ab-angle->ABCF B

Percentage Accurate: 53.4% → 67.3%

Time: 6.5s

Alternatives: 19

Speedup: 4.3×

• 67.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 53.4% accurate, 1.0× speedup

Math

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

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: 67.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - a\\ t_1 := \pi \cdot \frac{\left|angle\right|}{180}\\ t_2 := a + \left|b\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 1.2 \cdot 10^{+80}:\\ \;\;\;\;\left(t\_2 \cdot \left(t\_0 \cdot \left(\sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left|angle\right|\right) \cdot 2\right)\right)\right) \cdot \cos t\_1\\ \mathbf{elif}\;\left|angle\right| \leq 5.4 \cdot 10^{+242}:\\ \;\;\;\;\left(\left(t\_2 \cdot \left(t\_0 \cdot 2\right)\right) \cdot \sin t\_1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot \left(t\_0 \cdot \left(\sin \left(\left(0.005555555555555556 \cdot \left|angle\right|\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\pi \cdot \left|angle\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (fabs b) a))
        (t_1 (* PI (/ (fabs angle) 180.0)))
        (t_2 (+ a (fabs b))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 1.2e+80)
      (*
       (*
        t_2
        (* t_0 (* (sin (* (* PI 0.005555555555555556) (fabs angle))) 2.0)))
       (cos t_1))
      (if (<= (fabs angle) 5.4e+242)
        (* (* (* t_2 (* t_0 2.0)) (sin t_1)) 1.0)
        (*
         (*
          t_2
          (* t_0 (* (sin (* (* 0.005555555555555556 (fabs angle)) PI)) 2.0)))
         (cos (* -0.005555555555555556 (* PI (fabs angle))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - a;
	double t_1 = ((double) M_PI) * (fabs(angle) / 180.0);
	double t_2 = a + fabs(b);
	double tmp;
	if (fabs(angle) <= 1.2e+80) {
		tmp = (t_2 * (t_0 * (sin(((((double) M_PI) * 0.005555555555555556) * fabs(angle))) * 2.0))) * cos(t_1);
	} else if (fabs(angle) <= 5.4e+242) {
		tmp = ((t_2 * (t_0 * 2.0)) * sin(t_1)) * 1.0;
	} else {
		tmp = (t_2 * (t_0 * (sin(((0.005555555555555556 * fabs(angle)) * ((double) M_PI))) * 2.0))) * cos((-0.005555555555555556 * (((double) M_PI) * fabs(angle))));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - a;
	double t_1 = Math.PI * (Math.abs(angle) / 180.0);
	double t_2 = a + Math.abs(b);
	double tmp;
	if (Math.abs(angle) <= 1.2e+80) {
		tmp = (t_2 * (t_0 * (Math.sin(((Math.PI * 0.005555555555555556) * Math.abs(angle))) * 2.0))) * Math.cos(t_1);
	} else if (Math.abs(angle) <= 5.4e+242) {
		tmp = ((t_2 * (t_0 * 2.0)) * Math.sin(t_1)) * 1.0;
	} else {
		tmp = (t_2 * (t_0 * (Math.sin(((0.005555555555555556 * Math.abs(angle)) * Math.PI)) * 2.0))) * Math.cos((-0.005555555555555556 * (Math.PI * Math.abs(angle))));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - a
	t_1 = math.pi * (math.fabs(angle) / 180.0)
	t_2 = a + math.fabs(b)
	tmp = 0
	if math.fabs(angle) <= 1.2e+80:
		tmp = (t_2 * (t_0 * (math.sin(((math.pi * 0.005555555555555556) * math.fabs(angle))) * 2.0))) * math.cos(t_1)
	elif math.fabs(angle) <= 5.4e+242:
		tmp = ((t_2 * (t_0 * 2.0)) * math.sin(t_1)) * 1.0
	else:
		tmp = (t_2 * (t_0 * (math.sin(((0.005555555555555556 * math.fabs(angle)) * math.pi)) * 2.0))) * math.cos((-0.005555555555555556 * (math.pi * math.fabs(angle))))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - a)
	t_1 = Float64(pi * Float64(abs(angle) / 180.0))
	t_2 = Float64(a + abs(b))
	tmp = 0.0
	if (abs(angle) <= 1.2e+80)
		tmp = Float64(Float64(t_2 * Float64(t_0 * Float64(sin(Float64(Float64(pi * 0.005555555555555556) * abs(angle))) * 2.0))) * cos(t_1));
	elseif (abs(angle) <= 5.4e+242)
		tmp = Float64(Float64(Float64(t_2 * Float64(t_0 * 2.0)) * sin(t_1)) * 1.0);
	else
		tmp = Float64(Float64(t_2 * Float64(t_0 * Float64(sin(Float64(Float64(0.005555555555555556 * abs(angle)) * pi)) * 2.0))) * cos(Float64(-0.005555555555555556 * Float64(pi * abs(angle)))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - a;
	t_1 = pi * (abs(angle) / 180.0);
	t_2 = a + abs(b);
	tmp = 0.0;
	if (abs(angle) <= 1.2e+80)
		tmp = (t_2 * (t_0 * (sin(((pi * 0.005555555555555556) * abs(angle))) * 2.0))) * cos(t_1);
	elseif (abs(angle) <= 5.4e+242)
		tmp = ((t_2 * (t_0 * 2.0)) * sin(t_1)) * 1.0;
	else
		tmp = (t_2 * (t_0 * (sin(((0.005555555555555556 * abs(angle)) * pi)) * 2.0))) * cos((-0.005555555555555556 * (pi * abs(angle))));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 1.2e+80], N[(N[(t$95$2 * N[(t$95$0 * N[(N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 5.4e+242], N[(N[(N[(t$95$2 * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(t$95$2 * N[(t$95$0 * N[(N[Sin[N[(N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.005555555555555556 * N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if angle < 1.1999999999999999e80

   1. Initial program 53.4%

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 67.0%

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

   3. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 66.8%

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

   5. Applied rewrites 66.8%

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

   if 1.1999999999999999e80 < angle < 5.39999999999999968e242

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 57.3%

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

   3. Applied rewrites 57.3%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 56.3%

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

   if 5.39999999999999968e242 < angle

   1. Initial program 53.4%

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 67.0%

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

   3. Applied rewrites 66.7%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. lower-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 66.6%

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

   5. Applied rewrites 66.6%

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

Alternative 2: 67.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \frac{\left|angle\right|}{180}\\ t_1 := \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot \left|angle\right|\right) \cdot \pi\right) \cdot 2\right)\right)\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 1.65 \cdot 10^{+60}:\\ \;\;\;\;t\_1 \cdot \cos t\_0\\ \mathbf{elif}\;\left|angle\right| \leq 5.4 \cdot 10^{+242}:\\ \;\;\;\;\left(\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin t\_0\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \cos \left(-0.005555555555555556 \cdot \left(\pi \cdot \left|angle\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ (fabs angle) 180.0)))
        (t_1
         (*
          (+ a b)
          (*
           (- b a)
           (* (sin (* (* 0.005555555555555556 (fabs angle)) PI)) 2.0)))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 1.65e+60)
      (* t_1 (cos t_0))
      (if (<= (fabs angle) 5.4e+242)
        (* (* (* (+ a b) (* (- b a) 2.0)) (sin t_0)) 1.0)
        (* t_1 (cos (* -0.005555555555555556 (* PI (fabs angle))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (fabs(angle) / 180.0);
	double t_1 = (a + b) * ((b - a) * (sin(((0.005555555555555556 * fabs(angle)) * ((double) M_PI))) * 2.0));
	double tmp;
	if (fabs(angle) <= 1.65e+60) {
		tmp = t_1 * cos(t_0);
	} else if (fabs(angle) <= 5.4e+242) {
		tmp = (((a + b) * ((b - a) * 2.0)) * sin(t_0)) * 1.0;
	} else {
		tmp = t_1 * cos((-0.005555555555555556 * (((double) M_PI) * fabs(angle))));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (Math.abs(angle) / 180.0);
	double t_1 = (a + b) * ((b - a) * (Math.sin(((0.005555555555555556 * Math.abs(angle)) * Math.PI)) * 2.0));
	double tmp;
	if (Math.abs(angle) <= 1.65e+60) {
		tmp = t_1 * Math.cos(t_0);
	} else if (Math.abs(angle) <= 5.4e+242) {
		tmp = (((a + b) * ((b - a) * 2.0)) * Math.sin(t_0)) * 1.0;
	} else {
		tmp = t_1 * Math.cos((-0.005555555555555556 * (Math.PI * Math.abs(angle))));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (math.fabs(angle) / 180.0)
	t_1 = (a + b) * ((b - a) * (math.sin(((0.005555555555555556 * math.fabs(angle)) * math.pi)) * 2.0))
	tmp = 0
	if math.fabs(angle) <= 1.65e+60:
		tmp = t_1 * math.cos(t_0)
	elif math.fabs(angle) <= 5.4e+242:
		tmp = (((a + b) * ((b - a) * 2.0)) * math.sin(t_0)) * 1.0
	else:
		tmp = t_1 * math.cos((-0.005555555555555556 * (math.pi * math.fabs(angle))))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(abs(angle) / 180.0))
	t_1 = Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(sin(Float64(Float64(0.005555555555555556 * abs(angle)) * pi)) * 2.0)))
	tmp = 0.0
	if (abs(angle) <= 1.65e+60)
		tmp = Float64(t_1 * cos(t_0));
	elseif (abs(angle) <= 5.4e+242)
		tmp = Float64(Float64(Float64(Float64(a + b) * Float64(Float64(b - a) * 2.0)) * sin(t_0)) * 1.0);
	else
		tmp = Float64(t_1 * cos(Float64(-0.005555555555555556 * Float64(pi * abs(angle)))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (abs(angle) / 180.0);
	t_1 = (a + b) * ((b - a) * (sin(((0.005555555555555556 * abs(angle)) * pi)) * 2.0));
	tmp = 0.0;
	if (abs(angle) <= 1.65e+60)
		tmp = t_1 * cos(t_0);
	elseif (abs(angle) <= 5.4e+242)
		tmp = (((a + b) * ((b - a) * 2.0)) * sin(t_0)) * 1.0;
	else
		tmp = t_1 * cos((-0.005555555555555556 * (pi * abs(angle))));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 1.65e+60], N[(t$95$1 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 5.4e+242], N[(N[(N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(t$95$1 * N[Cos[N[(-0.005555555555555556 * N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 1.6499999999999999e60

   1. Initial program 53.4%

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 67.0%

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

   3. Applied rewrites 66.7%

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

   if 1.6499999999999999e60 < angle < 5.39999999999999968e242

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 57.3%

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

   3. Applied rewrites 57.3%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 56.3%

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

   if 5.39999999999999968e242 < angle

   1. Initial program 53.4%

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 67.0%

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

   3. Applied rewrites 66.7%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. lower-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 66.6%

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

   5. Applied rewrites 66.6%

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

Alternative 3: 67.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.011111111111111112 \cdot \left(\left|angle\right| \cdot \pi\right)\right)\\ t_1 := \left|b\right| - \left|a\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 5.6 \cdot 10^{+57}:\\ \;\;\;\;t\_1 \cdot \left(\left(\left|b\right| + \left|a\right|\right) \cdot t\_0\right)\\ \mathbf{elif}\;\left|angle\right| \leq 3.6 \cdot 10^{+242}:\\ \;\;\;\;\left(\left(\left(\left|a\right| + \left|b\right|\right) \cdot \left(t\_1 \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{\left|angle\right|}{180}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{\left|a\right|}{\left|b\right|}\right) \cdot \left|b\right|\right) \cdot \left(\left|b\right| \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sin (* 0.011111111111111112 (* (fabs angle) PI))))
        (t_1 (- (fabs b) (fabs a))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 5.6e+57)
      (* t_1 (* (+ (fabs b) (fabs a)) t_0))
      (if (<= (fabs angle) 3.6e+242)
        (*
         (*
          (* (+ (fabs a) (fabs b)) (* t_1 2.0))
          (sin (* PI (/ (fabs angle) 180.0))))
         1.0)
        (* (* (+ 1.0 (/ (fabs a) (fabs b))) (fabs b)) (* (fabs b) t_0)))))))

C

double code(double a, double b, double angle) {
	double t_0 = sin((0.011111111111111112 * (fabs(angle) * ((double) M_PI))));
	double t_1 = fabs(b) - fabs(a);
	double tmp;
	if (fabs(angle) <= 5.6e+57) {
		tmp = t_1 * ((fabs(b) + fabs(a)) * t_0);
	} else if (fabs(angle) <= 3.6e+242) {
		tmp = (((fabs(a) + fabs(b)) * (t_1 * 2.0)) * sin((((double) M_PI) * (fabs(angle) / 180.0)))) * 1.0;
	} else {
		tmp = ((1.0 + (fabs(a) / fabs(b))) * fabs(b)) * (fabs(b) * t_0);
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sin((0.011111111111111112 * (Math.abs(angle) * Math.PI)));
	double t_1 = Math.abs(b) - Math.abs(a);
	double tmp;
	if (Math.abs(angle) <= 5.6e+57) {
		tmp = t_1 * ((Math.abs(b) + Math.abs(a)) * t_0);
	} else if (Math.abs(angle) <= 3.6e+242) {
		tmp = (((Math.abs(a) + Math.abs(b)) * (t_1 * 2.0)) * Math.sin((Math.PI * (Math.abs(angle) / 180.0)))) * 1.0;
	} else {
		tmp = ((1.0 + (Math.abs(a) / Math.abs(b))) * Math.abs(b)) * (Math.abs(b) * t_0);
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.sin((0.011111111111111112 * (math.fabs(angle) * math.pi)))
	t_1 = math.fabs(b) - math.fabs(a)
	tmp = 0
	if math.fabs(angle) <= 5.6e+57:
		tmp = t_1 * ((math.fabs(b) + math.fabs(a)) * t_0)
	elif math.fabs(angle) <= 3.6e+242:
		tmp = (((math.fabs(a) + math.fabs(b)) * (t_1 * 2.0)) * math.sin((math.pi * (math.fabs(angle) / 180.0)))) * 1.0
	else:
		tmp = ((1.0 + (math.fabs(a) / math.fabs(b))) * math.fabs(b)) * (math.fabs(b) * t_0)
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = sin(Float64(0.011111111111111112 * Float64(abs(angle) * pi)))
	t_1 = Float64(abs(b) - abs(a))
	tmp = 0.0
	if (abs(angle) <= 5.6e+57)
		tmp = Float64(t_1 * Float64(Float64(abs(b) + abs(a)) * t_0));
	elseif (abs(angle) <= 3.6e+242)
		tmp = Float64(Float64(Float64(Float64(abs(a) + abs(b)) * Float64(t_1 * 2.0)) * sin(Float64(pi * Float64(abs(angle) / 180.0)))) * 1.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(abs(a) / abs(b))) * abs(b)) * Float64(abs(b) * t_0));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = sin((0.011111111111111112 * (abs(angle) * pi)));
	t_1 = abs(b) - abs(a);
	tmp = 0.0;
	if (abs(angle) <= 5.6e+57)
		tmp = t_1 * ((abs(b) + abs(a)) * t_0);
	elseif (abs(angle) <= 3.6e+242)
		tmp = (((abs(a) + abs(b)) * (t_1 * 2.0)) * sin((pi * (abs(angle) / 180.0)))) * 1.0;
	else
		tmp = ((1.0 + (abs(a) / abs(b))) * abs(b)) * (abs(b) * t_0);
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 5.6e+57], N[(t$95$1 * N[(N[(N[Abs[b], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 3.6e+242], N[(N[(N[(N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(1.0 + N[(N[Abs[a], $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 5.59999999999999999e57

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 67.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 67.0%

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 67.0%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 67.0%

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

   5. Applied rewrites 67.0%

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

   if 5.59999999999999999e57 < angle < 3.59999999999999995e242

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 57.3%

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

   3. Applied rewrites 57.3%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 56.3%

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

   if 3.59999999999999995e242 < angle

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 67.0%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 41.6%

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

   6. Applied rewrites 41.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound-+.f64 N/A

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

      6. lower-unsound-/.f64 44.8%

         \[\leadsto \left(\left(1 + \color{blue}{\frac{a}{b}}\right) \cdot b\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   8. Applied rewrites 44.8%

      \[\leadsto \color{blue}{\left(\left(1 + \frac{a}{b}\right) \cdot b\right)} \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 66.7% accurate, 1.6× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 6.5 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(\left|a\right| + \left|b\right|\right) \cdot \left(\left(\left|b\right| - \left|a\right|\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot \left|angle\right|\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot 1\\ \mathbf{elif}\;\left|angle\right| \leq 4.2 \cdot 10^{+207}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \mathsf{fma}\left(-\left|a\right|, \left|a\right|, \left|b\right| \cdot \left|b\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{\left|a\right|}{\left|b\right|}\right) \cdot \left|b\right|\right) \cdot \left(\left|b\right| \cdot \sin \left(0.011111111111111112 \cdot \left(\left|angle\right| \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 6.5e+130)
    (*
     (*
      (+ (fabs a) (fabs b))
      (*
       (- (fabs b) (fabs a))
       (* (sin (* (* 0.005555555555555556 (fabs angle)) PI)) 2.0)))
     1.0)
    (if (<= (fabs angle) 4.2e+207)
      (*
       0.011111111111111112
       (*
        (fabs angle)
        (* PI (fma (- (fabs a)) (fabs a) (* (fabs b) (fabs b))))))
      (*
       (* (+ 1.0 (/ (fabs a) (fabs b))) (fabs b))
       (* (fabs b) (sin (* 0.011111111111111112 (* (fabs angle) PI)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 6.5e+130) {
		tmp = ((fabs(a) + fabs(b)) * ((fabs(b) - fabs(a)) * (sin(((0.005555555555555556 * fabs(angle)) * ((double) M_PI))) * 2.0))) * 1.0;
	} else if (fabs(angle) <= 4.2e+207) {
		tmp = 0.011111111111111112 * (fabs(angle) * (((double) M_PI) * fma(-fabs(a), fabs(a), (fabs(b) * fabs(b)))));
	} else {
		tmp = ((1.0 + (fabs(a) / fabs(b))) * fabs(b)) * (fabs(b) * sin((0.011111111111111112 * (fabs(angle) * ((double) M_PI)))));
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 6.5e+130)
		tmp = Float64(Float64(Float64(abs(a) + abs(b)) * Float64(Float64(abs(b) - abs(a)) * Float64(sin(Float64(Float64(0.005555555555555556 * abs(angle)) * pi)) * 2.0))) * 1.0);
	elseif (abs(angle) <= 4.2e+207)
		tmp = Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * fma(Float64(-abs(a)), abs(a), Float64(abs(b) * abs(b))))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(abs(a) / abs(b))) * abs(b)) * Float64(abs(b) * sin(Float64(0.011111111111111112 * Float64(abs(angle) * pi)))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 6.5e+130], N[(N[(N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 4.2e+207], N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[((-N[Abs[a], $MachinePrecision]) * N[Abs[a], $MachinePrecision] + N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[Abs[a], $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 6.5e130

   1. Initial program 53.4%

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 67.0%

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

   3. Applied rewrites 66.7%

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

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 66.0%

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

   if 6.5e130 < angle < 4.1999999999999999e207

   1. Initial program 53.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 53.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-a, a, {b}^{2}\right)\right)\right) \]

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 53.2%

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

   6. Applied rewrites 53.2%

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

   if 4.1999999999999999e207 < angle

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 67.0%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 41.6%

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

   6. Applied rewrites 41.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound-+.f64 N/A

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

      6. lower-unsound-/.f64 44.8%

         \[\leadsto \left(\left(1 + \color{blue}{\frac{a}{b}}\right) \cdot b\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   8. Applied rewrites 44.8%

      \[\leadsto \color{blue}{\left(\left(1 + \frac{a}{b}\right) \cdot b\right)} \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.011111111111111112 \cdot \left(\left|angle\right| \cdot \pi\right)\right)\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 2.1 \cdot 10^{+110}:\\ \;\;\;\;\left(\left|b\right| - \left|a\right|\right) \cdot \left(\left(\left|b\right| + \left|a\right|\right) \cdot t\_0\right)\\ \mathbf{elif}\;\left|angle\right| \leq 4.2 \cdot 10^{+207}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \mathsf{fma}\left(-\left|a\right|, \left|a\right|, \left|b\right| \cdot \left|b\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{\left|a\right|}{\left|b\right|}\right) \cdot \left|b\right|\right) \cdot \left(\left|b\right| \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sin (* 0.011111111111111112 (* (fabs angle) PI)))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 2.1e+110)
      (* (- (fabs b) (fabs a)) (* (+ (fabs b) (fabs a)) t_0))
      (if (<= (fabs angle) 4.2e+207)
        (*
         0.011111111111111112
         (*
          (fabs angle)
          (* PI (fma (- (fabs a)) (fabs a) (* (fabs b) (fabs b))))))
        (* (* (+ 1.0 (/ (fabs a) (fabs b))) (fabs b)) (* (fabs b) t_0)))))))

C

double code(double a, double b, double angle) {
	double t_0 = sin((0.011111111111111112 * (fabs(angle) * ((double) M_PI))));
	double tmp;
	if (fabs(angle) <= 2.1e+110) {
		tmp = (fabs(b) - fabs(a)) * ((fabs(b) + fabs(a)) * t_0);
	} else if (fabs(angle) <= 4.2e+207) {
		tmp = 0.011111111111111112 * (fabs(angle) * (((double) M_PI) * fma(-fabs(a), fabs(a), (fabs(b) * fabs(b)))));
	} else {
		tmp = ((1.0 + (fabs(a) / fabs(b))) * fabs(b)) * (fabs(b) * t_0);
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	t_0 = sin(Float64(0.011111111111111112 * Float64(abs(angle) * pi)))
	tmp = 0.0
	if (abs(angle) <= 2.1e+110)
		tmp = Float64(Float64(abs(b) - abs(a)) * Float64(Float64(abs(b) + abs(a)) * t_0));
	elseif (abs(angle) <= 4.2e+207)
		tmp = Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * fma(Float64(-abs(a)), abs(a), Float64(abs(b) * abs(b))))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(abs(a) / abs(b))) * abs(b)) * Float64(abs(b) * t_0));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 2.1e+110], N[(N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 4.2e+207], N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[((-N[Abs[a], $MachinePrecision]) * N[Abs[a], $MachinePrecision] + N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[Abs[a], $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if angle < 2.10000000000000015e110

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 67.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 67.0%

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 67.0%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 67.0%

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

   5. Applied rewrites 67.0%

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

   if 2.10000000000000015e110 < angle < 4.1999999999999999e207

   1. Initial program 53.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 53.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-a, a, {b}^{2}\right)\right)\right) \]

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 53.2%

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

   6. Applied rewrites 53.2%

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

   if 4.1999999999999999e207 < angle

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 67.0%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 41.6%

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

   6. Applied rewrites 41.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound-+.f64 N/A

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

      6. lower-unsound-/.f64 44.8%

         \[\leadsto \left(\left(1 + \color{blue}{\frac{a}{b}}\right) \cdot b\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   8. Applied rewrites 44.8%

      \[\leadsto \color{blue}{\left(\left(1 + \frac{a}{b}\right) \cdot b\right)} \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 66.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|a\right| + b\right) \cdot \left(\left(b - \left|a\right|\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right) \cdot 2\right)\right)\\ \mathbf{if}\;\left|a\right| \leq 6 \cdot 10^{+159}:\\ \;\;\;\;t\_0 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0
         (*
          (+ (fabs a) b)
          (*
           (- b (fabs a))
           (* (sin (/ 0.005555555555555556 (/ 1.0 (* angle PI)))) 2.0)))))
   (if (<= (fabs a) 6e+159)
     (* t_0 (cos (* (* PI angle) 0.005555555555555556)))
     (* t_0 1.0))))

C

double code(double a, double b, double angle) {
	double t_0 = (fabs(a) + b) * ((b - fabs(a)) * (sin((0.005555555555555556 / (1.0 / (angle * ((double) M_PI))))) * 2.0));
	double tmp;
	if (fabs(a) <= 6e+159) {
		tmp = t_0 * cos(((((double) M_PI) * angle) * 0.005555555555555556));
	} else {
		tmp = t_0 * 1.0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (Math.abs(a) + b) * ((b - Math.abs(a)) * (Math.sin((0.005555555555555556 / (1.0 / (angle * Math.PI)))) * 2.0));
	double tmp;
	if (Math.abs(a) <= 6e+159) {
		tmp = t_0 * Math.cos(((Math.PI * angle) * 0.005555555555555556));
	} else {
		tmp = t_0 * 1.0;
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (math.fabs(a) + b) * ((b - math.fabs(a)) * (math.sin((0.005555555555555556 / (1.0 / (angle * math.pi)))) * 2.0))
	tmp = 0
	if math.fabs(a) <= 6e+159:
		tmp = t_0 * math.cos(((math.pi * angle) * 0.005555555555555556))
	else:
		tmp = t_0 * 1.0
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(abs(a) + b) * Float64(Float64(b - abs(a)) * Float64(sin(Float64(0.005555555555555556 / Float64(1.0 / Float64(angle * pi)))) * 2.0)))
	tmp = 0.0
	if (abs(a) <= 6e+159)
		tmp = Float64(t_0 * cos(Float64(Float64(pi * angle) * 0.005555555555555556)));
	else
		tmp = Float64(t_0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (abs(a) + b) * ((b - abs(a)) * (sin((0.005555555555555556 / (1.0 / (angle * pi)))) * 2.0));
	tmp = 0.0;
	if (abs(a) <= 6e+159)
		tmp = t_0 * cos(((pi * angle) * 0.005555555555555556));
	else
		tmp = t_0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(N[Abs[a], $MachinePrecision] + b), $MachinePrecision] * N[(N[(b - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(0.005555555555555556 / N[(1.0 / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 6e+159], N[(t$95$0 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.0000000000000004e159

   1. Initial program 53.4%

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 67.0%

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

   3. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

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

      7. div-flip N/A

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

      8. *-commutative N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-unsound-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 66.7%

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

   5. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 67.0%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\color{blue}{\frac{1}{\pi \cdot angle}}}\right) \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 67.0%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\frac{1}{\color{blue}{angle \cdot \pi}}}\right) \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Applied rewrites 67.0%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right)} \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 66.8%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 66.8%

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

   9. Applied rewrites 66.8%

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

   if 6.0000000000000004e159 < a

   1. Initial program 53.4%

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 67.0%

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

   3. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

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

      7. div-flip N/A

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

      8. *-commutative N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-unsound-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 66.7%

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

   5. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 67.0%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\color{blue}{\frac{1}{\pi \cdot angle}}}\right) \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 67.0%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\frac{1}{\color{blue}{angle \cdot \pi}}}\right) \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Applied rewrites 67.0%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right)} \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right) \cdot 2\right)\right)\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.3%

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

Alternative 7: 66.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := a + \left|b\right|\\ t_1 := \left|b\right| - a\\ \mathbf{if}\;\left|b\right| \leq 6.5 \cdot 10^{+257}:\\ \;\;\;\;\left(t\_0 \cdot \left(t\_1 \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(t\_1 \cdot \left(\sin \left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right) \cdot 2\right)\right)\right) \cdot 1\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (+ a (fabs b))) (t_1 (- (fabs b) a)))
   (if (<= (fabs b) 6.5e+257)
     (*
      (* t_0 (* t_1 (* (sin (* (* 0.005555555555555556 angle) PI)) 2.0)))
      (cos (* -0.005555555555555556 (* PI angle))))
     (*
      (*
       t_0
       (* t_1 (* (sin (/ 0.005555555555555556 (/ 1.0 (* angle PI)))) 2.0)))
      1.0))))

C

double code(double a, double b, double angle) {
	double t_0 = a + fabs(b);
	double t_1 = fabs(b) - a;
	double tmp;
	if (fabs(b) <= 6.5e+257) {
		tmp = (t_0 * (t_1 * (sin(((0.005555555555555556 * angle) * ((double) M_PI))) * 2.0))) * cos((-0.005555555555555556 * (((double) M_PI) * angle)));
	} else {
		tmp = (t_0 * (t_1 * (sin((0.005555555555555556 / (1.0 / (angle * ((double) M_PI))))) * 2.0))) * 1.0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = a + Math.abs(b);
	double t_1 = Math.abs(b) - a;
	double tmp;
	if (Math.abs(b) <= 6.5e+257) {
		tmp = (t_0 * (t_1 * (Math.sin(((0.005555555555555556 * angle) * Math.PI)) * 2.0))) * Math.cos((-0.005555555555555556 * (Math.PI * angle)));
	} else {
		tmp = (t_0 * (t_1 * (Math.sin((0.005555555555555556 / (1.0 / (angle * Math.PI)))) * 2.0))) * 1.0;
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = a + math.fabs(b)
	t_1 = math.fabs(b) - a
	tmp = 0
	if math.fabs(b) <= 6.5e+257:
		tmp = (t_0 * (t_1 * (math.sin(((0.005555555555555556 * angle) * math.pi)) * 2.0))) * math.cos((-0.005555555555555556 * (math.pi * angle)))
	else:
		tmp = (t_0 * (t_1 * (math.sin((0.005555555555555556 / (1.0 / (angle * math.pi)))) * 2.0))) * 1.0
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(a + abs(b))
	t_1 = Float64(abs(b) - a)
	tmp = 0.0
	if (abs(b) <= 6.5e+257)
		tmp = Float64(Float64(t_0 * Float64(t_1 * Float64(sin(Float64(Float64(0.005555555555555556 * angle) * pi)) * 2.0))) * cos(Float64(-0.005555555555555556 * Float64(pi * angle))));
	else
		tmp = Float64(Float64(t_0 * Float64(t_1 * Float64(sin(Float64(0.005555555555555556 / Float64(1.0 / Float64(angle * pi)))) * 2.0))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = a + abs(b);
	t_1 = abs(b) - a;
	tmp = 0.0;
	if (abs(b) <= 6.5e+257)
		tmp = (t_0 * (t_1 * (sin(((0.005555555555555556 * angle) * pi)) * 2.0))) * cos((-0.005555555555555556 * (pi * angle)));
	else
		tmp = (t_0 * (t_1 * (sin((0.005555555555555556 / (1.0 / (angle * pi)))) * 2.0))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 6.5e+257], N[(N[(t$95$0 * N[(t$95$1 * N[(N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(t$95$1 * N[(N[Sin[N[(0.005555555555555556 / N[(1.0 / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 6.50000000000000026e257

   1. Initial program 53.4%

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 67.0%

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

   3. Applied rewrites 66.7%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. lower-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 66.6%

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

   5. Applied rewrites 66.6%

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

   if 6.50000000000000026e257 < b

   1. Initial program 53.4%

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 67.0%

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

   3. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

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

      7. div-flip N/A

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

      8. *-commutative N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-unsound-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 66.7%

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

   5. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 67.0%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\color{blue}{\frac{1}{\pi \cdot angle}}}\right) \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 67.0%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\frac{1}{\color{blue}{angle \cdot \pi}}}\right) \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Applied rewrites 67.0%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right)} \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right) \cdot 2\right)\right)\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.3%

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

Alternative 8: 65.7% accurate, 1.2× speedup

Math

\[\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right) \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (*
   (+ a b)
   (* (- b a) (* (sin (/ 0.005555555555555556 (/ 1.0 (* angle PI)))) 2.0)))
  (cos (* PI (/ angle 180.0)))))

C

double code(double a, double b, double angle) {
	return ((a + b) * ((b - a) * (sin((0.005555555555555556 / (1.0 / (angle * ((double) M_PI))))) * 2.0))) * cos((((double) M_PI) * (angle / 180.0)));
}

Java

public static double code(double a, double b, double angle) {
	return ((a + b) * ((b - a) * (Math.sin((0.005555555555555556 / (1.0 / (angle * Math.PI)))) * 2.0))) * Math.cos((Math.PI * (angle / 180.0)));
}

Python

def code(a, b, angle):
	return ((a + b) * ((b - a) * (math.sin((0.005555555555555556 / (1.0 / (angle * math.pi)))) * 2.0))) * math.cos((math.pi * (angle / 180.0)))

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(sin(Float64(0.005555555555555556 / Float64(1.0 / Float64(angle * pi)))) * 2.0))) * cos(Float64(pi * Float64(angle / 180.0))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a + b) * ((b - a) * (sin((0.005555555555555556 / (1.0 / (angle * pi)))) * 2.0))) * cos((pi * (angle / 180.0)));
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(0.005555555555555556 / N[(1.0 / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift--.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. difference-of-squares N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lower-*.f64 67.0%

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

3. Applied rewrites 66.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. mult-flip-rev N/A

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

   7. div-flip N/A

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

   8. *-commutative N/A

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

   9. lower-unsound-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-unsound-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 66.7%

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

5. Applied rewrites 66.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 67.0%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\color{blue}{\frac{1}{\pi \cdot angle}}}\right) \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 67.0%

       \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\frac{1}{\color{blue}{angle \cdot \pi}}}\right) \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

7. Applied rewrites 67.0%

   \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right)} \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
8. Add Preprocessing

Alternative 9: 65.5% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 5 \cdot 10^{-42}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \left|angle\right|\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \cdot \left(b - a\right)\\ \mathbf{elif}\;\left|angle\right| \leq 2.1 \cdot 10^{+110}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(\left(\left|angle\right| \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 5e-42)
    (* (* (* (* 0.011111111111111112 (fabs angle)) (+ b a)) PI) (- b a))
    (if (<= (fabs angle) 2.1e+110)
      (*
       (* (- b a) (+ a b))
       (sin (* (* (fabs angle) PI) 0.011111111111111112)))
      (*
       0.011111111111111112
       (* (fabs angle) (* PI (fma (- a) a (* b b)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 5e-42) {
		tmp = (((0.011111111111111112 * fabs(angle)) * (b + a)) * ((double) M_PI)) * (b - a);
	} else if (fabs(angle) <= 2.1e+110) {
		tmp = ((b - a) * (a + b)) * sin(((fabs(angle) * ((double) M_PI)) * 0.011111111111111112));
	} else {
		tmp = 0.011111111111111112 * (fabs(angle) * (((double) M_PI) * fma(-a, a, (b * b))));
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 5e-42)
		tmp = Float64(Float64(Float64(Float64(0.011111111111111112 * abs(angle)) * Float64(b + a)) * pi) * Float64(b - a));
	elseif (abs(angle) <= 2.1e+110)
		tmp = Float64(Float64(Float64(b - a) * Float64(a + b)) * sin(Float64(Float64(abs(angle) * pi) * 0.011111111111111112)));
	else
		tmp = Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * fma(Float64(-a), a, Float64(b * b)))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 5e-42], N[(N[(N[(N[(0.011111111111111112 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 2.1e+110], N[(N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[((-a) * a + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 5.00000000000000003e-42

   1. Initial program 53.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

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

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 54.4%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 54.4%

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

   6. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 54.4%

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

   8. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 62.3%

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 62.3%

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

   10. Applied rewrites 62.3%

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

   if 5.00000000000000003e-42 < angle < 2.10000000000000015e110

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 57.3%

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

   if 2.10000000000000015e110 < angle

   1. Initial program 53.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 53.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-a, a, {b}^{2}\right)\right)\right) \]

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 53.2%

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

   6. Applied rewrites 53.2%

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

Alternative 10: 65.5% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 2.1 \cdot 10^{+110}:\\ \;\;\;\;\left(\left|b\right| - a\right) \cdot \left(\left(\left|b\right| + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\left|angle\right| \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \mathsf{fma}\left(-a, a, \left|b\right| \cdot \left|b\right|\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 2.1e+110)
    (*
     (- (fabs b) a)
     (* (+ (fabs b) a) (sin (* 0.011111111111111112 (* (fabs angle) PI)))))
    (*
     0.011111111111111112
     (* (fabs angle) (* PI (fma (- a) a (* (fabs b) (fabs b)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 2.1e+110) {
		tmp = (fabs(b) - a) * ((fabs(b) + a) * sin((0.011111111111111112 * (fabs(angle) * ((double) M_PI)))));
	} else {
		tmp = 0.011111111111111112 * (fabs(angle) * (((double) M_PI) * fma(-a, a, (fabs(b) * fabs(b)))));
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 2.1e+110)
		tmp = Float64(Float64(abs(b) - a) * Float64(Float64(abs(b) + a) * sin(Float64(0.011111111111111112 * Float64(abs(angle) * pi)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * fma(Float64(-a), a, Float64(abs(b) * abs(b))))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 2.1e+110], N[(N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] + a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[((-a) * a + N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 2.10000000000000015e110

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 67.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 67.0%

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 67.0%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 67.0%

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

   5. Applied rewrites 67.0%

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

   if 2.10000000000000015e110 < angle

   1. Initial program 53.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 53.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-a, a, {b}^{2}\right)\right)\right) \]

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 53.2%

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

   6. Applied rewrites 53.2%

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

Alternative 11: 65.3% accurate, 2.0× speedup

Math

\[\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right) \cdot 2\right)\right)\right) \cdot 1 \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (*
   (+ a b)
   (* (- b a) (* (sin (/ 0.005555555555555556 (/ 1.0 (* angle PI)))) 2.0)))
  1.0))

C

double code(double a, double b, double angle) {
	return ((a + b) * ((b - a) * (sin((0.005555555555555556 / (1.0 / (angle * ((double) M_PI))))) * 2.0))) * 1.0;
}

Java

public static double code(double a, double b, double angle) {
	return ((a + b) * ((b - a) * (Math.sin((0.005555555555555556 / (1.0 / (angle * Math.PI)))) * 2.0))) * 1.0;
}

Python

def code(a, b, angle):
	return ((a + b) * ((b - a) * (math.sin((0.005555555555555556 / (1.0 / (angle * math.pi)))) * 2.0))) * 1.0

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(sin(Float64(0.005555555555555556 / Float64(1.0 / Float64(angle * pi)))) * 2.0))) * 1.0)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a + b) * ((b - a) * (sin((0.005555555555555556 / (1.0 / (angle * pi)))) * 2.0))) * 1.0;
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(0.005555555555555556 / N[(1.0 / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift--.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. difference-of-squares N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lower-*.f64 67.0%

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

3. Applied rewrites 66.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. mult-flip-rev N/A

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

   7. div-flip N/A

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

   8. *-commutative N/A

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

   9. lower-unsound-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-unsound-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 66.7%

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

5. Applied rewrites 66.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 67.0%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\color{blue}{\frac{1}{\pi \cdot angle}}}\right) \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 67.0%

       \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\frac{1}{\color{blue}{angle \cdot \pi}}}\right) \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

7. Applied rewrites 67.0%

   \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right)} \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

8. Taylor expanded in angle around 0

   \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right) \cdot 2\right)\right)\right) \cdot \color{blue}{1} \]
9. Step-by-step derivation

10. Applied rewrites 65.3%

    \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right) \cdot 2\right)\right)\right) \cdot \color{blue}{1} \]
11. Add Preprocessing

Alternative 12: 64.7% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (* (+ a b) (* (- b a) (sin (* (* 0.011111111111111112 angle) PI)))))

C

double code(double a, double b, double angle) {
	return (a + b) * ((b - a) * sin(((0.011111111111111112 * angle) * ((double) M_PI))));
}

Java

public static double code(double a, double b, double angle) {
	return (a + b) * ((b - a) * Math.sin(((0.011111111111111112 * angle) * Math.PI)));
}

Python

def code(a, b, angle):
	return (a + b) * ((b - a) * math.sin(((0.011111111111111112 * angle) * math.pi)))

Julia

function code(a, b, angle)
	return Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(Float64(0.011111111111111112 * angle) * pi))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a + b) * ((b - a) * sin(((0.011111111111111112 * angle) * pi)));
end

Wolfram

code[a_, b_, angle_] := N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. 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. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. 2-sin N/A

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

   10. count-2 N/A

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

3. Applied rewrites 67.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 66.6%

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

5. Applied rewrites 66.6%

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

Alternative 13: 62.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| + \left|a\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 2.2 \cdot 10^{+190}:\\ \;\;\;\;\left(\left(\left|angle\right| \cdot t\_0\right) \cdot \left(\left|b\right| - \left|a\right|\right)\right) \cdot \left(0.011111111111111112 \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left|angle\right|\right) \cdot \left(t\_0 \cdot \left(\left|b\right| \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (+ (fabs b) (fabs a))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 2.2e+190)
      (*
       (* (* (fabs angle) t_0) (- (fabs b) (fabs a)))
       (* 0.011111111111111112 PI))
      (* (* 0.011111111111111112 (fabs angle)) (* t_0 (* (fabs b) PI)))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) + fabs(a);
	double tmp;
	if (fabs(angle) <= 2.2e+190) {
		tmp = ((fabs(angle) * t_0) * (fabs(b) - fabs(a))) * (0.011111111111111112 * ((double) M_PI));
	} else {
		tmp = (0.011111111111111112 * fabs(angle)) * (t_0 * (fabs(b) * ((double) M_PI)));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) + Math.abs(a);
	double tmp;
	if (Math.abs(angle) <= 2.2e+190) {
		tmp = ((Math.abs(angle) * t_0) * (Math.abs(b) - Math.abs(a))) * (0.011111111111111112 * Math.PI);
	} else {
		tmp = (0.011111111111111112 * Math.abs(angle)) * (t_0 * (Math.abs(b) * Math.PI));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) + math.fabs(a)
	tmp = 0
	if math.fabs(angle) <= 2.2e+190:
		tmp = ((math.fabs(angle) * t_0) * (math.fabs(b) - math.fabs(a))) * (0.011111111111111112 * math.pi)
	else:
		tmp = (0.011111111111111112 * math.fabs(angle)) * (t_0 * (math.fabs(b) * math.pi))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) + abs(a))
	tmp = 0.0
	if (abs(angle) <= 2.2e+190)
		tmp = Float64(Float64(Float64(abs(angle) * t_0) * Float64(abs(b) - abs(a))) * Float64(0.011111111111111112 * pi));
	else
		tmp = Float64(Float64(0.011111111111111112 * abs(angle)) * Float64(t_0 * Float64(abs(b) * pi)));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) + abs(a);
	tmp = 0.0;
	if (abs(angle) <= 2.2e+190)
		tmp = ((abs(angle) * t_0) * (abs(b) - abs(a))) * (0.011111111111111112 * pi);
	else
		tmp = (0.011111111111111112 * abs(angle)) * (t_0 * (abs(b) * pi));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 2.2e+190], N[(N[(N[(N[Abs[angle], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.2e190

   1. Initial program 53.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

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

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.4%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.4%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.4%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \cdot \frac{1}{90} \]

      4. associate-*l* N/A

         \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{90}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{90}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(angle \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\pi}\right) \]

      13. lower-*.f64 62.3%

          \[\leadsto \left(\left(angle \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \color{blue}{\pi}\right) \]

   8. Applied rewrites 62.3%

      \[\leadsto \left(\left(angle \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \pi\right)} \]

   if 2.2e190 < angle

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.4%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.4%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.4%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)} \cdot \pi\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      12. lower-*.f64 54.4%

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right)\right) \]

   8. Applied rewrites 54.4%

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \color{blue}{\pi}\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \]

       2. lower-PI.f64 37.3%

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \pi\right)\right) \]

   11. Applied rewrites 37.3%

       \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \color{blue}{\pi}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 62.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| + \left|a\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 2.2 \cdot 10^{+190}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\left|b\right| - \left|a\right|\right)\right)\right) \cdot t\_0\right) \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left|angle\right|\right) \cdot \left(t\_0 \cdot \left(\left|b\right| \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (+ (fabs b) (fabs a))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 2.2e+190)
      (*
       (* (* 0.011111111111111112 (* (fabs angle) (- (fabs b) (fabs a)))) t_0)
       PI)
      (* (* 0.011111111111111112 (fabs angle)) (* t_0 (* (fabs b) PI)))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) + fabs(a);
	double tmp;
	if (fabs(angle) <= 2.2e+190) {
		tmp = ((0.011111111111111112 * (fabs(angle) * (fabs(b) - fabs(a)))) * t_0) * ((double) M_PI);
	} else {
		tmp = (0.011111111111111112 * fabs(angle)) * (t_0 * (fabs(b) * ((double) M_PI)));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) + Math.abs(a);
	double tmp;
	if (Math.abs(angle) <= 2.2e+190) {
		tmp = ((0.011111111111111112 * (Math.abs(angle) * (Math.abs(b) - Math.abs(a)))) * t_0) * Math.PI;
	} else {
		tmp = (0.011111111111111112 * Math.abs(angle)) * (t_0 * (Math.abs(b) * Math.PI));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) + math.fabs(a)
	tmp = 0
	if math.fabs(angle) <= 2.2e+190:
		tmp = ((0.011111111111111112 * (math.fabs(angle) * (math.fabs(b) - math.fabs(a)))) * t_0) * math.pi
	else:
		tmp = (0.011111111111111112 * math.fabs(angle)) * (t_0 * (math.fabs(b) * math.pi))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) + abs(a))
	tmp = 0.0
	if (abs(angle) <= 2.2e+190)
		tmp = Float64(Float64(Float64(0.011111111111111112 * Float64(abs(angle) * Float64(abs(b) - abs(a)))) * t_0) * pi);
	else
		tmp = Float64(Float64(0.011111111111111112 * abs(angle)) * Float64(t_0 * Float64(abs(b) * pi)));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) + abs(a);
	tmp = 0.0;
	if (abs(angle) <= 2.2e+190)
		tmp = ((0.011111111111111112 * (abs(angle) * (abs(b) - abs(a)))) * t_0) * pi;
	else
		tmp = (0.011111111111111112 * abs(angle)) * (t_0 * (abs(b) * pi));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 2.2e+190], N[(N[(N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * Pi), $MachinePrecision], N[(N[(0.011111111111111112 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.2e190

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.4%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.4%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.4%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \color{blue}{\pi} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \color{blue}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \pi \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \pi \]

      7. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\right) \cdot \pi \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

      11. lower-*.f64 62.3%

          \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

   8. Applied rewrites 62.3%

      \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\pi} \]

   if 2.2e190 < angle

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.4%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.4%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.4%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)} \cdot \pi\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      12. lower-*.f64 54.4%

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right)\right) \]

   8. Applied rewrites 54.4%

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \color{blue}{\pi}\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \]

       2. lower-PI.f64 37.3%

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \pi\right)\right) \]

   11. Applied rewrites 37.3%

       \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \color{blue}{\pi}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 56.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 3.8 \cdot 10^{+195}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(\left(\left(\left|b\right| - \left|a\right|\right) \cdot \left(\left|b\right| + \left|a\right|\right)\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left|a\right| + \left|b\right|\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\left|b\right| \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (fabs b) 3.8e+195)
   (*
    (* PI angle)
    (* (* (- (fabs b) (fabs a)) (+ (fabs b) (fabs a))) 0.011111111111111112))
   (*
    (+ (fabs a) (fabs b))
    (* 0.011111111111111112 (* angle (* (fabs b) PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(b) <= 3.8e+195) {
		tmp = (((double) M_PI) * angle) * (((fabs(b) - fabs(a)) * (fabs(b) + fabs(a))) * 0.011111111111111112);
	} else {
		tmp = (fabs(a) + fabs(b)) * (0.011111111111111112 * (angle * (fabs(b) * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(b) <= 3.8e+195) {
		tmp = (Math.PI * angle) * (((Math.abs(b) - Math.abs(a)) * (Math.abs(b) + Math.abs(a))) * 0.011111111111111112);
	} else {
		tmp = (Math.abs(a) + Math.abs(b)) * (0.011111111111111112 * (angle * (Math.abs(b) * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(b) <= 3.8e+195:
		tmp = (math.pi * angle) * (((math.fabs(b) - math.fabs(a)) * (math.fabs(b) + math.fabs(a))) * 0.011111111111111112)
	else:
		tmp = (math.fabs(a) + math.fabs(b)) * (0.011111111111111112 * (angle * (math.fabs(b) * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(b) <= 3.8e+195)
		tmp = Float64(Float64(pi * angle) * Float64(Float64(Float64(abs(b) - abs(a)) * Float64(abs(b) + abs(a))) * 0.011111111111111112));
	else
		tmp = Float64(Float64(abs(a) + abs(b)) * Float64(0.011111111111111112 * Float64(angle * Float64(abs(b) * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(b) <= 3.8e+195)
		tmp = (pi * angle) * (((abs(b) - abs(a)) * (abs(b) + abs(a))) * 0.011111111111111112);
	else
		tmp = (abs(a) + abs(b)) * (0.011111111111111112 * (angle * (abs(b) * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[b], $MachinePrecision], 3.8e+195], N[(N[(Pi * angle), $MachinePrecision] * N[(N[(N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.8e195

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90} \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. associate-*l* N/A

         \[\leadsto \left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \frac{1}{90}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \frac{1}{90}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \frac{1}{90}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \frac{1}{90}\right) \]

      10. lower-*.f64 50.6%

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{0.011111111111111112}\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \frac{1}{90}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \frac{1}{90}\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - a \cdot a\right) \cdot \frac{1}{90}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left({b}^{2} - a \cdot a\right) \cdot \frac{1}{90}\right) \]

      15. unpow2 N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \frac{1}{90}\right) \]

      16. difference-of-squares-rev N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90}\right) \]

      17. +-commutative N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90}\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90}\right) \]

      19. lift--.f64 N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90}\right) \]

      20. *-commutative N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \frac{1}{90}\right) \]

      21. lower-*.f64 54.4%

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot 0.011111111111111112\right) \]

      22. lift-+.f64 N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \frac{1}{90}\right) \]

      23. +-commutative N/A

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \frac{1}{90}\right) \]

      24. lower-+.f64 54.4%

          \[\leadsto \left(\pi \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot 0.011111111111111112\right) \]

   6. Applied rewrites 54.4%

      \[\leadsto \left(\pi \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot 0.011111111111111112\right)} \]

   if 3.8e195 < b

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift-sin.f64 N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-cos.f64 N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

      9. 2-sin N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      10. count-2 N/A

          \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)} \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   4. Taylor expanded in a around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-PI.f64 41.6%

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   6. Applied rewrites 41.6%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi\right)\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(b \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-PI.f64 39.9%

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

   9. Applied rewrites 39.9%

      \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 56.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 3.8 \cdot 10^{+195}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(\left|b\right| + \left|a\right|\right) \cdot \left(\left(\left|b\right| - \left|a\right|\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left|a\right| + \left|b\right|\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\left|b\right| \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (fabs b) 3.8e+195)
   (*
    (* 0.011111111111111112 angle)
    (* (+ (fabs b) (fabs a)) (* (- (fabs b) (fabs a)) PI)))
   (*
    (+ (fabs a) (fabs b))
    (* 0.011111111111111112 (* angle (* (fabs b) PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(b) <= 3.8e+195) {
		tmp = (0.011111111111111112 * angle) * ((fabs(b) + fabs(a)) * ((fabs(b) - fabs(a)) * ((double) M_PI)));
	} else {
		tmp = (fabs(a) + fabs(b)) * (0.011111111111111112 * (angle * (fabs(b) * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(b) <= 3.8e+195) {
		tmp = (0.011111111111111112 * angle) * ((Math.abs(b) + Math.abs(a)) * ((Math.abs(b) - Math.abs(a)) * Math.PI));
	} else {
		tmp = (Math.abs(a) + Math.abs(b)) * (0.011111111111111112 * (angle * (Math.abs(b) * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(b) <= 3.8e+195:
		tmp = (0.011111111111111112 * angle) * ((math.fabs(b) + math.fabs(a)) * ((math.fabs(b) - math.fabs(a)) * math.pi))
	else:
		tmp = (math.fabs(a) + math.fabs(b)) * (0.011111111111111112 * (angle * (math.fabs(b) * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(b) <= 3.8e+195)
		tmp = Float64(Float64(0.011111111111111112 * angle) * Float64(Float64(abs(b) + abs(a)) * Float64(Float64(abs(b) - abs(a)) * pi)));
	else
		tmp = Float64(Float64(abs(a) + abs(b)) * Float64(0.011111111111111112 * Float64(angle * Float64(abs(b) * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(b) <= 3.8e+195)
		tmp = (0.011111111111111112 * angle) * ((abs(b) + abs(a)) * ((abs(b) - abs(a)) * pi));
	else
		tmp = (abs(a) + abs(b)) * (0.011111111111111112 * (angle * (abs(b) * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[b], $MachinePrecision], 3.8e+195], N[(N[(0.011111111111111112 * angle), $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.8e195

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.4%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.4%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.4%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)} \cdot \pi\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      12. lower-*.f64 54.4%

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right)\right) \]

   8. Applied rewrites 54.4%

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)} \]

   if 3.8e195 < b

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift-sin.f64 N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-cos.f64 N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

      9. 2-sin N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      10. count-2 N/A

          \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)} \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   4. Taylor expanded in a around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-PI.f64 41.6%

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   6. Applied rewrites 41.6%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi\right)\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(b \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-PI.f64 39.9%

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

   9. Applied rewrites 39.9%

      \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 56.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 3.8 \cdot 10^{+195}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(\left|b\right| - \left|a\right|\right) \cdot \left(\left|b\right| + \left|a\right|\right)\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left|a\right| + \left|b\right|\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\left|b\right| \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (fabs b) 3.8e+195)
   (*
    0.011111111111111112
    (* (* angle (* (- (fabs b) (fabs a)) (+ (fabs b) (fabs a)))) PI))
   (*
    (+ (fabs a) (fabs b))
    (* 0.011111111111111112 (* angle (* (fabs b) PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(b) <= 3.8e+195) {
		tmp = 0.011111111111111112 * ((angle * ((fabs(b) - fabs(a)) * (fabs(b) + fabs(a)))) * ((double) M_PI));
	} else {
		tmp = (fabs(a) + fabs(b)) * (0.011111111111111112 * (angle * (fabs(b) * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(b) <= 3.8e+195) {
		tmp = 0.011111111111111112 * ((angle * ((Math.abs(b) - Math.abs(a)) * (Math.abs(b) + Math.abs(a)))) * Math.PI);
	} else {
		tmp = (Math.abs(a) + Math.abs(b)) * (0.011111111111111112 * (angle * (Math.abs(b) * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(b) <= 3.8e+195:
		tmp = 0.011111111111111112 * ((angle * ((math.fabs(b) - math.fabs(a)) * (math.fabs(b) + math.fabs(a)))) * math.pi)
	else:
		tmp = (math.fabs(a) + math.fabs(b)) * (0.011111111111111112 * (angle * (math.fabs(b) * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(b) <= 3.8e+195)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle * Float64(Float64(abs(b) - abs(a)) * Float64(abs(b) + abs(a)))) * pi));
	else
		tmp = Float64(Float64(abs(a) + abs(b)) * Float64(0.011111111111111112 * Float64(angle * Float64(abs(b) * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(b) <= 3.8e+195)
		tmp = 0.011111111111111112 * ((angle * ((abs(b) - abs(a)) * (abs(b) + abs(a)))) * pi);
	else
		tmp = (abs(a) + abs(b)) * (0.011111111111111112 * (angle * (abs(b) * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[b], $MachinePrecision], 3.8e+195], N[(0.011111111111111112 * N[(N[(angle * N[(N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.8e195

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.4%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.4%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.4%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

   if 3.8e195 < b

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift-sin.f64 N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-cos.f64 N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

      9. 2-sin N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      10. count-2 N/A

          \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)} \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   4. Taylor expanded in a around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-PI.f64 41.6%

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   6. Applied rewrites 41.6%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi\right)\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(b \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-PI.f64 39.9%

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

   9. Applied rewrites 39.9%

      \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 40.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \pi\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 4 \cdot 10^{+108}:\\ \;\;\;\;\left(\left|a\right| + \left|b\right|\right) \cdot \left(0.011111111111111112 \cdot \left(\left|angle\right| \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left|angle\right|\right) \cdot \left(\left(\left|b\right| + \left|a\right|\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (fabs b) PI)))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 4e+108)
      (* (+ (fabs a) (fabs b)) (* 0.011111111111111112 (* (fabs angle) t_0)))
      (*
       (* 0.011111111111111112 (fabs angle))
       (* (+ (fabs b) (fabs a)) t_0))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) * ((double) M_PI);
	double tmp;
	if (fabs(angle) <= 4e+108) {
		tmp = (fabs(a) + fabs(b)) * (0.011111111111111112 * (fabs(angle) * t_0));
	} else {
		tmp = (0.011111111111111112 * fabs(angle)) * ((fabs(b) + fabs(a)) * t_0);
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) * Math.PI;
	double tmp;
	if (Math.abs(angle) <= 4e+108) {
		tmp = (Math.abs(a) + Math.abs(b)) * (0.011111111111111112 * (Math.abs(angle) * t_0));
	} else {
		tmp = (0.011111111111111112 * Math.abs(angle)) * ((Math.abs(b) + Math.abs(a)) * t_0);
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) * math.pi
	tmp = 0
	if math.fabs(angle) <= 4e+108:
		tmp = (math.fabs(a) + math.fabs(b)) * (0.011111111111111112 * (math.fabs(angle) * t_0))
	else:
		tmp = (0.011111111111111112 * math.fabs(angle)) * ((math.fabs(b) + math.fabs(a)) * t_0)
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) * pi)
	tmp = 0.0
	if (abs(angle) <= 4e+108)
		tmp = Float64(Float64(abs(a) + abs(b)) * Float64(0.011111111111111112 * Float64(abs(angle) * t_0)));
	else
		tmp = Float64(Float64(0.011111111111111112 * abs(angle)) * Float64(Float64(abs(b) + abs(a)) * t_0));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) * pi;
	tmp = 0.0;
	if (abs(angle) <= 4e+108)
		tmp = (abs(a) + abs(b)) * (0.011111111111111112 * (abs(angle) * t_0));
	else
		tmp = (0.011111111111111112 * abs(angle)) * ((abs(b) + abs(a)) * t_0);
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 4e+108], N[(N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.0000000000000001e108

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift-sin.f64 N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-cos.f64 N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

      9. 2-sin N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      10. count-2 N/A

          \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)} \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   4. Taylor expanded in a around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-PI.f64 41.6%

         \[\leadsto \left(a + b\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   6. Applied rewrites 41.6%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi\right)\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(b \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-PI.f64 39.9%

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

   9. Applied rewrites 39.9%

      \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi\right)\right)}\right) \]

   if 4.0000000000000001e108 < angle

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.5%

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.4%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.4%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.4%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)} \cdot \pi\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      12. lower-*.f64 54.4%

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right)\right) \]

   8. Applied rewrites 54.4%

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \color{blue}{\pi}\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \]

       2. lower-PI.f64 37.3%

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \pi\right)\right) \]

   11. Applied rewrites 37.3%

       \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \color{blue}{\pi}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 37.4% accurate, 7.2× speedup

Math

\[\left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + \left|a\right|\right) \cdot \left(b \cdot \pi\right)\right) \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* (* 0.011111111111111112 angle) (* (+ b (fabs a)) (* b PI))))

C

double code(double a, double b, double angle) {
	return (0.011111111111111112 * angle) * ((b + fabs(a)) * (b * ((double) M_PI)));
}

Java

public static double code(double a, double b, double angle) {
	return (0.011111111111111112 * angle) * ((b + Math.abs(a)) * (b * Math.PI));
}

Python

def code(a, b, angle):
	return (0.011111111111111112 * angle) * ((b + math.fabs(a)) * (b * math.pi))

Julia

function code(a, b, angle)
	return Float64(Float64(0.011111111111111112 * angle) * Float64(Float64(b + abs(a)) * Float64(b * pi)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (0.011111111111111112 * angle) * ((b + abs(a)) * (b * pi));
end

Wolfram

code[a_, b_, angle_] := N[(N[(0.011111111111111112 * angle), $MachinePrecision] * N[(N[(b + N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 50.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 50.5%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   6. lower-*.f64 50.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

   12. difference-of-squares-rev N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   13. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   14. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   15. lift--.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   16. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   17. lower-*.f64 54.4%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   18. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   19. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   20. lower-+.f64 54.4%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

6. Applied rewrites 54.4%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   4. associate-*l* N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)}\right) \]

   5. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)} \cdot \pi\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

   10. associate-*l* N/A

       \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

   12. lower-*.f64 54.4%

       \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right)\right) \]

8. Applied rewrites 54.4%

   \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)} \]

9. Taylor expanded in a around 0

   \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \color{blue}{\pi}\right)\right) \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \]

    2. lower-PI.f64 37.3%

       \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \pi\right)\right) \]

11. Applied rewrites 37.3%

    \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b \cdot \color{blue}{\pi}\right)\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025179 
(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)))))