ab-angle->ABCF B

Percentage Accurate: 54.4% → 67.2%

Time: 8.0s

Alternatives: 8

Speedup: N/A×

• 32.5% 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: 54.4% 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: 67.2% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\\ t_1 := \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\\ t_2 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 6.6 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sin t\_2 \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right), \cos t\_0, \left(\left(\sin t\_0 \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right)\right)\right)\\ \mathbf{elif}\;angle\_m \leq 5.7 \cdot 10^{+84}:\\ \;\;\;\;\left(\left(2 \cdot \left(e^{\log b\_m \cdot 2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot \sin \left(-1 \cdot t\_2 + \frac{\pi}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b\_m + a, b\_m, a \cdot \left(b\_m - a\right)\right) \cdot \left(t\_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{angle\_m}{180}, \frac{\pi}{2}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle_m) PI))
        (t_1 (sin (* PI (/ angle_m 180.0))))
        (t_2 (* (* PI angle_m) 0.005555555555555556)))
   (*
    angle_s
    (if (<= angle_m 6.6e+23)
      (fma
       (* (* (sin t_2) (+ a b_m)) (- b_m a))
       (cos t_0)
       (*
        (* (* (sin t_0) (+ a b_m)) (- b_m a))
        (sin (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0)))))
      (if (<= angle_m 5.7e+84)
        (*
         (* (* 2.0 (- (exp (* (log b_m) 2.0)) (pow a 2.0))) t_1)
         (sin (+ (* -1.0 t_2) (/ PI 2.0))))
        (*
         (fma (+ b_m a) b_m (* a (- b_m a)))
         (* t_1 (sin (fma PI (/ angle_m 180.0) (/ PI 2.0))))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (0.005555555555555556 * angle_m) * ((double) M_PI);
	double t_1 = sin((((double) M_PI) * (angle_m / 180.0)));
	double t_2 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	double tmp;
	if (angle_m <= 6.6e+23) {
		tmp = fma(((sin(t_2) * (a + b_m)) * (b_m - a)), cos(t_0), (((sin(t_0) * (a + b_m)) * (b_m - a)) * sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0)))));
	} else if (angle_m <= 5.7e+84) {
		tmp = ((2.0 * (exp((log(b_m) * 2.0)) - pow(a, 2.0))) * t_1) * sin(((-1.0 * t_2) + (((double) M_PI) / 2.0)));
	} else {
		tmp = fma((b_m + a), b_m, (a * (b_m - a))) * (t_1 * sin(fma(((double) M_PI), (angle_m / 180.0), (((double) M_PI) / 2.0))));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(0.005555555555555556 * angle_m) * pi)
	t_1 = sin(Float64(pi * Float64(angle_m / 180.0)))
	t_2 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	tmp = 0.0
	if (angle_m <= 6.6e+23)
		tmp = fma(Float64(Float64(sin(t_2) * Float64(a + b_m)) * Float64(b_m - a)), cos(t_0), Float64(Float64(Float64(sin(t_0) * Float64(a + b_m)) * Float64(b_m - a)) * sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0)))));
	elseif (angle_m <= 5.7e+84)
		tmp = Float64(Float64(Float64(2.0 * Float64(exp(Float64(log(b_m) * 2.0)) - (a ^ 2.0))) * t_1) * sin(Float64(Float64(-1.0 * t_2) + Float64(pi / 2.0))));
	else
		tmp = Float64(fma(Float64(b_m + a), b_m, Float64(a * Float64(b_m - a))) * Float64(t_1 * sin(fma(pi, Float64(angle_m / 180.0), Float64(pi / 2.0)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 6.6e+23], N[(N[(N[(N[Sin[t$95$2], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision] + N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 5.7e+84], N[(N[(N[(2.0 * N[(N[Exp[N[(N[Log[b$95$m], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sin[N[(N[(-1.0 * t$95$2), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m + a), $MachinePrecision] * b$95$m + N[(a * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if angle < 6.60000000000000059e23

   1. Initial program 61.3%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

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

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

      15. lower-sin.f64 N/A

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in angle around inf

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 73.5

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

   10. Applied rewrites 73.5%

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

       1. lift-sin.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. sin-+PI/2 N/A

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

       8. lower-cos.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-PI.f64 72.9

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

   12. Applied rewrites 72.9%

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

   if 6.60000000000000059e23 < angle < 5.6999999999999997e84

   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 inf

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

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

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

      2. lower-sin.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 23.0

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right) \]

   5. Applied rewrites 23.0%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)} \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sin-+PI/2 N/A

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

      8. associate-*r* N/A

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

      9. cos-neg-rev N/A

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

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

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

      11. lower-sin.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-+.f64 N/A

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

   7. Applied rewrites 72.9%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) + \frac{\pi}{2}\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 31.2

         \[\leadsto \left(\left(2 \cdot \left(e^{\color{blue}{\log b} \cdot 2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) + \frac{\pi}{2}\right) \]

   9. Applied rewrites 31.2%

      \[\leadsto \left(\left(2 \cdot \left(\color{blue}{e^{\log b \cdot 2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) + \frac{\pi}{2}\right) \]

   if 5.6999999999999997e84 < angle

   1. Initial program 25.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 40.4%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 42.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 42.7%

       \[\leadsto \mathsf{fma}\left(b + a, \color{blue}{b}, a \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{angle}{180}, \frac{\pi}{2}\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 6.6 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right), \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)\right)\\ \mathbf{elif}\;angle \leq 5.7 \cdot 10^{+84}:\\ \;\;\;\;\left(\left(2 \cdot \left(e^{\log b \cdot 2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(-1 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) + \frac{\pi}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b + a, b, a \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{angle}{180}, \frac{\pi}{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.7% accurate, N/A× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1 (sin (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0)))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         (- INFINITY))
      (* (* (* 0.011111111111111112 angle_m) (* (+ a b_m) PI)) (- b_m a))
      (fma
       (*
        (* (sin (* (* PI angle_m) 0.005555555555555556)) (+ a b_m))
        (- b_m a))
       t_1
       (*
        (*
         (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a b_m))
         (- b_m a))
        t_1))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0)));
	double tmp;
	if ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= -((double) INFINITY)) {
		tmp = ((0.011111111111111112 * angle_m) * ((a + b_m) * ((double) M_PI))) * (b_m - a);
	} else {
		tmp = fma(((sin(((((double) M_PI) * angle_m) * 0.005555555555555556)) * (a + b_m)) * (b_m - a)), t_1, (((sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a + b_m)) * (b_m - a)) * t_1));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0)))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * Float64(Float64(a + b_m) * pi)) * Float64(b_m - a));
	else
		tmp = fma(Float64(Float64(sin(Float64(Float64(pi * angle_m) * 0.005555555555555556)) * Float64(a + b_m)) * Float64(b_m - a)), t_1, Float64(Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a + b_m)) * Float64(b_m - a)) * t_1));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * 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], (-Infinity)], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * t$95$1), $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))))) < -inf.0

   1. Initial program 48.0%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

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

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

      15. lower-sin.f64 N/A

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

   5. Applied rewrites 9.6%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*l* N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. pow2 N/A

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

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

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

      9. +-commutative N/A

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

   8. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lift--.f64 77.7

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

   10. Applied rewrites 77.7%

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

   if -inf.0 < (*.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 55.5%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

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

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

      15. lower-sin.f64 N/A

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in angle around inf

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

   8. Applied rewrites 63.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 62.9

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

   10. Applied rewrites 62.9%

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

Alternative 3: 65.9% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\\ t_2 := \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(\left(a + b\_m\right) \cdot \pi\right)\right) \cdot \left(b\_m - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot \left(t\_1 + \frac{b\_m \cdot t\_1}{a}\right)\right) \cdot \left(b\_m - a\right), t\_2, \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right) \cdot t\_2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1 (sin (* 0.005555555555555556 (* angle_m PI))))
        (t_2 (sin (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0)))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         (- INFINITY))
      (* (* (* 0.011111111111111112 angle_m) (* (+ a b_m) PI)) (- b_m a))
      (fma
       (* (* a (+ t_1 (/ (* b_m t_1) a))) (- b_m a))
       t_2
       (*
        (*
         (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a b_m))
         (- b_m a))
        t_2))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = sin((0.005555555555555556 * (angle_m * ((double) M_PI))));
	double t_2 = sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0)));
	double tmp;
	if ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= -((double) INFINITY)) {
		tmp = ((0.011111111111111112 * angle_m) * ((a + b_m) * ((double) M_PI))) * (b_m - a);
	} else {
		tmp = fma(((a * (t_1 + ((b_m * t_1) / a))) * (b_m - a)), t_2, (((sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a + b_m)) * (b_m - a)) * t_2));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))
	t_2 = sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0)))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * Float64(Float64(a + b_m) * pi)) * Float64(b_m - a));
	else
		tmp = fma(Float64(Float64(a * Float64(t_1 + Float64(Float64(b_m * t_1) / a))) * Float64(b_m - a)), t_2, Float64(Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a + b_m)) * Float64(b_m - a)) * t_2));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * 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], (-Infinity)], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(t$95$1 + N[(N[(b$95$m * t$95$1), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * t$95$2), $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))))) < -inf.0

   1. Initial program 48.0%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

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

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

      15. lower-sin.f64 N/A

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

   5. Applied rewrites 9.6%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*l* N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. pow2 N/A

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

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

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

      9. +-commutative N/A

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

   8. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lift--.f64 77.7

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

   10. Applied rewrites 77.7%

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

   if -inf.0 < (*.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 55.5%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

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

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

      15. lower-sin.f64 N/A

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in angle around inf

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

   8. Applied rewrites 63.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 62.9

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

   10. Applied rewrites 62.9%

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

   11. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sin.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-PI.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-PI.f64 62.4

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

   13. Applied rewrites 62.4%

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

Alternative 4: 65.3% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right)\right)\\ t_1 := \left(\sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\\ angle\_s \cdot \mathsf{fma}\left(t\_1, t\_0, t\_1 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (sin (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0))))
        (t_1
         (*
          (* (sin (* (* PI angle_m) 0.005555555555555556)) (+ a b_m))
          (- b_m a))))
   (* angle_s (fma t_1 t_0 (* t_1 t_0)))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0)));
	double t_1 = (sin(((((double) M_PI) * angle_m) * 0.005555555555555556)) * (a + b_m)) * (b_m - a);
	return angle_s * fma(t_1, t_0, (t_1 * t_0));
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0)))
	t_1 = Float64(Float64(sin(Float64(Float64(pi * angle_m) * 0.005555555555555556)) * Float64(a + b_m)) * Float64(b_m - a))
	return Float64(angle_s * fma(t_1, t_0, Float64(t_1 * t_0)))
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * N[(t$95$1 * t$95$0 + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in a around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

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

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

   15. lower-sin.f64 N/A

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

5. Applied rewrites 32.9%

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

6. Taylor expanded in angle around inf

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

8. Applied rewrites 64.0%

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

Alternative 5: 64.1% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(\left(\pi \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ t_1 := 2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(angle\_m \cdot \pi\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\_m\right) \cdot \pi\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right), a, \left(\left(\left(b\_m \cdot b\_m\right) \cdot \pi\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;t\_1 \leq 20000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), angle\_m \cdot \pi, \mathsf{fma}\left(\left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle\_m, 0.011111111111111112, \left(\left(\pi \cdot b\_m\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right) \cdot b\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0
         (* (* 0.011111111111111112 angle_m) (* (* PI (+ a b_m)) (- b_m a))))
        (t_1 (* 2.0 (- (pow b_m 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= t_1 (- INFINITY))
      (fma
       (fma
        (* (* angle_m PI) a)
        -0.011111111111111112
        (* (* (* (* 0.0 b_m) PI) angle_m) 0.011111111111111112))
       a
       (* (* (* (* b_m b_m) PI) angle_m) 0.011111111111111112))
      (if (<= t_1 20000000000000.0)
        t_0
        (if (<= t_1 INFINITY)
          (fma
           (* -0.011111111111111112 (* a a))
           (* angle_m PI)
           (*
            (fma
             (* (* (* 0.0 a) PI) angle_m)
             0.011111111111111112
             (* (* (* PI b_m) angle_m) 0.011111111111111112))
            b_m))
          t_0))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (0.011111111111111112 * angle_m) * ((((double) M_PI) * (a + b_m)) * (b_m - a));
	double t_1 = 2.0 * (pow(b_m, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(((angle_m * ((double) M_PI)) * a), -0.011111111111111112, ((((0.0 * b_m) * ((double) M_PI)) * angle_m) * 0.011111111111111112)), a, ((((b_m * b_m) * ((double) M_PI)) * angle_m) * 0.011111111111111112));
	} else if (t_1 <= 20000000000000.0) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((-0.011111111111111112 * (a * a)), (angle_m * ((double) M_PI)), (fma((((0.0 * a) * ((double) M_PI)) * angle_m), 0.011111111111111112, (((((double) M_PI) * b_m) * angle_m) * 0.011111111111111112)) * b_m));
	} else {
		tmp = t_0;
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(0.011111111111111112 * angle_m) * Float64(Float64(pi * Float64(a + b_m)) * Float64(b_m - a)))
	t_1 = Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(fma(Float64(Float64(angle_m * pi) * a), -0.011111111111111112, Float64(Float64(Float64(Float64(0.0 * b_m) * pi) * angle_m) * 0.011111111111111112)), a, Float64(Float64(Float64(Float64(b_m * b_m) * pi) * angle_m) * 0.011111111111111112));
	elseif (t_1 <= 20000000000000.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = fma(Float64(-0.011111111111111112 * Float64(a * a)), Float64(angle_m * pi), Float64(fma(Float64(Float64(Float64(0.0 * a) * pi) * angle_m), 0.011111111111111112, Float64(Float64(Float64(pi * b_m) * angle_m) * 0.011111111111111112)) * b_m));
	else
		tmp = t_0;
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(N[(Pi * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112 + N[(N[(N[(N[(0.0 * b$95$m), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * a + N[(N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 20000000000000.0], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision] + N[(N[(N[(N[(N[(0.0 * a), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112 + N[(N[(N[(Pi * b$95$m), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $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)))) < -inf.0

   1. Initial program 45.3%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

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

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

      15. lower-sin.f64 N/A

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*l* N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. pow2 N/A

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

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

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

      9. +-commutative N/A

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

   8. Applied rewrites 49.4%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

   11. Applied rewrites 81.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(angle \cdot \pi\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\left(b \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if -inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 2e13 or +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 55.0%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

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

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

      15. lower-sin.f64 N/A

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*l* N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. pow2 N/A

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

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

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

      9. +-commutative N/A

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

   8. Applied rewrites 58.3%

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

   if 2e13 < (*.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 57.3%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

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

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

      15. lower-sin.f64 N/A

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*l* N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. pow2 N/A

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

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

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

      9. +-commutative N/A

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

   8. Applied rewrites 59.8%

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

   9. Taylor expanded in b around 0

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

       1. associate-*r* N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

   11. Applied rewrites 71.4%

       \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \color{blue}{angle \cdot \pi}, \mathsf{fma}\left(\left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle, 0.011111111111111112, \left(\left(\pi \cdot b\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 62.3% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.25 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(\left(a + b\_m\right) \cdot \pi\right)\right) \cdot \left(b\_m - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(\left(\pi \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 1.25e-14)
    (* (* (* 0.011111111111111112 angle_m) (* (+ a b_m) PI)) (- b_m a))
    (* (* 0.011111111111111112 angle_m) (* (* PI (+ a b_m)) (- b_m a))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 1.25e-14) {
		tmp = ((0.011111111111111112 * angle_m) * ((a + b_m) * ((double) M_PI))) * (b_m - a);
	} else {
		tmp = (0.011111111111111112 * angle_m) * ((((double) M_PI) * (a + b_m)) * (b_m - a));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 1.25e-14) {
		tmp = ((0.011111111111111112 * angle_m) * ((a + b_m) * Math.PI)) * (b_m - a);
	} else {
		tmp = (0.011111111111111112 * angle_m) * ((Math.PI * (a + b_m)) * (b_m - a));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if angle_m <= 1.25e-14:
		tmp = ((0.011111111111111112 * angle_m) * ((a + b_m) * math.pi)) * (b_m - a)
	else:
		tmp = (0.011111111111111112 * angle_m) * ((math.pi * (a + b_m)) * (b_m - a))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (angle_m <= 1.25e-14)
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * Float64(Float64(a + b_m) * pi)) * Float64(b_m - a));
	else
		tmp = Float64(Float64(0.011111111111111112 * angle_m) * Float64(Float64(pi * Float64(a + b_m)) * Float64(b_m - a)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (angle_m <= 1.25e-14)
		tmp = ((0.011111111111111112 * angle_m) * ((a + b_m) * pi)) * (b_m - a);
	else
		tmp = (0.011111111111111112 * angle_m) * ((pi * (a + b_m)) * (b_m - a));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1.25e-14], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(N[(Pi * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.25e-14

   1. Initial program 61.5%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

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

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

      15. lower-sin.f64 N/A

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

   5. Applied rewrites 34.2%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*l* N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. pow2 N/A

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

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

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

      9. +-commutative N/A

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

   8. Applied rewrites 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lift--.f64 76.8

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

   10. Applied rewrites 76.8%

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

   if 1.25e-14 < angle

   1. Initial program 32.1%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

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

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

      15. lower-sin.f64 N/A

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

   5. Applied rewrites 28.9%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*l* N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. pow2 N/A

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

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

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

      9. +-commutative N/A

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

   8. Applied rewrites 32.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 56.0% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.16 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), angle\_m \cdot \pi, \mathsf{fma}\left(\left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle\_m, 0.011111111111111112, \left(\left(\pi \cdot b\_m\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right) \cdot b\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(angle\_m \cdot \pi\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\_m\right) \cdot \pi\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right), a, \left(\left(\left(b\_m \cdot b\_m\right) \cdot \pi\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 1.16e+154)
    (fma
     (* -0.011111111111111112 (* a a))
     (* angle_m PI)
     (*
      (fma
       (* (* (* 0.0 a) PI) angle_m)
       0.011111111111111112
       (* (* (* PI b_m) angle_m) 0.011111111111111112))
      b_m))
    (fma
     (fma
      (* (* angle_m PI) a)
      -0.011111111111111112
      (* (* (* (* 0.0 b_m) PI) angle_m) 0.011111111111111112))
     a
     (* (* (* (* b_m b_m) PI) angle_m) 0.011111111111111112)))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 1.16e+154) {
		tmp = fma((-0.011111111111111112 * (a * a)), (angle_m * ((double) M_PI)), (fma((((0.0 * a) * ((double) M_PI)) * angle_m), 0.011111111111111112, (((((double) M_PI) * b_m) * angle_m) * 0.011111111111111112)) * b_m));
	} else {
		tmp = fma(fma(((angle_m * ((double) M_PI)) * a), -0.011111111111111112, ((((0.0 * b_m) * ((double) M_PI)) * angle_m) * 0.011111111111111112)), a, ((((b_m * b_m) * ((double) M_PI)) * angle_m) * 0.011111111111111112));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (a <= 1.16e+154)
		tmp = fma(Float64(-0.011111111111111112 * Float64(a * a)), Float64(angle_m * pi), Float64(fma(Float64(Float64(Float64(0.0 * a) * pi) * angle_m), 0.011111111111111112, Float64(Float64(Float64(pi * b_m) * angle_m) * 0.011111111111111112)) * b_m));
	else
		tmp = fma(fma(Float64(Float64(angle_m * pi) * a), -0.011111111111111112, Float64(Float64(Float64(Float64(0.0 * b_m) * pi) * angle_m) * 0.011111111111111112)), a, Float64(Float64(Float64(Float64(b_m * b_m) * pi) * angle_m) * 0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 1.16e+154], N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision] + N[(N[(N[(N[(N[(0.0 * a), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112 + N[(N[(N[(Pi * b$95$m), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112 + N[(N[(N[(N[(0.0 * b$95$m), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * a + N[(N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.16000000000000001e154

   1. Initial program 56.7%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

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

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

      15. lower-sin.f64 N/A

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

   5. Applied rewrites 32.5%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*l* N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. pow2 N/A

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

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

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

      9. +-commutative N/A

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

   8. Applied rewrites 58.0%

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

   9. Taylor expanded in b around 0

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

       1. associate-*r* N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

   11. Applied rewrites 57.4%

       \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \color{blue}{angle \cdot \pi}, \mathsf{fma}\left(\left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle, 0.011111111111111112, \left(\left(\pi \cdot b\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\right) \]

   if 1.16000000000000001e154 < a

   1. Initial program 35.5%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

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

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

      15. lower-sin.f64 N/A

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*l* N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. pow2 N/A

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

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

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

      9. +-commutative N/A

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

   8. Applied rewrites 50.6%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

   11. Applied rewrites 66.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(angle \cdot \pi\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\left(b \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 53.6% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), angle\_m \cdot \pi, \mathsf{fma}\left(\left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle\_m, 0.011111111111111112, \left(\left(\pi \cdot b\_m\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right) \cdot b\_m\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (fma
   (* -0.011111111111111112 (* a a))
   (* angle_m PI)
   (*
    (fma
     (* (* (* 0.0 a) PI) angle_m)
     0.011111111111111112
     (* (* (* PI b_m) angle_m) 0.011111111111111112))
    b_m))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * fma((-0.011111111111111112 * (a * a)), (angle_m * ((double) M_PI)), (fma((((0.0 * a) * ((double) M_PI)) * angle_m), 0.011111111111111112, (((((double) M_PI) * b_m) * angle_m) * 0.011111111111111112)) * b_m));
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	return Float64(angle_s * fma(Float64(-0.011111111111111112 * Float64(a * a)), Float64(angle_m * pi), Float64(fma(Float64(Float64(Float64(0.0 * a) * pi) * angle_m), 0.011111111111111112, Float64(Float64(Float64(pi * b_m) * angle_m) * 0.011111111111111112)) * b_m)))
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision] + N[(N[(N[(N[(N[(0.0 * a), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112 + N[(N[(N[(Pi * b$95$m), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in a around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

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

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

   15. lower-sin.f64 N/A

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

5. Applied rewrites 32.9%

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

6. Taylor expanded in angle around 0

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

   1. associate-*l* N/A

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

   2. metadata-eval N/A

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

   3. pow-flip N/A

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

   4. associate-*l* N/A

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

   5. associate-*r* N/A

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

   6. unpow2 N/A

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

   7. pow2 N/A

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

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

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

   9. +-commutative N/A

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

8. Applied rewrites 57.1%

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

9. Taylor expanded in b around 0

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

    1. associate-*r* N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

11. Applied rewrites 55.0%

    \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \color{blue}{angle \cdot \pi}, \mathsf{fma}\left(\left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle, 0.011111111111111112, \left(\left(\pi \cdot b\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot b\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(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)))))