rewrites 45.4%

       \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 54.3

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   12. Applied rewrites 54.3%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]

   if -4.0000000000000002e-209 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 50.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 59.2

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{90} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      6. pow2 N/A

         \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      7. lift-*.f64 56.5

         \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112 \]

   8. Applied rewrites 56.5%

      \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 66.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 8.3 \cdot 10^{+18}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \mathbf{elif}\;angle \leq 1.02 \cdot 10^{+61}:\\ \;\;\;\;\left(\left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right) \cdot 2\right) \cdot \left(\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \sin \left(\left(-\pi\right) \cdot \frac{angle}{180} + \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(a + b\_m\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 8.3e+18)
   (*
    2.0
    (* (* (sin (* (* PI angle) 0.005555555555555556)) (+ a b_m)) (- b_m a)))
   (if (<= angle 1.02e+61)
     (*
      (* (* (+ b_m a) (- b_m a)) 2.0)
      (*
       (* (* angle PI) 0.005555555555555556)
       (sin (+ (* (- PI) (/ angle 180.0)) (/ PI 2.0)))))
     (* (* (- b_m a) (+ a b_m)) (sin (* 2.0 (* PI (/ angle 180.0))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 8.3e+18) {
		tmp = 2.0 * ((sin(((((double) M_PI) * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a));
	} else if (angle <= 1.02e+61) {
		tmp = (((b_m + a) * (b_m - a)) * 2.0) * (((angle * ((double) M_PI)) * 0.005555555555555556) * sin(((-((double) M_PI) * (angle / 180.0)) + (((double) M_PI) / 2.0))));
	} else {
		tmp = ((b_m - a) * (a + b_m)) * sin((2.0 * (((double) M_PI) * (angle / 180.0))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 8.3e+18) {
		tmp = 2.0 * ((Math.sin(((Math.PI * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a));
	} else if (angle <= 1.02e+61) {
		tmp = (((b_m + a) * (b_m - a)) * 2.0) * (((angle * Math.PI) * 0.005555555555555556) * Math.sin(((-Math.PI * (angle / 180.0)) + (Math.PI / 2.0))));
	} else {
		tmp = ((b_m - a) * (a + b_m)) * Math.sin((2.0 * (Math.PI * (angle / 180.0))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if angle <= 8.3e+18:
		tmp = 2.0 * ((math.sin(((math.pi * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a))
	elif angle <= 1.02e+61:
		tmp = (((b_m + a) * (b_m - a)) * 2.0) * (((angle * math.pi) * 0.005555555555555556) * math.sin(((-math.pi * (angle / 180.0)) + (math.pi / 2.0))))
	else:
		tmp = ((b_m - a) * (a + b_m)) * math.sin((2.0 * (math.pi * (angle / 180.0))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 8.3e+18)
		tmp = Float64(2.0 * Float64(Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(a + b_m)) * Float64(b_m - a)));
	elseif (angle <= 1.02e+61)
		tmp = Float64(Float64(Float64(Float64(b_m + a) * Float64(b_m - a)) * 2.0) * Float64(Float64(Float64(angle * pi) * 0.005555555555555556) * sin(Float64(Float64(Float64(-pi) * Float64(angle / 180.0)) + Float64(pi / 2.0)))));
	else
		tmp = Float64(Float64(Float64(b_m - a) * Float64(a + b_m)) * sin(Float64(2.0 * Float64(pi * Float64(angle / 180.0)))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if (angle <= 8.3e+18)
		tmp = 2.0 * ((sin(((pi * angle) * 0.005555555555555556)) * (a + b_m)) * (b_m - a));
	elseif (angle <= 1.02e+61)
		tmp = (((b_m + a) * (b_m - a)) * 2.0) * (((angle * pi) * 0.005555555555555556) * sin(((-pi * (angle / 180.0)) + (pi / 2.0))));
	else
		tmp = ((b_m - a) * (a + b_m)) * sin((2.0 * (pi * (angle / 180.0))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 8.3e+18], N[(2.0 * N[(N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 1.02e+61], N[(N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * N[Sin[N[(N[((-Pi) * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 8.3e18

   1. Initial program 58.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      2. difference-of-squares-rev N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   7. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]

   8. Taylor expanded in angle around 0

      \[\leadsto 2 \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 73.2%

       \[\leadsto 2 \cdot \left(\color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

   if 8.3e18 < angle < 1.01999999999999999e61

   1. Initial program 23.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 33.8%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\color{blue}{\left(-\pi \cdot \frac{angle}{180}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

      8. lift-PI.f64 55.1

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \frac{\color{blue}{\pi}}{2}\right)\right) \]

   6. Applied rewrites 55.1%

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \frac{\pi}{2}\right)}\right) \]

   7. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \frac{\pi}{2}\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right) \cdot \sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \frac{\pi}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \color{blue}{\frac{1}{180}}\right) \cdot \sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \frac{\pi}{2}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \color{blue}{\frac{1}{180}}\right) \cdot \sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \frac{\pi}{2}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \frac{\pi}{2}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \frac{\pi}{2}\right)\right) \]

      6. lift-PI.f64 70.7

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \frac{\pi}{2}\right)\right) \]

   9. Applied rewrites 70.7%

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)} \cdot \sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \frac{\pi}{2}\right)\right) \]

   if 1.01999999999999999e61 < angle

   1. Initial program 31.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 35.2%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \]

   6. Applied rewrites 35.2%

      \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 8.3 \cdot 10^{+18}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{elif}\;angle \leq 1.02 \cdot 10^{+61}:\\ \;\;\;\;\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \sin \left(\left(-\pi\right) \cdot \frac{angle}{180} + \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 63.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 4.1 \cdot 10^{+68}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(a + b\_m\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 4.1e+68)
   (* (* (* (* PI angle) (+ a b_m)) (- b_m a)) 0.011111111111111112)
   (* (* (- b_m a) (+ a b_m)) (sin (* 2.0 (* PI (/ angle 180.0)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 4.1e+68) {
		tmp = (((((double) M_PI) * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	} else {
		tmp = ((b_m - a) * (a + b_m)) * sin((2.0 * (((double) M_PI) * (angle / 180.0))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 4.1e+68) {
		tmp = (((Math.PI * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	} else {
		tmp = ((b_m - a) * (a + b_m)) * Math.sin((2.0 * (Math.PI * (angle / 180.0))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if angle <= 4.1e+68:
		tmp = (((math.pi * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112
	else:
		tmp = ((b_m - a) * (a + b_m)) * math.sin((2.0 * (math.pi * (angle / 180.0))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 4.1e+68)
		tmp = Float64(Float64(Float64(Float64(pi * angle) * Float64(a + b_m)) * Float64(b_m - a)) * 0.011111111111111112);
	else
		tmp = Float64(Float64(Float64(b_m - a) * Float64(a + b_m)) * sin(Float64(2.0 * Float64(pi * Float64(angle / 180.0)))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if (angle <= 4.1e+68)
		tmp = (((pi * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	else
		tmp = ((b_m - a) * (a + b_m)) * sin((2.0 * (pi * (angle / 180.0))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 4.1e+68], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.0999999999999999e68

   1. Initial program 56.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 60.6

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      16. lift--.f64 69.5

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112 \]

   7. Applied rewrites 69.5%

      \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112 \]

   if 4.0999999999999999e68 < angle

   1. Initial program 31.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \]

   6. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 62.8% accurate, 11.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right) \cdot 2\right) \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 1.6e-162)
   (* (* (* (+ b_m a) (- b_m a)) 2.0) (* (* PI angle) 0.005555555555555556))
   (* (* (* (* PI angle) (+ a b_m)) (- b_m a)) 0.011111111111111112)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 1.6e-162) {
		tmp = (((b_m + a) * (b_m - a)) * 2.0) * ((((double) M_PI) * angle) * 0.005555555555555556);
	} else {
		tmp = (((((double) M_PI) * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 1.6e-162) {
		tmp = (((b_m + a) * (b_m - a)) * 2.0) * ((Math.PI * angle) * 0.005555555555555556);
	} else {
		tmp = (((Math.PI * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if b_m <= 1.6e-162:
		tmp = (((b_m + a) * (b_m - a)) * 2.0) * ((math.pi * angle) * 0.005555555555555556)
	else:
		tmp = (((math.pi * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (b_m <= 1.6e-162)
		tmp = Float64(Float64(Float64(Float64(b_m + a) * Float64(b_m - a)) * 2.0) * Float64(Float64(pi * angle) * 0.005555555555555556));
	else
		tmp = Float64(Float64(Float64(Float64(pi * angle) * Float64(a + b_m)) * Float64(b_m - a)) * 0.011111111111111112);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if (b_m <= 1.6e-162)
		tmp = (((b_m + a) * (b_m - a)) * 2.0) * ((pi * angle) * 0.005555555555555556);
	else
		tmp = (((pi * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[b$95$m, 1.6e-162], N[(N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.59999999999999988e-162

   1. Initial program 56.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   4. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\pi \cdot angle\right) \cdot \color{blue}{\frac{1}{180}}\right) \]

      5. lift-*.f64 58.2

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\pi \cdot angle\right) \cdot \color{blue}{0.005555555555555556}\right) \]

   7. Applied rewrites 58.2%

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \]

   if 1.59999999999999988e-162 < b

   1. Initial program 41.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 45.4

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      16. lift--.f64 59.7

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112 \]

   7. Applied rewrites 59.7%

      \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 62.8% accurate, 13.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.8 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 2.8e-162)
   (* (* (* (* a a) angle) PI) -0.011111111111111112)
   (* (* (* (* PI angle) (+ a b_m)) (- b_m a)) 0.011111111111111112)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 2.8e-162) {
		tmp = (((a * a) * angle) * ((double) M_PI)) * -0.011111111111111112;
	} else {
		tmp = (((((double) M_PI) * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 2.8e-162) {
		tmp = (((a * a) * angle) * Math.PI) * -0.011111111111111112;
	} else {
		tmp = (((Math.PI * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if b_m <= 2.8e-162:
		tmp = (((a * a) * angle) * math.pi) * -0.011111111111111112
	else:
		tmp = (((math.pi * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (b_m <= 2.8e-162)
		tmp = Float64(Float64(Float64(Float64(a * a) * angle) * pi) * -0.011111111111111112);
	else
		tmp = Float64(Float64(Float64(Float64(pi * angle) * Float64(a + b_m)) * Float64(b_m - a)) * 0.011111111111111112);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if (b_m <= 2.8e-162)
		tmp = (((a * a) * angle) * pi) * -0.011111111111111112;
	else
		tmp = (((pi * angle) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[b$95$m, 2.8e-162], N[(N[(N[(N[(a * a), $MachinePrecision] * angle), $MachinePrecision] * Pi), $MachinePrecision] * -0.011111111111111112), $MachinePrecision], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.80000000000000022e-162

   1. Initial program 56.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 58.1

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 38.5

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   8. Applied rewrites 38.5%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(\pi \cdot angle\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{-1}{90} \]

      10. lower-*.f64 N/A

          \[\leadsto \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{-1}{90} \]

      11. associate-*r* N/A

          \[\leadsto \left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90} \]

      14. pow2 N/A

          \[\leadsto \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90} \]

      16. lift-PI.f64 38.6

          \[\leadsto \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right) \cdot -0.011111111111111112 \]

   10. Applied rewrites 38.6%

       \[\leadsto \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right) \cdot -0.011111111111111112 \]

   if 2.80000000000000022e-162 < b

   1. Initial program 41.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 45.4

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90} \]

      16. lift--.f64 59.7

          \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112 \]

   7. Applied rewrites 59.7%

      \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot 0.011111111111111112 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 38.7% accurate, 16.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 1.15 \cdot 10^{-83}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 1.15e-83)
   (* (* -0.011111111111111112 a) (* (* angle PI) a))
   (* (* -0.011111111111111112 (* a a)) (* PI angle))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 1.15e-83) {
		tmp = (-0.011111111111111112 * a) * ((angle * ((double) M_PI)) * a);
	} else {
		tmp = (-0.011111111111111112 * (a * a)) * (((double) M_PI) * angle);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 1.15e-83) {
		tmp = (-0.011111111111111112 * a) * ((angle * Math.PI) * a);
	} else {
		tmp = (-0.011111111111111112 * (a * a)) * (Math.PI * angle);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if angle <= 1.15e-83:
		tmp = (-0.011111111111111112 * a) * ((angle * math.pi) * a)
	else:
		tmp = (-0.011111111111111112 * (a * a)) * (math.pi * angle)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 1.15e-83)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle * pi) * a));
	else
		tmp = Float64(Float64(-0.011111111111111112 * Float64(a * a)) * Float64(pi * angle));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if (angle <= 1.15e-83)
		tmp = (-0.011111111111111112 * a) * ((angle * pi) * a);
	else
		tmp = (-0.011111111111111112 * (a * a)) * (pi * angle);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 1.15e-83], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.14999999999999995e-83

   1. Initial program 56.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 59.8

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 32.3

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   8. Applied rewrites 32.3%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 32.3

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   10. Applied rewrites 32.3%

       \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 36.5

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   12. Applied rewrites 36.5%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]

   if 1.14999999999999995e-83 < angle

   1. Initial program 39.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 40.9

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   5. Applied rewrites 40.9%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 31.3

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   8. Applied rewrites 31.3%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 38.0% accurate, 21.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (* (* -0.011111111111111112 a) (* (* angle PI) a)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	return (-0.011111111111111112 * a) * ((angle * ((double) M_PI)) * a);
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	return (-0.011111111111111112 * a) * ((angle * Math.PI) * a);
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	return (-0.011111111111111112 * a) * ((angle * math.pi) * a)

Julia

b_m = abs(b)
function code(a, b_m, angle)
	return Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle * pi) * a))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle)
	tmp = (-0.011111111111111112 * a) * ((angle * pi) * a);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   5. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   8. unpow2 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   9. unpow2 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

   10. difference-of-squares N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   13. lower--.f64 53.6

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

5. Applied rewrites 53.6%

   \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

6. Taylor expanded in a around inf

   \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

   8. lift-PI.f64 31.9

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

8. Applied rewrites 31.9%

   \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   5. lower-*.f64 31.9

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

10. Applied rewrites 31.9%

    \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

    2. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

    4. lift-PI.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

    5. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

    6. associate-*l* N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

    7. *-commutative N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

    8. lower-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

    9. lift-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

    11. lower-*.f64 N/A

        \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

    12. lift-*.f64 N/A

        \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

    13. lift-PI.f64 34.1

        \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

12. Applied rewrites 34.1%

    \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025072 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))