ab-angle->ABCF B

Percentage Accurate: 54.1% → 68.2%

Time: 4.0s

Alternatives: 18

Speedup: 3.3×

• 66.9% 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))))
  (* (* (* 2 (- (pow b 2) (pow a 2))) (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), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(2 * N[(N[Power[b, 2], $MachinePrecision] - N[Power[a, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 54.1% 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))))
  (* (* (* 2 (- (pow b 2) (pow a 2))) (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), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(2 * N[(N[Power[b, 2], $MachinePrecision] - N[Power[a, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := b - \left|a\right|\\ t_1 := \frac{-1}{2} \cdot \pi - \frac{-1}{2} \cdot \pi\\ t_2 := \left(\frac{1}{90} \cdot \left|angle\right|\right) \cdot \pi + \pi\\ t_3 := \left|a\right| + b\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 109999999999999994813382992798917661634827554997553248915876435581884342024512697523016464072704:\\ \;\;\;\;t\_3 \cdot \left(t\_0 \cdot \sin \left(\left(\left({\pi}^{\frac{2}{3}} \cdot \left|angle\right|\right) \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\sin \left(\frac{t\_1 - t\_2}{2}\right) \cdot \cos \left(\frac{t\_1 + t\_2}{2}\right)\right)\right) \cdot \left(\left(2 \cdot t\_3\right) \cdot t\_0\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (- b (fabs a)))
       (t_1 (- (* -1/2 PI) (* -1/2 PI)))
       (t_2 (+ (* (* 1/90 (fabs angle)) PI) PI))
       (t_3 (+ (fabs a) b)))
  (*
   (copysign 1 angle)
   (if (<=
        (fabs angle)
        109999999999999994813382992798917661634827554997553248915876435581884342024512697523016464072704)
     (*
      t_3
      (*
       t_0
       (sin (* (* (* (pow PI 2/3) (fabs angle)) (cbrt PI)) 1/90))))
     (/
      (*
       (* 2 (* (sin (/ (- t_1 t_2) 2)) (cos (/ (+ t_1 t_2) 2))))
       (* (* 2 t_3) t_0))
      2)))))

C

double code(double a, double b, double angle) {
	double t_0 = b - fabs(a);
	double t_1 = (-0.5 * ((double) M_PI)) - (-0.5 * ((double) M_PI));
	double t_2 = ((0.011111111111111112 * fabs(angle)) * ((double) M_PI)) + ((double) M_PI);
	double t_3 = fabs(a) + b;
	double tmp;
	if (fabs(angle) <= 1.1e+95) {
		tmp = t_3 * (t_0 * sin((((pow(((double) M_PI), 0.6666666666666666) * fabs(angle)) * cbrt(((double) M_PI))) * 0.011111111111111112)));
	} else {
		tmp = ((2.0 * (sin(((t_1 - t_2) / 2.0)) * cos(((t_1 + t_2) / 2.0)))) * ((2.0 * t_3) * t_0)) / 2.0;
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = b - Math.abs(a);
	double t_1 = (-0.5 * Math.PI) - (-0.5 * Math.PI);
	double t_2 = ((0.011111111111111112 * Math.abs(angle)) * Math.PI) + Math.PI;
	double t_3 = Math.abs(a) + b;
	double tmp;
	if (Math.abs(angle) <= 1.1e+95) {
		tmp = t_3 * (t_0 * Math.sin((((Math.pow(Math.PI, 0.6666666666666666) * Math.abs(angle)) * Math.cbrt(Math.PI)) * 0.011111111111111112)));
	} else {
		tmp = ((2.0 * (Math.sin(((t_1 - t_2) / 2.0)) * Math.cos(((t_1 + t_2) / 2.0)))) * ((2.0 * t_3) * t_0)) / 2.0;
	}
	return Math.copySign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(b - abs(a))
	t_1 = Float64(Float64(-0.5 * pi) - Float64(-0.5 * pi))
	t_2 = Float64(Float64(Float64(0.011111111111111112 * abs(angle)) * pi) + pi)
	t_3 = Float64(abs(a) + b)
	tmp = 0.0
	if (abs(angle) <= 1.1e+95)
		tmp = Float64(t_3 * Float64(t_0 * sin(Float64(Float64(Float64((pi ^ 0.6666666666666666) * abs(angle)) * cbrt(pi)) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(sin(Float64(Float64(t_1 - t_2) / 2.0)) * cos(Float64(Float64(t_1 + t_2) / 2.0)))) * Float64(Float64(2.0 * t_3) * t_0)) / 2.0);
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(b - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1/2 * Pi), $MachinePrecision] - N[(-1/2 * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(1/90 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] + Pi), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[a], $MachinePrecision] + b), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 109999999999999994813382992798917661634827554997553248915876435581884342024512697523016464072704], N[(t$95$3 * N[(t$95$0 * N[Sin[N[(N[(N[(N[Power[Pi, 2/3], $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision] * 1/90), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2 * N[(N[Sin[N[(N[(t$95$1 - t$95$2), $MachinePrecision] / 2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / 2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2 * t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / 2), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.0999999999999999e95

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \frac{1}{90}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \frac{1}{90}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left(angle \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\color{blue}{\pi}}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      9. cbrt-unprod N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \color{blue}{\sqrt[3]{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)} \cdot \pi}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      11. lift-PI.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \sqrt[3]{\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      12. lower-cbrt.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \sqrt[3]{\color{blue}{\pi} \cdot \mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      14. lift-PI.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \sqrt[3]{\pi \cdot \color{blue}{\pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \sqrt[3]{\color{blue}{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \frac{1}{90}\right)\right) \]

      17. lower-cbrt.f64 68.6%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right) \cdot \frac{1}{90}\right)\right) \]

   5. Applied rewrites 68.6%

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

      1. lift-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \color{blue}{\sqrt[3]{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \sqrt[3]{\color{blue}{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

      3. cbrt-prod N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)}\right) \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

      4. lift-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot \sqrt[3]{\pi}\right)\right) \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

      5. lift-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right) \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

      6. pow2 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}}\right) \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

      7. lower-pow.f64 67.7%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}}\right) \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

   7. Applied rewrites 67.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left(angle \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)} \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot angle\right)} \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

      3. lower-*.f64 67.7%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot angle\right)} \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot angle\right) \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

      5. lift-cbrt.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left({\color{blue}{\left(\sqrt[3]{\pi}\right)}}^{2} \cdot angle\right) \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

      6. pow-cbrt N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(\color{blue}{{\pi}^{\left(\frac{2}{3}\right)}} \cdot angle\right) \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(\color{blue}{{\pi}^{\left(\frac{2}{3}\right)}} \cdot angle\right) \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

      8. metadata-eval 68.6%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot angle\right) \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

   9. Applied rewrites 68.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot angle\right)} \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

   if 1.0999999999999999e95 < angle

   1. Initial program 54.1%

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

   3. Applied rewrites 27.1%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(0 - \frac{1}{2} \cdot \pi\right) - \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90} - \pi \cdot \frac{-1}{2}\right)\right) \cdot \left(\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}{2}} \]

   5. Applied rewrites 30.0%

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

   7. Applied rewrites 28.8%

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\sin \left(\frac{\left(\frac{-1}{2} \cdot \pi - \frac{-1}{2} \cdot \pi\right) - \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi + \pi\right)}{2}\right) \cdot \cos \left(\frac{\left(\frac{-1}{2} \cdot \pi - \frac{-1}{2} \cdot \pi\right) + \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi + \pi\right)}{2}\right)\right)\right)} \cdot \left(\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 67.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := b - \left|a\right|\\ t_1 := \left|a\right| + b\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 122000000000000006913119480225967436871409575765767305984376187617974147724947271645625313132544:\\ \;\;\;\;t\_1 \cdot \left(t\_0 \cdot \sin \left(\left(\left|angle\right| \cdot \pi\right) \cdot \frac{1}{90}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sin \pi - \sin \left(\left(\frac{1}{90} \cdot \left|angle\right|\right) \cdot \pi + \pi\right)\right) \cdot \left(t\_1 \cdot 2\right)\right) \cdot \left(t\_0 \cdot \frac{1}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (- b (fabs a))) (t_1 (+ (fabs a) b)))
  (*
   (copysign 1 angle)
   (if (<=
        (fabs angle)
        122000000000000006913119480225967436871409575765767305984376187617974147724947271645625313132544)
     (* t_1 (* t_0 (sin (* (* (fabs angle) PI) 1/90))))
     (*
      (*
       (- (sin PI) (sin (+ (* (* 1/90 (fabs angle)) PI) PI)))
       (* t_1 2))
      (* t_0 1/2))))))

C

double code(double a, double b, double angle) {
	double t_0 = b - fabs(a);
	double t_1 = fabs(a) + b;
	double tmp;
	if (fabs(angle) <= 1.22e+95) {
		tmp = t_1 * (t_0 * sin(((fabs(angle) * ((double) M_PI)) * 0.011111111111111112)));
	} else {
		tmp = ((sin(((double) M_PI)) - sin((((0.011111111111111112 * fabs(angle)) * ((double) M_PI)) + ((double) M_PI)))) * (t_1 * 2.0)) * (t_0 * 0.5);
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = b - Math.abs(a);
	double t_1 = Math.abs(a) + b;
	double tmp;
	if (Math.abs(angle) <= 1.22e+95) {
		tmp = t_1 * (t_0 * Math.sin(((Math.abs(angle) * Math.PI) * 0.011111111111111112)));
	} else {
		tmp = ((Math.sin(Math.PI) - Math.sin((((0.011111111111111112 * Math.abs(angle)) * Math.PI) + Math.PI))) * (t_1 * 2.0)) * (t_0 * 0.5);
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = b - math.fabs(a)
	t_1 = math.fabs(a) + b
	tmp = 0
	if math.fabs(angle) <= 1.22e+95:
		tmp = t_1 * (t_0 * math.sin(((math.fabs(angle) * math.pi) * 0.011111111111111112)))
	else:
		tmp = ((math.sin(math.pi) - math.sin((((0.011111111111111112 * math.fabs(angle)) * math.pi) + math.pi))) * (t_1 * 2.0)) * (t_0 * 0.5)
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(b - abs(a))
	t_1 = Float64(abs(a) + b)
	tmp = 0.0
	if (abs(angle) <= 1.22e+95)
		tmp = Float64(t_1 * Float64(t_0 * sin(Float64(Float64(abs(angle) * pi) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(Float64(sin(pi) - sin(Float64(Float64(Float64(0.011111111111111112 * abs(angle)) * pi) + pi))) * Float64(t_1 * 2.0)) * Float64(t_0 * 0.5));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = b - abs(a);
	t_1 = abs(a) + b;
	tmp = 0.0;
	if (abs(angle) <= 1.22e+95)
		tmp = t_1 * (t_0 * sin(((abs(angle) * pi) * 0.011111111111111112)));
	else
		tmp = ((sin(pi) - sin((((0.011111111111111112 * abs(angle)) * pi) + pi))) * (t_1 * 2.0)) * (t_0 * 0.5);
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(b - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[a], $MachinePrecision] + b), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 122000000000000006913119480225967436871409575765767305984376187617974147724947271645625313132544], N[(t$95$1 * N[(t$95$0 * N[Sin[N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * 1/90), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[Pi], $MachinePrecision] - N[Sin[N[(N[(N[(1/90 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * 2), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * 1/2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.2200000000000001e95

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   if 1.2200000000000001e95 < angle

   1. Initial program 54.1%

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

   3. Applied rewrites 27.1%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(0 - \frac{1}{2} \cdot \pi\right) - \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90} - \pi \cdot \frac{-1}{2}\right)\right) \cdot \left(\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}{2}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\cos \left(0 - \frac{1}{2} \cdot \pi\right) - \cos \left(\color{blue}{\left(angle \cdot \pi\right) \cdot \frac{1}{90}} - \pi \cdot \frac{-1}{2}\right)\right) \cdot \left(\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}{2} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\cos \left(0 - \frac{1}{2} \cdot \pi\right) - \cos \left(\color{blue}{\frac{1}{90} \cdot \left(angle \cdot \pi\right)} - \pi \cdot \frac{-1}{2}\right)\right) \cdot \left(\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\cos \left(0 - \frac{1}{2} \cdot \pi\right) - \cos \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \pi\right)} - \pi \cdot \frac{-1}{2}\right)\right) \cdot \left(\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}{2} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(\cos \left(0 - \frac{1}{2} \cdot \pi\right) - \cos \left(\color{blue}{\left(\frac{1}{90} \cdot angle\right) \cdot \pi} - \pi \cdot \frac{-1}{2}\right)\right) \cdot \left(\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(\cos \left(0 - \frac{1}{2} \cdot \pi\right) - \cos \left(\color{blue}{\left(\frac{1}{90} \cdot angle\right) \cdot \pi} - \pi \cdot \frac{-1}{2}\right)\right) \cdot \left(\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}{2} \]

      6. lower-*.f64 26.9%

         \[\leadsto \frac{\left(\cos \left(0 - \frac{1}{2} \cdot \pi\right) - \cos \left(\color{blue}{\left(\frac{1}{90} \cdot angle\right)} \cdot \pi - \pi \cdot \frac{-1}{2}\right)\right) \cdot \left(\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}{2} \]

   5. Applied rewrites 26.9%

      \[\leadsto \frac{\left(\cos \left(0 - \frac{1}{2} \cdot \pi\right) - \cos \left(\color{blue}{\left(\frac{1}{90} \cdot angle\right) \cdot \pi} - \pi \cdot \frac{-1}{2}\right)\right) \cdot \left(\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}{2} \]

   7. Applied rewrites 27.2%

      \[\leadsto \color{blue}{\left(\left(\sin \pi - \sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi + \pi\right)\right) \cdot \left(\left(a + b\right) \cdot 2\right)\right) \cdot \left(\left(b - a\right) \cdot \frac{1}{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := b - \left|a\right|\\ t_1 := \left(\frac{1}{90} \cdot \left|angle\right|\right) \cdot \pi + \pi\\ t_2 := \left|a\right| + b\\ t_3 := \frac{-1}{2} \cdot \pi - \frac{-1}{2} \cdot \pi\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 122000000000000006913119480225967436871409575765767305984376187617974147724947271645625313132544:\\ \;\;\;\;t\_2 \cdot \left(t\_0 \cdot \sin \left(\left(\left|angle\right| \cdot \pi\right) \cdot \frac{1}{90}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\sin \left(\frac{t\_3 - t\_1}{2}\right) \cdot \cos \left(\frac{t\_3 + t\_1}{2}\right)\right)\right) \cdot \left(\left(2 \cdot t\_2\right) \cdot t\_0\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (- b (fabs a)))
       (t_1 (+ (* (* 1/90 (fabs angle)) PI) PI))
       (t_2 (+ (fabs a) b))
       (t_3 (- (* -1/2 PI) (* -1/2 PI))))
  (*
   (copysign 1 angle)
   (if (<=
        (fabs angle)
        122000000000000006913119480225967436871409575765767305984376187617974147724947271645625313132544)
     (* t_2 (* t_0 (sin (* (* (fabs angle) PI) 1/90))))
     (/
      (*
       (* 2 (* (sin (/ (- t_3 t_1) 2)) (cos (/ (+ t_3 t_1) 2))))
       (* (* 2 t_2) t_0))
      2)))))

C

double code(double a, double b, double angle) {
	double t_0 = b - fabs(a);
	double t_1 = ((0.011111111111111112 * fabs(angle)) * ((double) M_PI)) + ((double) M_PI);
	double t_2 = fabs(a) + b;
	double t_3 = (-0.5 * ((double) M_PI)) - (-0.5 * ((double) M_PI));
	double tmp;
	if (fabs(angle) <= 1.22e+95) {
		tmp = t_2 * (t_0 * sin(((fabs(angle) * ((double) M_PI)) * 0.011111111111111112)));
	} else {
		tmp = ((2.0 * (sin(((t_3 - t_1) / 2.0)) * cos(((t_3 + t_1) / 2.0)))) * ((2.0 * t_2) * t_0)) / 2.0;
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = b - Math.abs(a);
	double t_1 = ((0.011111111111111112 * Math.abs(angle)) * Math.PI) + Math.PI;
	double t_2 = Math.abs(a) + b;
	double t_3 = (-0.5 * Math.PI) - (-0.5 * Math.PI);
	double tmp;
	if (Math.abs(angle) <= 1.22e+95) {
		tmp = t_2 * (t_0 * Math.sin(((Math.abs(angle) * Math.PI) * 0.011111111111111112)));
	} else {
		tmp = ((2.0 * (Math.sin(((t_3 - t_1) / 2.0)) * Math.cos(((t_3 + t_1) / 2.0)))) * ((2.0 * t_2) * t_0)) / 2.0;
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = b - math.fabs(a)
	t_1 = ((0.011111111111111112 * math.fabs(angle)) * math.pi) + math.pi
	t_2 = math.fabs(a) + b
	t_3 = (-0.5 * math.pi) - (-0.5 * math.pi)
	tmp = 0
	if math.fabs(angle) <= 1.22e+95:
		tmp = t_2 * (t_0 * math.sin(((math.fabs(angle) * math.pi) * 0.011111111111111112)))
	else:
		tmp = ((2.0 * (math.sin(((t_3 - t_1) / 2.0)) * math.cos(((t_3 + t_1) / 2.0)))) * ((2.0 * t_2) * t_0)) / 2.0
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(b - abs(a))
	t_1 = Float64(Float64(Float64(0.011111111111111112 * abs(angle)) * pi) + pi)
	t_2 = Float64(abs(a) + b)
	t_3 = Float64(Float64(-0.5 * pi) - Float64(-0.5 * pi))
	tmp = 0.0
	if (abs(angle) <= 1.22e+95)
		tmp = Float64(t_2 * Float64(t_0 * sin(Float64(Float64(abs(angle) * pi) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(sin(Float64(Float64(t_3 - t_1) / 2.0)) * cos(Float64(Float64(t_3 + t_1) / 2.0)))) * Float64(Float64(2.0 * t_2) * t_0)) / 2.0);
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = b - abs(a);
	t_1 = ((0.011111111111111112 * abs(angle)) * pi) + pi;
	t_2 = abs(a) + b;
	t_3 = (-0.5 * pi) - (-0.5 * pi);
	tmp = 0.0;
	if (abs(angle) <= 1.22e+95)
		tmp = t_2 * (t_0 * sin(((abs(angle) * pi) * 0.011111111111111112)));
	else
		tmp = ((2.0 * (sin(((t_3 - t_1) / 2.0)) * cos(((t_3 + t_1) / 2.0)))) * ((2.0 * t_2) * t_0)) / 2.0;
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(b - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1/90 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] + Pi), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[a], $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-1/2 * Pi), $MachinePrecision] - N[(-1/2 * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 122000000000000006913119480225967436871409575765767305984376187617974147724947271645625313132544], N[(t$95$2 * N[(t$95$0 * N[Sin[N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * 1/90), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2 * N[(N[Sin[N[(N[(t$95$3 - t$95$1), $MachinePrecision] / 2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$3 + t$95$1), $MachinePrecision] / 2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2 * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / 2), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.2200000000000001e95

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   if 1.2200000000000001e95 < angle

   1. Initial program 54.1%

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

   3. Applied rewrites 27.1%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(0 - \frac{1}{2} \cdot \pi\right) - \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90} - \pi \cdot \frac{-1}{2}\right)\right) \cdot \left(\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}{2}} \]

   5. Applied rewrites 30.0%

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

   7. Applied rewrites 28.8%

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\sin \left(\frac{\left(\frac{-1}{2} \cdot \pi - \frac{-1}{2} \cdot \pi\right) - \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi + \pi\right)}{2}\right) \cdot \cos \left(\frac{\left(\frac{-1}{2} \cdot \pi - \frac{-1}{2} \cdot \pi\right) + \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi + \pi\right)}{2}\right)\right)\right)} \cdot \left(\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 67.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|a\right| \leq 549999999999999956603258195768086461654723523156372230507658473319512693915163480152926077793794477926070024461497827820572263214299772340635469176401651039300337687939013232287576626105770303862730433435560529687113014705508278468436865391708340224:\\ \;\;\;\;\left(\left|a\right| + b\right) \cdot \left(\left(b - \left|a\right|\right) \cdot \sin \left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{90} \cdot \left({\left(\left|a\right|\right)}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (if (<=
     (fabs a)
     549999999999999956603258195768086461654723523156372230507658473319512693915163480152926077793794477926070024461497827820572263214299772340635469176401651039300337687939013232287576626105770303862730433435560529687113014705508278468436865391708340224)
  (* (+ (fabs a) b) (* (- b (fabs a)) (sin (* (* 1/90 PI) angle))))
  (*
   (* -1/90 (* (pow (fabs a) 2) (* angle PI)))
   (cos (/ (* angle PI) 180)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(a) <= 5.5e+248) {
		tmp = (fabs(a) + b) * ((b - fabs(a)) * sin(((0.011111111111111112 * ((double) M_PI)) * angle)));
	} else {
		tmp = (-0.011111111111111112 * (pow(fabs(a), 2.0) * (angle * ((double) M_PI)))) * cos(((angle * ((double) M_PI)) / 180.0));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(a) <= 5.5e+248) {
		tmp = (Math.abs(a) + b) * ((b - Math.abs(a)) * Math.sin(((0.011111111111111112 * Math.PI) * angle)));
	} else {
		tmp = (-0.011111111111111112 * (Math.pow(Math.abs(a), 2.0) * (angle * Math.PI))) * Math.cos(((angle * Math.PI) / 180.0));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(a) <= 5.5e+248:
		tmp = (math.fabs(a) + b) * ((b - math.fabs(a)) * math.sin(((0.011111111111111112 * math.pi) * angle)))
	else:
		tmp = (-0.011111111111111112 * (math.pow(math.fabs(a), 2.0) * (angle * math.pi))) * math.cos(((angle * math.pi) / 180.0))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(a) <= 5.5e+248)
		tmp = Float64(Float64(abs(a) + b) * Float64(Float64(b - abs(a)) * sin(Float64(Float64(0.011111111111111112 * pi) * angle))));
	else
		tmp = Float64(Float64(-0.011111111111111112 * Float64((abs(a) ^ 2.0) * Float64(angle * pi))) * cos(Float64(Float64(angle * pi) / 180.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(a) <= 5.5e+248)
		tmp = (abs(a) + b) * ((b - abs(a)) * sin(((0.011111111111111112 * pi) * angle)));
	else
		tmp = (-0.011111111111111112 * ((abs(a) ^ 2.0) * (angle * pi))) * cos(((angle * pi) / 180.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[a], $MachinePrecision], 549999999999999956603258195768086461654723523156372230507658473319512693915163480152926077793794477926070024461497827820572263214299772340635469176401651039300337687939013232287576626105770303862730433435560529687113014705508278468436865391708340224], N[(N[(N[Abs[a], $MachinePrecision] + b), $MachinePrecision] * N[(N[(b - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(1/90 * Pi), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1/90 * N[(N[Power[N[Abs[a], $MachinePrecision], 2], $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle * Pi), $MachinePrecision] / 180), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.4999999999999996e248

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)} \]
   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. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 68.7%

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

   5. Applied rewrites 68.7%

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

   if 5.4999999999999996e248 < a

   1. Initial program 54.1%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 35.9%

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lower-PI.f64 34.9%

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Applied rewrites 34.9%

      \[\leadsto \left(\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\frac{angle}{180}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{\pi \cdot angle}{180}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\frac{\color{blue}{angle \cdot \pi}}{180}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\frac{\color{blue}{angle \cdot \pi}}{180}\right) \]

      6. lower-/.f64 35.1%

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)} \]

   9. Applied rewrites 35.1%

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 67.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| + b\\ t_1 := b - \left|a\right|\\ \mathbf{if}\;\left|a\right| \leq 294999999999999979715892151964708824211181979871733743190477131889411011138639576703702293249515424096579239073331843163821161847545577565121560611077028454405154333307039309919961205010578929076059141599838785369121243529216:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \sin \left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot t\_1\right) \cdot t\_0\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (+ (fabs a) b)) (t_1 (- b (fabs a))))
  (if (<=
       (fabs a)
       294999999999999979715892151964708824211181979871733743190477131889411011138639576703702293249515424096579239073331843163821161847545577565121560611077028454405154333307039309919961205010578929076059141599838785369121243529216)
    (* t_0 (* t_1 (sin (* (* 1/90 PI) angle))))
    (* 1/90 (* (* (* angle PI) t_1) t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(a) + b;
	double t_1 = b - fabs(a);
	double tmp;
	if (fabs(a) <= 2.95e+224) {
		tmp = t_0 * (t_1 * sin(((0.011111111111111112 * ((double) M_PI)) * angle)));
	} else {
		tmp = 0.011111111111111112 * (((angle * ((double) M_PI)) * t_1) * t_0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(a) + b;
	double t_1 = b - Math.abs(a);
	double tmp;
	if (Math.abs(a) <= 2.95e+224) {
		tmp = t_0 * (t_1 * Math.sin(((0.011111111111111112 * Math.PI) * angle)));
	} else {
		tmp = 0.011111111111111112 * (((angle * Math.PI) * t_1) * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(a) + b
	t_1 = b - math.fabs(a)
	tmp = 0
	if math.fabs(a) <= 2.95e+224:
		tmp = t_0 * (t_1 * math.sin(((0.011111111111111112 * math.pi) * angle)))
	else:
		tmp = 0.011111111111111112 * (((angle * math.pi) * t_1) * t_0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(a) + b)
	t_1 = Float64(b - abs(a))
	tmp = 0.0
	if (abs(a) <= 2.95e+224)
		tmp = Float64(t_0 * Float64(t_1 * sin(Float64(Float64(0.011111111111111112 * pi) * angle))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(angle * pi) * t_1) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(a) + b;
	t_1 = b - abs(a);
	tmp = 0.0;
	if (abs(a) <= 2.95e+224)
		tmp = t_0 * (t_1 * sin(((0.011111111111111112 * pi) * angle)));
	else
		tmp = 0.011111111111111112 * (((angle * pi) * t_1) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$1 = N[(b - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 294999999999999979715892151964708824211181979871733743190477131889411011138639576703702293249515424096579239073331843163821161847545577565121560611077028454405154333307039309919961205010578929076059141599838785369121243529216], N[(t$95$0 * N[(t$95$1 * N[Sin[N[(N[(1/90 * Pi), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1/90 * N[(N[(N[(angle * Pi), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.9499999999999998e224

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)} \]
   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. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 68.7%

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

   5. Applied rewrites 68.7%

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

   if 2.9499999999999998e224 < a

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 62.9%

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

   8. Applied rewrites 62.9%

      \[\leadsto \frac{1}{90} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(a + b\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 66.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right) \leq \frac{7435084542388915}{37175422711944576569951562453747514003281505041484861476394296482898516429010109338629207862403908764744968094754824326261364631138622642694770764527559865644574271011186496848567160245857443421755979894558785930151693717671947002203927473508716452075301127636353597820594133720367104}:\\ \;\;\;\;-1 \cdot \left(\left(t\_0 \cdot \left|a\right|\right) \cdot \left|a\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left|a\right| + \left|b\right|\right) \cdot \left(\left|b\right| \cdot t\_0\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (sin (* 1/90 (* angle PI)))))
  (if (<=
       (* 2 (- (pow (fabs b) 2) (pow (fabs a) 2)))
       7435084542388915/37175422711944576569951562453747514003281505041484861476394296482898516429010109338629207862403908764744968094754824326261364631138622642694770764527559865644574271011186496848567160245857443421755979894558785930151693717671947002203927473508716452075301127636353597820594133720367104)
    (* -1 (* (* t_0 (fabs a)) (fabs a)))
    (* (+ (fabs a) (fabs b)) (* (fabs b) t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = sin((0.011111111111111112 * (angle * ((double) M_PI))));
	double tmp;
	if ((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) <= 2e-268) {
		tmp = -1.0 * ((t_0 * fabs(a)) * fabs(a));
	} else {
		tmp = (fabs(a) + fabs(b)) * (fabs(b) * t_0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sin((0.011111111111111112 * (angle * Math.PI)));
	double tmp;
	if ((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) <= 2e-268) {
		tmp = -1.0 * ((t_0 * Math.abs(a)) * Math.abs(a));
	} else {
		tmp = (Math.abs(a) + Math.abs(b)) * (Math.abs(b) * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.sin((0.011111111111111112 * (angle * math.pi)))
	tmp = 0
	if (2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) <= 2e-268:
		tmp = -1.0 * ((t_0 * math.fabs(a)) * math.fabs(a))
	else:
		tmp = (math.fabs(a) + math.fabs(b)) * (math.fabs(b) * t_0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = sin(Float64(0.011111111111111112 * Float64(angle * pi)))
	tmp = 0.0
	if (Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) <= 2e-268)
		tmp = Float64(-1.0 * Float64(Float64(t_0 * abs(a)) * abs(a)));
	else
		tmp = Float64(Float64(abs(a) + abs(b)) * Float64(abs(b) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = sin((0.011111111111111112 * (angle * pi)));
	tmp = 0.0;
	if ((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) <= 2e-268)
		tmp = -1.0 * ((t_0 * abs(a)) * abs(a));
	else
		tmp = (abs(a) + abs(b)) * (abs(b) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(1/90 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(2 * N[(N[Power[N[Abs[b], $MachinePrecision], 2], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 7435084542388915/37175422711944576569951562453747514003281505041484861476394296482898516429010109338629207862403908764744968094754824326261364631138622642694770764527559865644574271011186496848567160245857443421755979894558785930151693717671947002203927473508716452075301127636353597820594133720367104], N[(-1 * N[(N[(t$95$0 * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.9999999999999999e-268

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 35.9%

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

   6. Applied rewrites 35.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 41.2%

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

   8. Applied rewrites 41.2%

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

   if 1.9999999999999999e-268 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   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 42.6%

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

   6. Applied rewrites 42.6%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| + \left|b\right|\\ \mathbf{if}\;2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right) \leq \frac{-1129605583483287}{22592111669665739975592870737637022906810831294812620197467215446901550642889587999246991367961839975767182923986271972624986374927027127581012424707895568851446368731861728964581056579941628221790058875830676242925879296}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(\left|b\right| - \left|a\right|\right)\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left|b\right| \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (+ (fabs a) (fabs b))))
  (if (<=
       (* 2 (- (pow (fabs b) 2) (pow (fabs a) 2)))
       -1129605583483287/22592111669665739975592870737637022906810831294812620197467215446901550642889587999246991367961839975767182923986271972624986374927027127581012424707895568851446368731861728964581056579941628221790058875830676242925879296)
    (* 1/90 (* (* (* angle PI) (- (fabs b) (fabs a))) t_0))
    (* t_0 (* (fabs b) (sin (* 1/90 (* angle PI))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(a) + fabs(b);
	double tmp;
	if ((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) <= -5e-206) {
		tmp = 0.011111111111111112 * (((angle * ((double) M_PI)) * (fabs(b) - fabs(a))) * t_0);
	} else {
		tmp = t_0 * (fabs(b) * sin((0.011111111111111112 * (angle * ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(a) + Math.abs(b);
	double tmp;
	if ((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) <= -5e-206) {
		tmp = 0.011111111111111112 * (((angle * Math.PI) * (Math.abs(b) - Math.abs(a))) * t_0);
	} else {
		tmp = t_0 * (Math.abs(b) * Math.sin((0.011111111111111112 * (angle * Math.PI))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(a) + math.fabs(b)
	tmp = 0
	if (2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) <= -5e-206:
		tmp = 0.011111111111111112 * (((angle * math.pi) * (math.fabs(b) - math.fabs(a))) * t_0)
	else:
		tmp = t_0 * (math.fabs(b) * math.sin((0.011111111111111112 * (angle * math.pi))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(a) + abs(b))
	tmp = 0.0
	if (Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) <= -5e-206)
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(angle * pi) * Float64(abs(b) - abs(a))) * t_0));
	else
		tmp = Float64(t_0 * Float64(abs(b) * sin(Float64(0.011111111111111112 * Float64(angle * pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(a) + abs(b);
	tmp = 0.0;
	if ((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) <= -5e-206)
		tmp = 0.011111111111111112 * (((angle * pi) * (abs(b) - abs(a))) * t_0);
	else
		tmp = t_0 * (abs(b) * sin((0.011111111111111112 * (angle * pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2 * N[(N[Power[N[Abs[b], $MachinePrecision], 2], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1129605583483287/22592111669665739975592870737637022906810831294812620197467215446901550642889587999246991367961839975767182923986271972624986374927027127581012424707895568851446368731861728964581056579941628221790058875830676242925879296], N[(1/90 * N[(N[(N[(angle * Pi), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] * N[Sin[N[(1/90 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -5e-206

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 62.9%

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

   8. Applied rewrites 62.9%

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

   if -5e-206 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   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 42.6%

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

   6. Applied rewrites 42.6%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 64.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| + \left|b\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq \frac{5072854620270127}{9223372036854775808}:\\ \;\;\;\;\left(\left(\left|angle\right| \cdot \left(t\_0 \cdot \pi\right)\right) \cdot \left(\left|b\right| - \left|a\right|\right)\right) \cdot \frac{1}{90}\\ \mathbf{elif}\;\left|angle\right| \leq 4000000000000000193386768462214636230113579383562057023488:\\ \;\;\;\;\left(\left(-\left|a\right|\right) \cdot \left|a\right|\right) \cdot \sin \left(\frac{1}{90} \cdot \left(\left|angle\right| \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(t\_0 \cdot \left(-1 \cdot \left|a\right|\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (+ (fabs a) (fabs b))))
  (*
   (copysign 1 angle)
   (if (<= (fabs angle) 5072854620270127/9223372036854775808)
     (* (* (* (fabs angle) (* t_0 PI)) (- (fabs b) (fabs a))) 1/90)
     (if (<=
          (fabs angle)
          4000000000000000193386768462214636230113579383562057023488)
       (*
        (* (- (fabs a)) (fabs a))
        (sin (* 1/90 (* (fabs angle) PI))))
       (* 1/90 (* (fabs angle) (* PI (* t_0 (* -1 (fabs a)))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(a) + fabs(b);
	double tmp;
	if (fabs(angle) <= 0.00055) {
		tmp = ((fabs(angle) * (t_0 * ((double) M_PI))) * (fabs(b) - fabs(a))) * 0.011111111111111112;
	} else if (fabs(angle) <= 4e+57) {
		tmp = (-fabs(a) * fabs(a)) * sin((0.011111111111111112 * (fabs(angle) * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (fabs(angle) * (((double) M_PI) * (t_0 * (-1.0 * fabs(a)))));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(a) + Math.abs(b);
	double tmp;
	if (Math.abs(angle) <= 0.00055) {
		tmp = ((Math.abs(angle) * (t_0 * Math.PI)) * (Math.abs(b) - Math.abs(a))) * 0.011111111111111112;
	} else if (Math.abs(angle) <= 4e+57) {
		tmp = (-Math.abs(a) * Math.abs(a)) * Math.sin((0.011111111111111112 * (Math.abs(angle) * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (Math.abs(angle) * (Math.PI * (t_0 * (-1.0 * Math.abs(a)))));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(a) + math.fabs(b)
	tmp = 0
	if math.fabs(angle) <= 0.00055:
		tmp = ((math.fabs(angle) * (t_0 * math.pi)) * (math.fabs(b) - math.fabs(a))) * 0.011111111111111112
	elif math.fabs(angle) <= 4e+57:
		tmp = (-math.fabs(a) * math.fabs(a)) * math.sin((0.011111111111111112 * (math.fabs(angle) * math.pi)))
	else:
		tmp = 0.011111111111111112 * (math.fabs(angle) * (math.pi * (t_0 * (-1.0 * math.fabs(a)))))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(a) + abs(b))
	tmp = 0.0
	if (abs(angle) <= 0.00055)
		tmp = Float64(Float64(Float64(abs(angle) * Float64(t_0 * pi)) * Float64(abs(b) - abs(a))) * 0.011111111111111112);
	elseif (abs(angle) <= 4e+57)
		tmp = Float64(Float64(Float64(-abs(a)) * abs(a)) * sin(Float64(0.011111111111111112 * Float64(abs(angle) * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * Float64(t_0 * Float64(-1.0 * abs(a))))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(a) + abs(b);
	tmp = 0.0;
	if (abs(angle) <= 0.00055)
		tmp = ((abs(angle) * (t_0 * pi)) * (abs(b) - abs(a))) * 0.011111111111111112;
	elseif (abs(angle) <= 4e+57)
		tmp = (-abs(a) * abs(a)) * sin((0.011111111111111112 * (abs(angle) * pi)));
	else
		tmp = 0.011111111111111112 * (abs(angle) * (pi * (t_0 * (-1.0 * abs(a)))));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 5072854620270127/9223372036854775808], N[(N[(N[(N[Abs[angle], $MachinePrecision] * N[(t$95$0 * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1/90), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 4000000000000000193386768462214636230113579383562057023488], N[(N[((-N[Abs[a], $MachinePrecision]) * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(1/90 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1/90 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[(t$95$0 * N[(-1 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if angle < 5.5000000000000003e-4

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.2%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 62.9%

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

   8. Applied rewrites 62.9%

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

   if 5.5000000000000003e-4 < angle < 4.0000000000000002e57

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 35.9%

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

   6. Applied rewrites 35.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 35.9%

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

   8. Applied rewrites 35.9%

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

   if 4.0000000000000002e57 < angle

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 37.0%

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

   9. Applied rewrites 37.0%

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(-1 \cdot \color{blue}{a}\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 64.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq \frac{-1129605583483287}{22592111669665739975592870737637022906810831294812620197467215446901550642889587999246991367961839975767182923986271972624986374927027127581012424707895568851446368731861728964581056579941628221790058875830676242925879296}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (if (<=
     (* 2 (- (pow b 2) (pow a 2)))
     -1129605583483287/22592111669665739975592870737637022906810831294812620197467215446901550642889587999246991367961839975767182923986271972624986374927027127581012424707895568851446368731861728964581056579941628221790058875830676242925879296)
  (* 1/90 (* (* (* angle PI) (- b a)) (+ a b)))
  (* b (* (- b a) (sin (* (* angle PI) 1/90))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= -5e-206) {
		tmp = 0.011111111111111112 * (((angle * ((double) M_PI)) * (b - a)) * (a + b));
	} else {
		tmp = b * ((b - a) * sin(((angle * ((double) M_PI)) * 0.011111111111111112)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= -5e-206) {
		tmp = 0.011111111111111112 * (((angle * Math.PI) * (b - a)) * (a + b));
	} else {
		tmp = b * ((b - a) * Math.sin(((angle * Math.PI) * 0.011111111111111112)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= -5e-206:
		tmp = 0.011111111111111112 * (((angle * math.pi) * (b - a)) * (a + b))
	else:
		tmp = b * ((b - a) * math.sin(((angle * math.pi) * 0.011111111111111112)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= -5e-206)
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(angle * pi) * Float64(b - a)) * Float64(a + b)));
	else
		tmp = Float64(b * Float64(Float64(b - a) * sin(Float64(Float64(angle * pi) * 0.011111111111111112))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= -5e-206)
		tmp = 0.011111111111111112 * (((angle * pi) * (b - a)) * (a + b));
	else
		tmp = b * ((b - a) * sin(((angle * pi) * 0.011111111111111112)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(2 * N[(N[Power[b, 2], $MachinePrecision] - N[Power[a, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1129605583483287/22592111669665739975592870737637022906810831294812620197467215446901550642889587999246991367961839975767182923986271972624986374927027127581012424707895568851446368731861728964581056579941628221790058875830676242925879296], N[(1/90 * N[(N[(N[(angle * Pi), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 1/90), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -5e-206

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 62.9%

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

   8. Applied rewrites 62.9%

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

   if -5e-206 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 42.6%

      \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| + \left|b\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq \frac{5718490662849961}{9223372036854775808}:\\ \;\;\;\;\left(\left(\left|angle\right| \cdot \left(t\_0 \cdot \pi\right)\right) \cdot \left(\left|b\right| - \left|a\right|\right)\right) \cdot \frac{1}{90}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(t\_0 \cdot \left(-1 \cdot \left|a\right|\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (+ (fabs a) (fabs b))))
  (*
   (copysign 1 angle)
   (if (<= (fabs angle) 5718490662849961/9223372036854775808)
     (* (* (* (fabs angle) (* t_0 PI)) (- (fabs b) (fabs a))) 1/90)
     (* 1/90 (* (fabs angle) (* PI (* t_0 (* -1 (fabs a))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(a) + fabs(b);
	double tmp;
	if (fabs(angle) <= 0.00062) {
		tmp = ((fabs(angle) * (t_0 * ((double) M_PI))) * (fabs(b) - fabs(a))) * 0.011111111111111112;
	} else {
		tmp = 0.011111111111111112 * (fabs(angle) * (((double) M_PI) * (t_0 * (-1.0 * fabs(a)))));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(a) + Math.abs(b);
	double tmp;
	if (Math.abs(angle) <= 0.00062) {
		tmp = ((Math.abs(angle) * (t_0 * Math.PI)) * (Math.abs(b) - Math.abs(a))) * 0.011111111111111112;
	} else {
		tmp = 0.011111111111111112 * (Math.abs(angle) * (Math.PI * (t_0 * (-1.0 * Math.abs(a)))));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(a) + math.fabs(b)
	tmp = 0
	if math.fabs(angle) <= 0.00062:
		tmp = ((math.fabs(angle) * (t_0 * math.pi)) * (math.fabs(b) - math.fabs(a))) * 0.011111111111111112
	else:
		tmp = 0.011111111111111112 * (math.fabs(angle) * (math.pi * (t_0 * (-1.0 * math.fabs(a)))))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(a) + abs(b))
	tmp = 0.0
	if (abs(angle) <= 0.00062)
		tmp = Float64(Float64(Float64(abs(angle) * Float64(t_0 * pi)) * Float64(abs(b) - abs(a))) * 0.011111111111111112);
	else
		tmp = Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * Float64(t_0 * Float64(-1.0 * abs(a))))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(a) + abs(b);
	tmp = 0.0;
	if (abs(angle) <= 0.00062)
		tmp = ((abs(angle) * (t_0 * pi)) * (abs(b) - abs(a))) * 0.011111111111111112;
	else
		tmp = 0.011111111111111112 * (abs(angle) * (pi * (t_0 * (-1.0 * abs(a)))));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 5718490662849961/9223372036854775808], N[(N[(N[(N[Abs[angle], $MachinePrecision] * N[(t$95$0 * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1/90), $MachinePrecision], N[(1/90 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[(t$95$0 * N[(-1 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 6.2e-4

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.2%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 62.9%

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

   8. Applied rewrites 62.9%

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

   if 6.2e-4 < angle

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 37.0%

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

   9. Applied rewrites 37.0%

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(-1 \cdot \color{blue}{a}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 64.1% accurate, 3.2× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq \frac{4789048565205903}{95780971304118053647396689196894323976171195136475136}:\\ \;\;\;\;\left(\left(\left|angle\right| \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{1}{90}\\ \mathbf{else}:\\ \;\;\;\;\left(\left|angle\right| \cdot \pi\right) \cdot \left(\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \frac{1}{90}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (*
 (copysign 1 angle)
 (if (<=
      (fabs angle)
      4789048565205903/95780971304118053647396689196894323976171195136475136)
   (* (* (* (fabs angle) (* (+ a b) PI)) (- b a)) 1/90)
   (* (* (fabs angle) PI) (* (* (- b a) (+ a b)) 1/90)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 5e-38) {
		tmp = ((fabs(angle) * ((a + b) * ((double) M_PI))) * (b - a)) * 0.011111111111111112;
	} else {
		tmp = (fabs(angle) * ((double) M_PI)) * (((b - a) * (a + b)) * 0.011111111111111112);
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(angle) <= 5e-38) {
		tmp = ((Math.abs(angle) * ((a + b) * Math.PI)) * (b - a)) * 0.011111111111111112;
	} else {
		tmp = (Math.abs(angle) * Math.PI) * (((b - a) * (a + b)) * 0.011111111111111112);
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(angle) <= 5e-38:
		tmp = ((math.fabs(angle) * ((a + b) * math.pi)) * (b - a)) * 0.011111111111111112
	else:
		tmp = (math.fabs(angle) * math.pi) * (((b - a) * (a + b)) * 0.011111111111111112)
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 5e-38)
		tmp = Float64(Float64(Float64(abs(angle) * Float64(Float64(a + b) * pi)) * Float64(b - a)) * 0.011111111111111112);
	else
		tmp = Float64(Float64(abs(angle) * pi) * Float64(Float64(Float64(b - a) * Float64(a + b)) * 0.011111111111111112));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(angle) <= 5e-38)
		tmp = ((abs(angle) * ((a + b) * pi)) * (b - a)) * 0.011111111111111112;
	else
		tmp = (abs(angle) * pi) * (((b - a) * (a + b)) * 0.011111111111111112);
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 4789048565205903/95780971304118053647396689196894323976171195136475136], N[(N[(N[(N[Abs[angle], $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * 1/90), $MachinePrecision], N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * 1/90), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 5.0000000000000003e-38

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.2%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 62.9%

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

   8. Applied rewrites 62.9%

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

   if 5.0000000000000003e-38 < angle

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

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

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 54.2%

      \[\leadsto \left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \frac{1}{90}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.6% accurate, 3.2× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq \frac{1461501637330903}{1461501637330902918203684832716283019655932542976}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left(\left(\left|angle\right| \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{90} \cdot \left|angle\right|\right) \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (*
 (copysign 1 angle)
 (if (<=
      (fabs angle)
      1461501637330903/1461501637330902918203684832716283019655932542976)
   (* 1/90 (* (* (* (fabs angle) PI) (- b a)) (+ a b)))
   (* (* (* 1/90 (fabs angle)) PI) (* (- b a) (+ a b))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 1e-33) {
		tmp = 0.011111111111111112 * (((fabs(angle) * ((double) M_PI)) * (b - a)) * (a + b));
	} else {
		tmp = ((0.011111111111111112 * fabs(angle)) * ((double) M_PI)) * ((b - a) * (a + b));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(angle) <= 1e-33) {
		tmp = 0.011111111111111112 * (((Math.abs(angle) * Math.PI) * (b - a)) * (a + b));
	} else {
		tmp = ((0.011111111111111112 * Math.abs(angle)) * Math.PI) * ((b - a) * (a + b));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(angle) <= 1e-33:
		tmp = 0.011111111111111112 * (((math.fabs(angle) * math.pi) * (b - a)) * (a + b))
	else:
		tmp = ((0.011111111111111112 * math.fabs(angle)) * math.pi) * ((b - a) * (a + b))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 1e-33)
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(abs(angle) * pi) * Float64(b - a)) * Float64(a + b)));
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * abs(angle)) * pi) * Float64(Float64(b - a) * Float64(a + b)));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(angle) <= 1e-33)
		tmp = 0.011111111111111112 * (((abs(angle) * pi) * (b - a)) * (a + b));
	else
		tmp = ((0.011111111111111112 * abs(angle)) * pi) * ((b - a) * (a + b));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 1461501637330903/1461501637330902918203684832716283019655932542976], N[(1/90 * N[(N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1/90 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.0000000000000001e-33

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 62.9%

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

   8. Applied rewrites 62.9%

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

   if 1.0000000000000001e-33 < angle

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift--.f64 N/A

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

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

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. unpow2 N/A

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

   8. Applied rewrites 54.2%

      \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 62.1% accurate, 3.2× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq \frac{1461501637330903}{1461501637330902918203684832716283019655932542976}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left(\left(\left|angle\right| \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (*
 (copysign 1 angle)
 (if (<=
      (fabs angle)
      1461501637330903/1461501637330902918203684832716283019655932542976)
   (* 1/90 (* (* (* (fabs angle) PI) (- b a)) (+ a b)))
   (* 1/90 (* (fabs angle) (* PI (* (+ a b) (- b a))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 1e-33) {
		tmp = 0.011111111111111112 * (((fabs(angle) * ((double) M_PI)) * (b - a)) * (a + b));
	} else {
		tmp = 0.011111111111111112 * (fabs(angle) * (((double) M_PI) * ((a + b) * (b - a))));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(angle) <= 1e-33) {
		tmp = 0.011111111111111112 * (((Math.abs(angle) * Math.PI) * (b - a)) * (a + b));
	} else {
		tmp = 0.011111111111111112 * (Math.abs(angle) * (Math.PI * ((a + b) * (b - a))));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(angle) <= 1e-33:
		tmp = 0.011111111111111112 * (((math.fabs(angle) * math.pi) * (b - a)) * (a + b))
	else:
		tmp = 0.011111111111111112 * (math.fabs(angle) * (math.pi * ((a + b) * (b - a))))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 1e-33)
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(abs(angle) * pi) * Float64(b - a)) * Float64(a + b)));
	else
		tmp = Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * Float64(Float64(a + b) * Float64(b - a)))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(angle) <= 1e-33)
		tmp = 0.011111111111111112 * (((abs(angle) * pi) * (b - a)) * (a + b));
	else
		tmp = 0.011111111111111112 * (abs(angle) * (pi * ((a + b) * (b - a))));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 1461501637330903/1461501637330902918203684832716283019655932542976], N[(1/90 * N[(N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1/90 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.0000000000000001e-33

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 62.9%

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

   8. Applied rewrites 62.9%

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

   if 1.0000000000000001e-33 < angle

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 58.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ \mathbf{if}\;2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right) \leq \frac{6490371073168535}{1298074214633706907132624082305024}:\\ \;\;\;\;\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\left|a\right| + \left|b\right|\right) \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left(\left|b\right| \cdot \pi\right) \cdot \left(t\_0 \cdot angle\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (- (fabs b) (fabs a))))
  (if (<=
       (* 2 (- (pow (fabs b) 2) (pow (fabs a) 2)))
       6490371073168535/1298074214633706907132624082305024)
    (* 1/90 (* angle (* PI (* (+ (fabs a) (fabs b)) t_0))))
    (* 1/90 (* (* (fabs b) PI) (* t_0 angle))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double tmp;
	if ((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) <= 5e-18) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * ((fabs(a) + fabs(b)) * t_0)));
	} else {
		tmp = 0.011111111111111112 * ((fabs(b) * ((double) M_PI)) * (t_0 * angle));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - Math.abs(a);
	double tmp;
	if ((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) <= 5e-18) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * ((Math.abs(a) + Math.abs(b)) * t_0)));
	} else {
		tmp = 0.011111111111111112 * ((Math.abs(b) * Math.PI) * (t_0 * angle));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	tmp = 0
	if (2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) <= 5e-18:
		tmp = 0.011111111111111112 * (angle * (math.pi * ((math.fabs(a) + math.fabs(b)) * t_0)))
	else:
		tmp = 0.011111111111111112 * ((math.fabs(b) * math.pi) * (t_0 * angle))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	tmp = 0.0
	if (Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) <= 5e-18)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(abs(a) + abs(b)) * t_0))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(abs(b) * pi) * Float64(t_0 * angle)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	tmp = 0.0;
	if ((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) <= 5e-18)
		tmp = 0.011111111111111112 * (angle * (pi * ((abs(a) + abs(b)) * t_0)));
	else
		tmp = 0.011111111111111112 * ((abs(b) * pi) * (t_0 * angle));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2 * N[(N[Power[N[Abs[b], $MachinePrecision], 2], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 6490371073168535/1298074214633706907132624082305024], N[(1/90 * N[(angle * N[(Pi * N[(N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1/90 * N[(N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision] * N[(t$95$0 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 5.0000000000000004e-18

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

   if 5.0000000000000004e-18 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.7%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 41.7%

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

   11. Applied rewrites 41.7%

       \[\leadsto \frac{1}{90} \cdot \left(\left(b \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot angle\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 43.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ t_1 := \pi \cdot \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right)\right) \cdot \sin t\_1\right) \cdot \cos t\_1 \leq 49999999999999996681683364986231121055509847158923091289463001947809936825071710129649256226662527266508888537465191395528952846213699856588865536:\\ \;\;\;\;\left(\frac{1}{90} \cdot \left(\left(\pi \cdot t\_0\right) \cdot \left|b\right|\right)\right) \cdot \left|angle\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left(\left|b\right| \cdot \pi\right) \cdot \left(t\_0 \cdot \left|angle\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (- (fabs b) (fabs a))) (t_1 (* PI (/ (fabs angle) 180))))
  (*
   (copysign 1 angle)
   (if (<=
        (*
         (* (* 2 (- (pow (fabs b) 2) (pow (fabs a) 2))) (sin t_1))
         (cos t_1))
        49999999999999996681683364986231121055509847158923091289463001947809936825071710129649256226662527266508888537465191395528952846213699856588865536)
     (* (* 1/90 (* (* PI t_0) (fabs b))) (fabs angle))
     (* 1/90 (* (* (fabs b) PI) (* t_0 (fabs angle))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double t_1 = ((double) M_PI) * (fabs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) * sin(t_1)) * cos(t_1)) <= 5e+145) {
		tmp = (0.011111111111111112 * ((((double) M_PI) * t_0) * fabs(b))) * fabs(angle);
	} else {
		tmp = 0.011111111111111112 * ((fabs(b) * ((double) M_PI)) * (t_0 * 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) - Math.abs(a);
	double t_1 = Math.PI * (Math.abs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) * Math.sin(t_1)) * Math.cos(t_1)) <= 5e+145) {
		tmp = (0.011111111111111112 * ((Math.PI * t_0) * Math.abs(b))) * Math.abs(angle);
	} else {
		tmp = 0.011111111111111112 * ((Math.abs(b) * Math.PI) * (t_0 * Math.abs(angle)));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	t_1 = math.pi * (math.fabs(angle) / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) * math.sin(t_1)) * math.cos(t_1)) <= 5e+145:
		tmp = (0.011111111111111112 * ((math.pi * t_0) * math.fabs(b))) * math.fabs(angle)
	else:
		tmp = 0.011111111111111112 * ((math.fabs(b) * math.pi) * (t_0 * math.fabs(angle)))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	t_1 = Float64(pi * Float64(abs(angle) / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 5e+145)
		tmp = Float64(Float64(0.011111111111111112 * Float64(Float64(pi * t_0) * abs(b))) * abs(angle));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(abs(b) * pi) * Float64(t_0 * abs(angle))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	t_1 = pi * (abs(angle) / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 5e+145)
		tmp = (0.011111111111111112 * ((pi * t_0) * abs(b))) * abs(angle);
	else
		tmp = 0.011111111111111112 * ((abs(b) * pi) * (t_0 * 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] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(2 * N[(N[Power[N[Abs[b], $MachinePrecision], 2], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 49999999999999996681683364986231121055509847158923091289463001947809936825071710129649256226662527266508888537465191395528952846213699856588865536], N[(N[(1/90 * N[(N[(Pi * t$95$0), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision], N[(1/90 * N[(N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision] * N[(t$95$0 * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 4.9999999999999997e145

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 37.7%

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 37.7%

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

   11. Applied rewrites 37.7%

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

   if 4.9999999999999997e145 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.7%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 41.7%

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

   11. Applied rewrites 41.7%

       \[\leadsto \frac{1}{90} \cdot \left(\left(b \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot angle\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 43.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ t_1 := \pi \cdot \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right)\right) \cdot \sin t\_1\right) \cdot \cos t\_1 \leq \frac{6518515124270355}{32592575621351777380295131014550050576823494298654980010178247189670100796213387298934358016}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left(\left(\pi \cdot t\_0\right) \cdot \left|b\right|\right) \cdot \left|angle\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left(\left|b\right| \cdot \pi\right) \cdot \left(t\_0 \cdot \left|angle\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (- (fabs b) (fabs a))) (t_1 (* PI (/ (fabs angle) 180))))
  (*
   (copysign 1 angle)
   (if (<=
        (*
         (* (* 2 (- (pow (fabs b) 2) (pow (fabs a) 2))) (sin t_1))
         (cos t_1))
        6518515124270355/32592575621351777380295131014550050576823494298654980010178247189670100796213387298934358016)
     (* 1/90 (* (* (* PI t_0) (fabs b)) (fabs angle)))
     (* 1/90 (* (* (fabs b) PI) (* t_0 (fabs angle))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double t_1 = ((double) M_PI) * (fabs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) * sin(t_1)) * cos(t_1)) <= 2e-76) {
		tmp = 0.011111111111111112 * (((((double) M_PI) * t_0) * fabs(b)) * fabs(angle));
	} else {
		tmp = 0.011111111111111112 * ((fabs(b) * ((double) M_PI)) * (t_0 * 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) - Math.abs(a);
	double t_1 = Math.PI * (Math.abs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) * Math.sin(t_1)) * Math.cos(t_1)) <= 2e-76) {
		tmp = 0.011111111111111112 * (((Math.PI * t_0) * Math.abs(b)) * Math.abs(angle));
	} else {
		tmp = 0.011111111111111112 * ((Math.abs(b) * Math.PI) * (t_0 * Math.abs(angle)));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	t_1 = math.pi * (math.fabs(angle) / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) * math.sin(t_1)) * math.cos(t_1)) <= 2e-76:
		tmp = 0.011111111111111112 * (((math.pi * t_0) * math.fabs(b)) * math.fabs(angle))
	else:
		tmp = 0.011111111111111112 * ((math.fabs(b) * math.pi) * (t_0 * math.fabs(angle)))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	t_1 = Float64(pi * Float64(abs(angle) / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 2e-76)
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(pi * t_0) * abs(b)) * abs(angle)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(abs(b) * pi) * Float64(t_0 * abs(angle))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	t_1 = pi * (abs(angle) / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 2e-76)
		tmp = 0.011111111111111112 * (((pi * t_0) * abs(b)) * abs(angle));
	else
		tmp = 0.011111111111111112 * ((abs(b) * pi) * (t_0 * 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] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(2 * N[(N[Power[N[Abs[b], $MachinePrecision], 2], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 6518515124270355/32592575621351777380295131014550050576823494298654980010178247189670100796213387298934358016], N[(1/90 * N[(N[(N[(Pi * t$95$0), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1/90 * N[(N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision] * N[(t$95$0 * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 1.9999999999999999e-76

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.7%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 37.7%

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 37.7%

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

   11. Applied rewrites 37.7%

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

   if 1.9999999999999999e-76 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.7%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 41.7%

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

   11. Applied rewrites 41.7%

       \[\leadsto \frac{1}{90} \cdot \left(\left(b \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot angle\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 43.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ t_1 := \pi \cdot \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right)\right) \cdot \sin t\_1\right) \cdot \cos t\_1 \leq \frac{6518515124270355}{32592575621351777380295131014550050576823494298654980010178247189670100796213387298934358016}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(\left|b\right| \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left(\left|b\right| \cdot \pi\right) \cdot \left(t\_0 \cdot \left|angle\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (- (fabs b) (fabs a))) (t_1 (* PI (/ (fabs angle) 180))))
  (*
   (copysign 1 angle)
   (if (<=
        (*
         (* (* 2 (- (pow (fabs b) 2) (pow (fabs a) 2))) (sin t_1))
         (cos t_1))
        6518515124270355/32592575621351777380295131014550050576823494298654980010178247189670100796213387298934358016)
     (* 1/90 (* (fabs angle) (* PI (* (fabs b) t_0))))
     (* 1/90 (* (* (fabs b) PI) (* t_0 (fabs angle))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double t_1 = ((double) M_PI) * (fabs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) * sin(t_1)) * cos(t_1)) <= 2e-76) {
		tmp = 0.011111111111111112 * (fabs(angle) * (((double) M_PI) * (fabs(b) * t_0)));
	} else {
		tmp = 0.011111111111111112 * ((fabs(b) * ((double) M_PI)) * (t_0 * 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) - Math.abs(a);
	double t_1 = Math.PI * (Math.abs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) * Math.sin(t_1)) * Math.cos(t_1)) <= 2e-76) {
		tmp = 0.011111111111111112 * (Math.abs(angle) * (Math.PI * (Math.abs(b) * t_0)));
	} else {
		tmp = 0.011111111111111112 * ((Math.abs(b) * Math.PI) * (t_0 * Math.abs(angle)));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	t_1 = math.pi * (math.fabs(angle) / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) * math.sin(t_1)) * math.cos(t_1)) <= 2e-76:
		tmp = 0.011111111111111112 * (math.fabs(angle) * (math.pi * (math.fabs(b) * t_0)))
	else:
		tmp = 0.011111111111111112 * ((math.fabs(b) * math.pi) * (t_0 * math.fabs(angle)))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	t_1 = Float64(pi * Float64(abs(angle) / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 2e-76)
		tmp = Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * Float64(abs(b) * t_0))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(abs(b) * pi) * Float64(t_0 * abs(angle))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	t_1 = pi * (abs(angle) / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 2e-76)
		tmp = 0.011111111111111112 * (abs(angle) * (pi * (abs(b) * t_0)));
	else
		tmp = 0.011111111111111112 * ((abs(b) * pi) * (t_0 * 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] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(2 * N[(N[Power[N[Abs[b], $MachinePrecision], 2], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 6518515124270355/32592575621351777380295131014550050576823494298654980010178247189670100796213387298934358016], N[(1/90 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[(N[Abs[b], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1/90 * N[(N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision] * N[(t$95$0 * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 1.9999999999999999e-76

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.7%

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

   if 1.9999999999999999e-76 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 54.2%

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.7%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 41.7%

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

   11. Applied rewrites 41.7%

       \[\leadsto \frac{1}{90} \cdot \left(\left(b \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot angle\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 38.8% accurate, 15.1× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (* 1/90 (* angle (* PI (* (fabs b) (- (fabs b) (fabs a)))))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (((double) M_PI) * (fabs(b) * (fabs(b) - fabs(a)))));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (Math.PI * (Math.abs(b) * (Math.abs(b) - Math.abs(a)))));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * (angle * (math.pi * (math.fabs(b) * (math.fabs(b) - math.fabs(a)))))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(abs(b) * Float64(abs(b) - abs(a))))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (angle * (pi * (abs(b) * (abs(b) - abs(a)))));
end

Wolfram

code[a_, b_, angle_] := N[(1/90 * N[(angle * N[(Pi * N[(N[Abs[b], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. 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--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-cos.f64 N/A

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

3. Applied rewrites 68.9%

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

4. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 54.2%

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

6. Applied rewrites 54.2%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 37.7%

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

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2 (- (pow b 2) (pow a 2))) (sin (* PI (/ angle 180)))) (cos (* PI (/ angle 180)))))