ab-angle->ABCF B

Percentage Accurate: 54.1% → 67.8%

Time: 56.0s

Alternatives: 17

Speedup: 14.6×

• 66.7% 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: 67.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := a + \left|b\right|\\ t_1 := \left|b\right| - a\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq \frac{4113761393303015}{102844034832575377634685573909834406561420991602098741459288064}:\\ \;\;\;\;t\_1 \cdot \left(\frac{1}{90} \cdot \left(\left|angle\right| \cdot \left(\pi \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot t\_0\right) \cdot \sin \left(\left(\left|angle\right| \cdot \pi\right) \cdot \frac{1}{90}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (+ a (fabs b))) (t_1 (- (fabs b) a)))
  (*
   (copysign 1 angle)
   (if (<=
        (fabs angle)
        4113761393303015/102844034832575377634685573909834406561420991602098741459288064)
     (* t_1 (* 1/90 (* (fabs angle) (* PI t_0))))
     (* (* t_1 t_0) (sin (* (* (fabs angle) PI) 1/90)))))))

C

double code(double a, double b, double angle) {
	double t_0 = a + fabs(b);
	double t_1 = fabs(b) - a;
	double tmp;
	if (fabs(angle) <= 4e-47) {
		tmp = t_1 * (0.011111111111111112 * (fabs(angle) * (((double) M_PI) * t_0)));
	} else {
		tmp = (t_1 * t_0) * sin(((fabs(angle) * ((double) M_PI)) * 0.011111111111111112));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = a + Math.abs(b);
	double t_1 = Math.abs(b) - a;
	double tmp;
	if (Math.abs(angle) <= 4e-47) {
		tmp = t_1 * (0.011111111111111112 * (Math.abs(angle) * (Math.PI * t_0)));
	} else {
		tmp = (t_1 * t_0) * Math.sin(((Math.abs(angle) * Math.PI) * 0.011111111111111112));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = a + math.fabs(b)
	t_1 = math.fabs(b) - a
	tmp = 0
	if math.fabs(angle) <= 4e-47:
		tmp = t_1 * (0.011111111111111112 * (math.fabs(angle) * (math.pi * t_0)))
	else:
		tmp = (t_1 * t_0) * math.sin(((math.fabs(angle) * math.pi) * 0.011111111111111112))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(a + abs(b))
	t_1 = Float64(abs(b) - a)
	tmp = 0.0
	if (abs(angle) <= 4e-47)
		tmp = Float64(t_1 * Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * t_0))));
	else
		tmp = Float64(Float64(t_1 * t_0) * sin(Float64(Float64(abs(angle) * pi) * 0.011111111111111112)));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = a + abs(b);
	t_1 = abs(b) - a;
	tmp = 0.0;
	if (abs(angle) <= 4e-47)
		tmp = t_1 * (0.011111111111111112 * (abs(angle) * (pi * t_0)));
	else
		tmp = (t_1 * t_0) * sin(((abs(angle) * pi) * 0.011111111111111112));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 4113761393303015/102844034832575377634685573909834406561420991602098741459288064], N[(t$95$1 * N[(1/90 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * t$95$0), $MachinePrecision] * N[Sin[N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * 1/90), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.9999999999999999e-47

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right), b, b, a, a\right)} \]
   4. Step-by-step derivation

      1. lift-134-z0z1z2z3z4 N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 67.8%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 67.8%

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-+.f64 62.6%

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

   8. Applied rewrites 62.6%

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

   if 3.9999999999999999e-47 < 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 57.9%

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

Alternative 2: 67.8% accurate, 3.5× speedup

Math

\[\mathsf{134\_z0z1z2z3z4}\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right), b, b, a, a\right) \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (134-z0z1z2z3z4 (sin (* (* angle PI) 1/90)) b b a a))

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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

3. Applied rewrites 67.8%

   \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right), b, b, a, a\right)} \]
4. Add Preprocessing

Alternative 3: 67.7% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (* (+ a b) (* (- b a) (sin (* (* angle PI) 1/90)))))

C

double code(double a, double b, double angle) {
	return (a + b) * ((b - a) * sin(((angle * ((double) M_PI)) * 0.011111111111111112)));
}

Java

public static double code(double a, double b, double angle) {
	return (a + b) * ((b - a) * Math.sin(((angle * Math.PI) * 0.011111111111111112)));
}

Python

def code(a, b, angle):
	return (a + b) * ((b - a) * math.sin(((angle * math.pi) * 0.011111111111111112)))

Julia

function code(a, b, angle)
	return Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(Float64(angle * pi) * 0.011111111111111112))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a + b) * ((b - a) * sin(((angle * pi) * 0.011111111111111112)));
end

Wolfram

code[a_, b_, angle_] := N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 1/90), $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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

3. Applied rewrites 67.8%

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

Alternative 4: 67.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := a + \left|b\right|\\ t_1 := \left|b\right| - a\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq \frac{3946757204148067}{127314748520905380391777855525586135065716774604121015664758778084648831235208544136462336}:\\ \;\;\;\;t\_1 \cdot \left(\frac{1}{90} \cdot \left(\left|angle\right| \cdot \left(\pi \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot t\_0\right) \cdot \sin \left(\left(\frac{1}{90} \cdot \pi\right) \cdot \left|angle\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (+ a (fabs b))) (t_1 (- (fabs b) a)))
  (*
   (copysign 1 angle)
   (if (<=
        (fabs angle)
        3946757204148067/127314748520905380391777855525586135065716774604121015664758778084648831235208544136462336)
     (* t_1 (* 1/90 (* (fabs angle) (* PI t_0))))
     (* (* t_1 t_0) (sin (* (* 1/90 PI) (fabs angle))))))))

C

double code(double a, double b, double angle) {
	double t_0 = a + fabs(b);
	double t_1 = fabs(b) - a;
	double tmp;
	if (fabs(angle) <= 3.1e-74) {
		tmp = t_1 * (0.011111111111111112 * (fabs(angle) * (((double) M_PI) * t_0)));
	} else {
		tmp = (t_1 * t_0) * sin(((0.011111111111111112 * ((double) M_PI)) * fabs(angle)));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = a + Math.abs(b);
	double t_1 = Math.abs(b) - a;
	double tmp;
	if (Math.abs(angle) <= 3.1e-74) {
		tmp = t_1 * (0.011111111111111112 * (Math.abs(angle) * (Math.PI * t_0)));
	} else {
		tmp = (t_1 * t_0) * Math.sin(((0.011111111111111112 * Math.PI) * Math.abs(angle)));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = a + math.fabs(b)
	t_1 = math.fabs(b) - a
	tmp = 0
	if math.fabs(angle) <= 3.1e-74:
		tmp = t_1 * (0.011111111111111112 * (math.fabs(angle) * (math.pi * t_0)))
	else:
		tmp = (t_1 * t_0) * math.sin(((0.011111111111111112 * math.pi) * math.fabs(angle)))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(a + abs(b))
	t_1 = Float64(abs(b) - a)
	tmp = 0.0
	if (abs(angle) <= 3.1e-74)
		tmp = Float64(t_1 * Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * t_0))));
	else
		tmp = Float64(Float64(t_1 * t_0) * sin(Float64(Float64(0.011111111111111112 * pi) * abs(angle))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = a + abs(b);
	t_1 = abs(b) - a;
	tmp = 0.0;
	if (abs(angle) <= 3.1e-74)
		tmp = t_1 * (0.011111111111111112 * (abs(angle) * (pi * t_0)));
	else
		tmp = (t_1 * t_0) * sin(((0.011111111111111112 * pi) * abs(angle)));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 3946757204148067/127314748520905380391777855525586135065716774604121015664758778084648831235208544136462336], N[(t$95$1 * N[(1/90 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * t$95$0), $MachinePrecision] * N[Sin[N[(N[(1/90 * Pi), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.1000000000000002e-74

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right), b, b, a, a\right)} \]
   4. Step-by-step derivation

      1. lift-134-z0z1z2z3z4 N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 67.8%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 67.8%

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-+.f64 62.6%

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

   8. Applied rewrites 62.6%

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

   if 3.1000000000000002e-74 < 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 57.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 57.9%

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

   5. Applied rewrites 57.9%

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

Alternative 5: 67.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ t_1 := \sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right) \leq \frac{-8254602048994769}{16509204097989538948510618278641143953713978938628797566498772157083559454897651858575539246489287121708574843539285271215073361234458923039079343076966861778387083874968908880272859927388326282907007278381599637629625884148486304338699903086469889890298080653054312448}:\\ \;\;\;\;t\_0 \cdot \left(\left|a\right| \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left|b\right| \cdot t\_1\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (- (fabs b) (fabs a))) (t_1 (sin (* 1/90 (* angle PI)))))
  (if (<=
       (* 2 (- (pow (fabs b) 2) (pow (fabs a) 2)))
       -8254602048994769/16509204097989538948510618278641143953713978938628797566498772157083559454897651858575539246489287121708574843539285271215073361234458923039079343076966861778387083874968908880272859927388326282907007278381599637629625884148486304338699903086469889890298080653054312448)
    (* t_0 (* (fabs a) t_1))
    (* t_0 (* (fabs b) t_1)))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double t_1 = sin((0.011111111111111112 * (angle * ((double) M_PI))));
	double tmp;
	if ((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) <= -5e-253) {
		tmp = t_0 * (fabs(a) * t_1);
	} else {
		tmp = t_0 * (fabs(b) * t_1);
	}
	return 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.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))) <= -5e-253) {
		tmp = t_0 * (Math.abs(a) * t_1);
	} else {
		tmp = t_0 * (Math.abs(b) * t_1);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	t_1 = 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))) <= -5e-253:
		tmp = t_0 * (math.fabs(a) * t_1)
	else:
		tmp = t_0 * (math.fabs(b) * t_1)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	t_1 = sin(Float64(0.011111111111111112 * Float64(angle * pi)))
	tmp = 0.0
	if (Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) <= -5e-253)
		tmp = Float64(t_0 * Float64(abs(a) * t_1));
	else
		tmp = Float64(t_0 * Float64(abs(b) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	t_1 = sin((0.011111111111111112 * (angle * pi)));
	tmp = 0.0;
	if ((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) <= -5e-253)
		tmp = t_0 * (abs(a) * t_1);
	else
		tmp = t_0 * (abs(b) * t_1);
	end
	tmp_2 = 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[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], -8254602048994769/16509204097989538948510618278641143953713978938628797566498772157083559454897651858575539246489287121708574843539285271215073361234458923039079343076966861778387083874968908880272859927388326282907007278381599637629625884148486304338699903086469889890298080653054312448], N[(t$95$0 * N[(N[Abs[a], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.9999999999999997e-253

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right), b, b, a, a\right)} \]
   4. Step-by-step derivation

      1. lift-134-z0z1z2z3z4 N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 67.8%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 67.8%

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 42.6%

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

   8. Applied rewrites 42.6%

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

   if -4.9999999999999997e-253 < (*.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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right), b, b, a, a\right)} \]
   4. Step-by-step derivation

      1. lift-134-z0z1z2z3z4 N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 67.8%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 67.8%

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 42.0%

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

   8. Applied rewrites 42.0%

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

Alternative 6: 66.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left({\left(\left|b\right|\right)}^{2} - {a}^{2}\right)\\ t_1 := a + \left|b\right|\\ t_2 := \left|b\right| - a\\ \mathbf{if}\;t\_0 \leq \frac{-8254602048994769}{16509204097989538948510618278641143953713978938628797566498772157083559454897651858575539246489287121708574843539285271215073361234458923039079343076966861778387083874968908880272859927388326282907007278381599637629625884148486304338699903086469889890298080653054312448}:\\ \;\;\;\;t\_2 \cdot \left(a \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 49999999999999996962677625276823109300201436100586624765953857856616022815066169514216546287202538742184280590280810862893585968713180152651178994204334413874936507208410055205338551265812204529218599012742757995383198412754109163297745561348039749026730174593312862032038021904229799310374521740690718720:\\ \;\;\;\;\left(\left|b\right| \cdot t\_1\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot t\_1\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* 2 (- (pow (fabs b) 2) (pow a 2))))
       (t_1 (+ a (fabs b)))
       (t_2 (- (fabs b) a)))
  (if (<=
       t_0
       -8254602048994769/16509204097989538948510618278641143953713978938628797566498772157083559454897651858575539246489287121708574843539285271215073361234458923039079343076966861778387083874968908880272859927388326282907007278381599637629625884148486304338699903086469889890298080653054312448)
    (* t_2 (* a (sin (* 1/90 (* angle PI)))))
    (if (<=
         t_0
         49999999999999996962677625276823109300201436100586624765953857856616022815066169514216546287202538742184280590280810862893585968713180152651178994204334413874936507208410055205338551265812204529218599012742757995383198412754109163297745561348039749026730174593312862032038021904229799310374521740690718720)
      (* (* (fabs b) t_1) (sin (* (* angle PI) 1/90)))
      (* t_2 (* 1/90 (* angle (* PI t_1))))))))

C

double code(double a, double b, double angle) {
	double t_0 = 2.0 * (pow(fabs(b), 2.0) - pow(a, 2.0));
	double t_1 = a + fabs(b);
	double t_2 = fabs(b) - a;
	double tmp;
	if (t_0 <= -5e-253) {
		tmp = t_2 * (a * sin((0.011111111111111112 * (angle * ((double) M_PI)))));
	} else if (t_0 <= 5e+304) {
		tmp = (fabs(b) * t_1) * sin(((angle * ((double) M_PI)) * 0.011111111111111112));
	} else {
		tmp = t_2 * (0.011111111111111112 * (angle * (((double) M_PI) * t_1)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(a, 2.0));
	double t_1 = a + Math.abs(b);
	double t_2 = Math.abs(b) - a;
	double tmp;
	if (t_0 <= -5e-253) {
		tmp = t_2 * (a * Math.sin((0.011111111111111112 * (angle * Math.PI))));
	} else if (t_0 <= 5e+304) {
		tmp = (Math.abs(b) * t_1) * Math.sin(((angle * Math.PI) * 0.011111111111111112));
	} else {
		tmp = t_2 * (0.011111111111111112 * (angle * (Math.PI * t_1)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = 2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(a, 2.0))
	t_1 = a + math.fabs(b)
	t_2 = math.fabs(b) - a
	tmp = 0
	if t_0 <= -5e-253:
		tmp = t_2 * (a * math.sin((0.011111111111111112 * (angle * math.pi))))
	elif t_0 <= 5e+304:
		tmp = (math.fabs(b) * t_1) * math.sin(((angle * math.pi) * 0.011111111111111112))
	else:
		tmp = t_2 * (0.011111111111111112 * (angle * (math.pi * t_1)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(2.0 * Float64((abs(b) ^ 2.0) - (a ^ 2.0)))
	t_1 = Float64(a + abs(b))
	t_2 = Float64(abs(b) - a)
	tmp = 0.0
	if (t_0 <= -5e-253)
		tmp = Float64(t_2 * Float64(a * sin(Float64(0.011111111111111112 * Float64(angle * pi)))));
	elseif (t_0 <= 5e+304)
		tmp = Float64(Float64(abs(b) * t_1) * sin(Float64(Float64(angle * pi) * 0.011111111111111112)));
	else
		tmp = Float64(t_2 * Float64(0.011111111111111112 * Float64(angle * Float64(pi * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = 2.0 * ((abs(b) ^ 2.0) - (a ^ 2.0));
	t_1 = a + abs(b);
	t_2 = abs(b) - a;
	tmp = 0.0;
	if (t_0 <= -5e-253)
		tmp = t_2 * (a * sin((0.011111111111111112 * (angle * pi))));
	elseif (t_0 <= 5e+304)
		tmp = (abs(b) * t_1) * sin(((angle * pi) * 0.011111111111111112));
	else
		tmp = t_2 * (0.011111111111111112 * (angle * (pi * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(2 * N[(N[Power[N[Abs[b], $MachinePrecision], 2], $MachinePrecision] - N[Power[a, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision]}, If[LessEqual[t$95$0, -8254602048994769/16509204097989538948510618278641143953713978938628797566498772157083559454897651858575539246489287121708574843539285271215073361234458923039079343076966861778387083874968908880272859927388326282907007278381599637629625884148486304338699903086469889890298080653054312448], N[(t$95$2 * N[(a * N[Sin[N[(1/90 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 49999999999999996962677625276823109300201436100586624765953857856616022815066169514216546287202538742184280590280810862893585968713180152651178994204334413874936507208410055205338551265812204529218599012742757995383198412754109163297745561348039749026730174593312862032038021904229799310374521740690718720], N[(N[(N[Abs[b], $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 1/90), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(1/90 * N[(angle * N[(Pi * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.9999999999999997e-253

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right), b, b, a, a\right)} \]
   4. Step-by-step derivation

      1. lift-134-z0z1z2z3z4 N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 67.8%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 67.8%

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 42.6%

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

   8. Applied rewrites 42.6%

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

   if -4.9999999999999997e-253 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 4.9999999999999997e304

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 57.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 37.6%

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

   if 4.9999999999999997e304 < (*.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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right), b, b, a, a\right)} \]
   4. Step-by-step derivation

      1. lift-134-z0z1z2z3z4 N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 67.8%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 67.8%

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-+.f64 62.6%

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

   8. Applied rewrites 62.6%

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

Alternative 7: 64.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| + \left|b\right|\\ t_1 := \left|b\right| - \left|a\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 33000:\\ \;\;\;\;t\_1 \cdot \left(\frac{1}{90} \cdot \left(\left|angle\right| \cdot \left(\pi \cdot t\_0\right)\right)\right)\\ \mathbf{elif}\;\left|angle\right| \leq 450000000000000016958053089881796353851548744843094964733530545522697246262216880657861102563284036476374305723951859506724049072546807429311473890536442725603127680570225934753847055352537767547108754807894734836540518200341427621717685623513184497194228104494423834156650142359422500864:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(\frac{t\_0 \cdot t\_1}{t\_1} \cdot \left|b\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(t\_0 \cdot \left(-1 \cdot \left|a\right|\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (+ (fabs a) (fabs b))) (t_1 (- (fabs b) (fabs a))))
  (*
   (copysign 1 angle)
   (if (<= (fabs angle) 33000)
     (* t_1 (* 1/90 (* (fabs angle) (* PI t_0))))
     (if (<=
          (fabs angle)
          450000000000000016958053089881796353851548744843094964733530545522697246262216880657861102563284036476374305723951859506724049072546807429311473890536442725603127680570225934753847055352537767547108754807894734836540518200341427621717685623513184497194228104494423834156650142359422500864)
       (*
        1/90
        (* (fabs angle) (* PI (* (/ (* t_0 t_1) t_1) (fabs b)))))
       (* 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 t_1 = fabs(b) - fabs(a);
	double tmp;
	if (fabs(angle) <= 33000.0) {
		tmp = t_1 * (0.011111111111111112 * (fabs(angle) * (((double) M_PI) * t_0)));
	} else if (fabs(angle) <= 4.5e+287) {
		tmp = 0.011111111111111112 * (fabs(angle) * (((double) M_PI) * (((t_0 * t_1) / t_1) * fabs(b))));
	} 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 t_1 = Math.abs(b) - Math.abs(a);
	double tmp;
	if (Math.abs(angle) <= 33000.0) {
		tmp = t_1 * (0.011111111111111112 * (Math.abs(angle) * (Math.PI * t_0)));
	} else if (Math.abs(angle) <= 4.5e+287) {
		tmp = 0.011111111111111112 * (Math.abs(angle) * (Math.PI * (((t_0 * t_1) / t_1) * Math.abs(b))));
	} 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)
	t_1 = math.fabs(b) - math.fabs(a)
	tmp = 0
	if math.fabs(angle) <= 33000.0:
		tmp = t_1 * (0.011111111111111112 * (math.fabs(angle) * (math.pi * t_0)))
	elif math.fabs(angle) <= 4.5e+287:
		tmp = 0.011111111111111112 * (math.fabs(angle) * (math.pi * (((t_0 * t_1) / t_1) * math.fabs(b))))
	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))
	t_1 = Float64(abs(b) - abs(a))
	tmp = 0.0
	if (abs(angle) <= 33000.0)
		tmp = Float64(t_1 * Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * t_0))));
	elseif (abs(angle) <= 4.5e+287)
		tmp = Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * Float64(Float64(Float64(t_0 * t_1) / t_1) * abs(b)))));
	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);
	t_1 = abs(b) - abs(a);
	tmp = 0.0;
	if (abs(angle) <= 33000.0)
		tmp = t_1 * (0.011111111111111112 * (abs(angle) * (pi * t_0)));
	elseif (abs(angle) <= 4.5e+287)
		tmp = 0.011111111111111112 * (abs(angle) * (pi * (((t_0 * t_1) / t_1) * abs(b))));
	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]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 33000], N[(t$95$1 * N[(1/90 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 450000000000000016958053089881796353851548744843094964733530545522697246262216880657861102563284036476374305723951859506724049072546807429311473890536442725603127680570225934753847055352537767547108754807894734836540518200341427621717685623513184497194228104494423834156650142359422500864], N[(1/90 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $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 < 33000

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right), b, b, a, a\right)} \]
   4. Step-by-step derivation

      1. lift-134-z0z1z2z3z4 N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 67.8%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 67.8%

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-+.f64 62.6%

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

   8. Applied rewrites 62.6%

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

   if 33000 < angle < 4.5000000000000002e287

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 57.9%

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

   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

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

   9. Applied rewrites 37.2%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. flip-+ N/A

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

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

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

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

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

       6. lower-unsound-*.f32 N/A

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

       7. lower-*.f32 N/A

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

       8. sqr-neg-rev N/A

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

       9. lower-*.f32 N/A

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

       10. lower-unsound-*.f32 N/A

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

       11. lower-unsound-*.f32 N/A

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

       12. lower-*.f32 N/A

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

       13. unpow2 N/A

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

       14. lift-pow.f64 N/A

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

       15. lower-unsound-*.f32 N/A

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

       16. lower-*.f32 N/A

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

       17. sqr-neg-rev N/A

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

       18. unpow2 N/A

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

       19. lift-pow.f64 N/A

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

       20. lower-unsound--.f32 29.3%

           \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\frac{\left( \left( {b}^{2} \right)_{\text{binary64}} - \left( {a}^{2} \right)_{\text{binary64}} \right)_{\text{binary32}}}{b - a} \cdot b\right)\right)\right) \]

       21. lower--.f32 N/A

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

       22. lift-pow.f64 N/A

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

       23. unpow2 N/A

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

       24. lift-pow.f64 N/A

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

       25. unpow2 N/A

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

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

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

       27. +-commutative N/A

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

       28. lift-+.f64 N/A

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

       29. lift--.f64 N/A

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

       30. lower-*.f64 41.1%

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

   11. Applied rewrites 41.1%

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

   if 4.5000000000000002e287 < 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 57.9%

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

   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 37.3%

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

   9. Applied rewrites 37.3%

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

Alternative 8: 63.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| + \left|b\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 35000:\\ \;\;\;\;\left(\left|b\right| - \left|a\right|\right) \cdot \left(\frac{1}{90} \cdot \left(\left|angle\right| \cdot \left(\pi \cdot t\_0\right)\right)\right)\\ \mathbf{elif}\;\left|angle\right| \leq 450000000000000016958053089881796353851548744843094964733530545522697246262216880657861102563284036476374305723951859506724049072546807429311473890536442725603127680570225934753847055352537767547108754807894734836540518200341427621717685623513184497194228104494423834156650142359422500864:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(\left(\left(1 + \frac{\left|a\right|}{\left|b\right|}\right) \cdot \left|b\right|\right) \cdot \left|b\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(t\_0 \cdot \left(-1 \cdot \left|a\right|\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (+ (fabs a) (fabs b))))
  (*
   (copysign 1 angle)
   (if (<= (fabs angle) 35000)
     (* (- (fabs b) (fabs a)) (* 1/90 (* (fabs angle) (* PI t_0))))
     (if (<=
          (fabs angle)
          450000000000000016958053089881796353851548744843094964733530545522697246262216880657861102563284036476374305723951859506724049072546807429311473890536442725603127680570225934753847055352537767547108754807894734836540518200341427621717685623513184497194228104494423834156650142359422500864)
       (*
        1/90
        (*
         (fabs angle)
         (*
          PI
          (* (* (+ 1 (/ (fabs a) (fabs b))) (fabs b)) (fabs b)))))
       (* 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) <= 35000.0) {
		tmp = (fabs(b) - fabs(a)) * (0.011111111111111112 * (fabs(angle) * (((double) M_PI) * t_0)));
	} else if (fabs(angle) <= 4.5e+287) {
		tmp = 0.011111111111111112 * (fabs(angle) * (((double) M_PI) * (((1.0 + (fabs(a) / fabs(b))) * fabs(b)) * fabs(b))));
	} 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) <= 35000.0) {
		tmp = (Math.abs(b) - Math.abs(a)) * (0.011111111111111112 * (Math.abs(angle) * (Math.PI * t_0)));
	} else if (Math.abs(angle) <= 4.5e+287) {
		tmp = 0.011111111111111112 * (Math.abs(angle) * (Math.PI * (((1.0 + (Math.abs(a) / Math.abs(b))) * Math.abs(b)) * Math.abs(b))));
	} 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) <= 35000.0:
		tmp = (math.fabs(b) - math.fabs(a)) * (0.011111111111111112 * (math.fabs(angle) * (math.pi * t_0)))
	elif math.fabs(angle) <= 4.5e+287:
		tmp = 0.011111111111111112 * (math.fabs(angle) * (math.pi * (((1.0 + (math.fabs(a) / math.fabs(b))) * math.fabs(b)) * math.fabs(b))))
	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) <= 35000.0)
		tmp = Float64(Float64(abs(b) - abs(a)) * Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * t_0))));
	elseif (abs(angle) <= 4.5e+287)
		tmp = Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * Float64(Float64(Float64(1.0 + Float64(abs(a) / abs(b))) * abs(b)) * abs(b)))));
	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) <= 35000.0)
		tmp = (abs(b) - abs(a)) * (0.011111111111111112 * (abs(angle) * (pi * t_0)));
	elseif (abs(angle) <= 4.5e+287)
		tmp = 0.011111111111111112 * (abs(angle) * (pi * (((1.0 + (abs(a) / abs(b))) * abs(b)) * abs(b))));
	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], 35000], N[(N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(1/90 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 450000000000000016958053089881796353851548744843094964733530545522697246262216880657861102563284036476374305723951859506724049072546807429311473890536442725603127680570225934753847055352537767547108754807894734836540518200341427621717685623513184497194228104494423834156650142359422500864], N[(1/90 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[(N[(N[(1 + N[(N[Abs[a], $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $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 < 35000

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right), b, b, a, a\right)} \]
   4. Step-by-step derivation

      1. lift-134-z0z1z2z3z4 N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 67.8%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 67.8%

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-+.f64 62.6%

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

   8. Applied rewrites 62.6%

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

   if 35000 < angle < 4.5000000000000002e287

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 57.9%

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

   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

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

   9. Applied rewrites 37.2%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. sum-to-mult N/A

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

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

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

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

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

       6. lower-unsound-/.f64 40.2%

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

   11. Applied rewrites 40.2%

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

   if 4.5000000000000002e287 < 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 57.9%

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

   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 37.3%

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

   9. Applied rewrites 37.3%

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

Alternative 9: 62.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| + \left|b\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 44999999999999997619501552954440605329731224537770508792680074270517912479690686752281910390554624:\\ \;\;\;\;\left(\left|b\right| - \left|a\right|\right) \cdot \left(\frac{1}{90} \cdot \left(\left|angle\right| \cdot \left(\pi \cdot t\_0\right)\right)\right)\\ \mathbf{elif}\;\left|angle\right| \leq 450000000000000016958053089881796353851548744843094964733530545522697246262216880657861102563284036476374305723951859506724049072546807429311473890536442725603127680570225934753847055352537767547108754807894734836540518200341427621717685623513184497194228104494423834156650142359422500864:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left(\left(\left|angle\right| \cdot \pi\right) \cdot t\_0\right) \cdot \left|b\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{90} \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(t\_0 \cdot \left(-1 \cdot \left|a\right|\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (+ (fabs a) (fabs b))))
  (*
   (copysign 1 angle)
   (if (<=
        (fabs angle)
        44999999999999997619501552954440605329731224537770508792680074270517912479690686752281910390554624)
     (* (- (fabs b) (fabs a)) (* 1/90 (* (fabs angle) (* PI t_0))))
     (if (<=
          (fabs angle)
          450000000000000016958053089881796353851548744843094964733530545522697246262216880657861102563284036476374305723951859506724049072546807429311473890536442725603127680570225934753847055352537767547108754807894734836540518200341427621717685623513184497194228104494423834156650142359422500864)
       (* 1/90 (* (* (* (fabs angle) PI) t_0) (fabs b)))
       (* 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) <= 4.5e+97) {
		tmp = (fabs(b) - fabs(a)) * (0.011111111111111112 * (fabs(angle) * (((double) M_PI) * t_0)));
	} else if (fabs(angle) <= 4.5e+287) {
		tmp = 0.011111111111111112 * (((fabs(angle) * ((double) M_PI)) * t_0) * fabs(b));
	} 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) <= 4.5e+97) {
		tmp = (Math.abs(b) - Math.abs(a)) * (0.011111111111111112 * (Math.abs(angle) * (Math.PI * t_0)));
	} else if (Math.abs(angle) <= 4.5e+287) {
		tmp = 0.011111111111111112 * (((Math.abs(angle) * Math.PI) * t_0) * Math.abs(b));
	} 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) <= 4.5e+97:
		tmp = (math.fabs(b) - math.fabs(a)) * (0.011111111111111112 * (math.fabs(angle) * (math.pi * t_0)))
	elif math.fabs(angle) <= 4.5e+287:
		tmp = 0.011111111111111112 * (((math.fabs(angle) * math.pi) * t_0) * math.fabs(b))
	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) <= 4.5e+97)
		tmp = Float64(Float64(abs(b) - abs(a)) * Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * t_0))));
	elseif (abs(angle) <= 4.5e+287)
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(abs(angle) * pi) * t_0) * abs(b)));
	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) <= 4.5e+97)
		tmp = (abs(b) - abs(a)) * (0.011111111111111112 * (abs(angle) * (pi * t_0)));
	elseif (abs(angle) <= 4.5e+287)
		tmp = 0.011111111111111112 * (((abs(angle) * pi) * t_0) * abs(b));
	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], 44999999999999997619501552954440605329731224537770508792680074270517912479690686752281910390554624], N[(N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(1/90 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 450000000000000016958053089881796353851548744843094964733530545522697246262216880657861102563284036476374305723951859506724049072546807429311473890536442725603127680570225934753847055352537767547108754807894734836540518200341427621717685623513184497194228104494423834156650142359422500864], N[(1/90 * N[(N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Abs[b], $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 < 4.4999999999999998e97

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right), b, b, a, a\right)} \]
   4. Step-by-step derivation

      1. lift-134-z0z1z2z3z4 N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 67.8%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.8%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 67.8%

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-+.f64 62.6%

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

   8. Applied rewrites 62.6%

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

   if 4.4999999999999998e97 < angle < 4.5000000000000002e287

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 57.9%

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

   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

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

   9. Applied rewrites 37.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. lower-*.f64 41.2%

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

   11. Applied rewrites 41.2%

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

   if 4.5000000000000002e287 < 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 57.9%

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

   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 37.3%

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

   9. Applied rewrites 37.3%

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

Alternative 10: 62.6% accurate, 14.6× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (* (- (fabs b) a) (* 1/90 (* angle (* PI (+ a (fabs b)))))))

C

double code(double a, double b, double angle) {
	return (fabs(b) - a) * (0.011111111111111112 * (angle * (((double) M_PI) * (a + fabs(b)))));
}

Java

public static double code(double a, double b, double angle) {
	return (Math.abs(b) - a) * (0.011111111111111112 * (angle * (Math.PI * (a + Math.abs(b)))));
}

Python

def code(a, b, angle):
	return (math.fabs(b) - a) * (0.011111111111111112 * (angle * (math.pi * (a + math.fabs(b)))))

Julia

function code(a, b, angle)
	return Float64(Float64(abs(b) - a) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(a + abs(b))))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (abs(b) - a) * (0.011111111111111112 * (angle * (pi * (a + abs(b)))));
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision] * N[(1/90 * N[(angle * N[(Pi * N[(a + N[Abs[b], $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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

3. Applied rewrites 67.8%

   \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right), b, b, a, a\right)} \]
4. Step-by-step derivation

   1. lift-134-z0z1z2z3z4 N/A

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

   2. *-commutative N/A

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

   3. difference-of-squares N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. lift--.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 67.8%

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 67.8%

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 67.8%

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

5. Applied rewrites 67.8%

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

6. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-+.f64 62.6%

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

8. Applied rewrites 62.6%

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

Alternative 11: 62.5% accurate, 14.6× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (* 1/90 (* (* PI (+ a (fabs b))) (* (- (fabs b) a) angle))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((((double) M_PI) * (a + fabs(b))) * ((fabs(b) - a) * angle));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((Math.PI * (a + Math.abs(b))) * ((Math.abs(b) - a) * angle));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * ((math.pi * (a + math.fabs(b))) * ((math.fabs(b) - a) * angle))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(pi * Float64(a + abs(b))) * Float64(Float64(abs(b) - a) * angle)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * ((pi * (a + abs(b))) * ((abs(b) - a) * angle));
end

Wolfram

code[a_, b_, angle_] := N[(1/90 * N[(N[(Pi * N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision] * angle), $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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

3. Applied rewrites 57.9%

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

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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 62.5%

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

8. Applied rewrites 62.5%

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

Alternative 12: 56.6% accurate, 10.5× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| + \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 920000000000000012194172518790057045799306659821299225420975556855664091087607883247395862037184893378553456937101100189413504230300998018716177162909772320919725008294726605769539584:\\ \;\;\;\;\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(t\_0 \cdot \left(\left|b\right| - \left|a\right|\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot \frac{1}{90}\right) \cdot \left(\pi \cdot t\_0\right)\right) \cdot \left|b\right|\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (+ (fabs a) (fabs b))))
  (if (<=
       (fabs b)
       920000000000000012194172518790057045799306659821299225420975556855664091087607883247395862037184893378553456937101100189413504230300998018716177162909772320919725008294726605769539584)
    (* 1/90 (* angle (* PI (* t_0 (- (fabs b) (fabs a))))))
    (* (* (* angle 1/90) (* PI t_0)) (fabs b)))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(a) + fabs(b);
	double tmp;
	if (fabs(b) <= 9.2e+182) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (t_0 * (fabs(b) - fabs(a)))));
	} else {
		tmp = ((angle * 0.011111111111111112) * (((double) M_PI) * t_0)) * fabs(b);
	}
	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 (Math.abs(b) <= 9.2e+182) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (t_0 * (Math.abs(b) - Math.abs(a)))));
	} else {
		tmp = ((angle * 0.011111111111111112) * (Math.PI * t_0)) * Math.abs(b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(a) + math.fabs(b)
	tmp = 0
	if math.fabs(b) <= 9.2e+182:
		tmp = 0.011111111111111112 * (angle * (math.pi * (t_0 * (math.fabs(b) - math.fabs(a)))))
	else:
		tmp = ((angle * 0.011111111111111112) * (math.pi * t_0)) * math.fabs(b)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(a) + abs(b))
	tmp = 0.0
	if (abs(b) <= 9.2e+182)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(t_0 * Float64(abs(b) - abs(a))))));
	else
		tmp = Float64(Float64(Float64(angle * 0.011111111111111112) * Float64(pi * t_0)) * abs(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(a) + abs(b);
	tmp = 0.0;
	if (abs(b) <= 9.2e+182)
		tmp = 0.011111111111111112 * (angle * (pi * (t_0 * (abs(b) - abs(a)))));
	else
		tmp = ((angle * 0.011111111111111112) * (pi * t_0)) * abs(b);
	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[Abs[b], $MachinePrecision], 920000000000000012194172518790057045799306659821299225420975556855664091087607883247395862037184893378553456937101100189413504230300998018716177162909772320919725008294726605769539584], N[(1/90 * N[(angle * N[(Pi * N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(angle * 1/90), $MachinePrecision] * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 9.2000000000000001e182

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 57.9%

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

   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 9.2000000000000001e182 < b

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

   3. Applied rewrites 57.9%

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

   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

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

   9. Applied rewrites 37.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 41.3%

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

   11. Applied rewrites 41.3%

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

Alternative 13: 40.7% accurate, 15.1× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (* (* (* angle (* PI (+ (fabs a) (fabs b)))) (fabs b)) 1/90))

C

double code(double a, double b, double angle) {
	return ((angle * (((double) M_PI) * (fabs(a) + fabs(b)))) * fabs(b)) * 0.011111111111111112;
}

Java

public static double code(double a, double b, double angle) {
	return ((angle * (Math.PI * (Math.abs(a) + Math.abs(b)))) * Math.abs(b)) * 0.011111111111111112;
}

Python

def code(a, b, angle):
	return ((angle * (math.pi * (math.fabs(a) + math.fabs(b)))) * math.fabs(b)) * 0.011111111111111112

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(angle * Float64(pi * Float64(abs(a) + abs(b)))) * abs(b)) * 0.011111111111111112)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((angle * (pi * (abs(a) + abs(b)))) * abs(b)) * 0.011111111111111112;
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(angle * N[(Pi * N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * 1/90), $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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

3. Applied rewrites 57.9%

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

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

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

9. Applied rewrites 37.2%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 37.2%

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. associate-*r* N/A

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

    8. associate-*r* N/A

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

    9. lower-*.f64 N/A

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

    10. lower-*.f64 N/A

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

    11. lower-*.f64 41.2%

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

11. Applied rewrites 41.2%

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

Alternative 14: 40.7% accurate, 15.1× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (* 1/90 (* (* (* angle PI) (+ (fabs a) (fabs b))) (fabs b))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * (((angle * ((double) M_PI)) * (fabs(a) + fabs(b))) * fabs(b));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (((angle * Math.PI) * (Math.abs(a) + Math.abs(b))) * Math.abs(b));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * (((angle * math.pi) * (math.fabs(a) + math.fabs(b))) * math.fabs(b))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(Float64(angle * pi) * Float64(abs(a) + abs(b))) * abs(b)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (((angle * pi) * (abs(a) + abs(b))) * abs(b));
end

Wolfram

code[a_, b_, angle_] := N[(1/90 * N[(N[(N[(angle * Pi), $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[b], $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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

3. Applied rewrites 57.9%

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

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

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

9. Applied rewrites 37.2%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*r* N/A

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

    4. lift-PI.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-*r* N/A

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

    7. lower-*.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. lift-PI.f64 N/A

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

    10. lower-*.f64 41.2%

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

11. Applied rewrites 41.2%

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

Alternative 15: 40.2% accurate, 16.2× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (* 1/90 (* (* PI (+ a (fabs b))) (* (fabs b) angle))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((((double) M_PI) * (a + fabs(b))) * (fabs(b) * angle));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((Math.PI * (a + Math.abs(b))) * (Math.abs(b) * angle));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * ((math.pi * (a + math.fabs(b))) * (math.fabs(b) * angle))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(pi * Float64(a + abs(b))) * Float64(abs(b) * angle)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * ((pi * (a + abs(b))) * (abs(b) * angle));
end

Wolfram

code[a_, b_, angle_] := N[(1/90 * N[(N[(Pi * N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * angle), $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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

3. Applied rewrites 57.9%

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

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

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

9. Applied rewrites 37.2%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. associate-*r* N/A

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

    6. associate-*l* N/A

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

    7. lower-*.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. lower-*.f64 40.0%

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

11. Applied rewrites 40.0%

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

Alternative 16: 37.4% accurate, 16.2× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (* 1/90 (* (* angle (* (fabs b) (+ a (fabs b)))) PI)))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((angle * (fabs(b) * (a + fabs(b)))) * ((double) M_PI));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((angle * (Math.abs(b) * (a + Math.abs(b)))) * Math.PI);
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * ((angle * (math.fabs(b) * (a + math.fabs(b)))) * math.pi)

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(angle * Float64(abs(b) * Float64(a + abs(b)))) * pi))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * ((angle * (abs(b) * (a + abs(b)))) * pi);
end

Wolfram

code[a_, b_, angle_] := N[(1/90 * N[(N[(angle * N[(N[Abs[b], $MachinePrecision] * N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

3. Applied rewrites 57.9%

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

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

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

9. Applied rewrites 37.2%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. lower-*.f64 N/A

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

    6. lower-*.f64 37.2%

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 37.2%

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

11. Applied rewrites 37.2%

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

Alternative 17: 37.4% accurate, 16.2× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (* 1/90 (* angle (* PI (* (+ a (fabs b)) (fabs b))))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (((double) M_PI) * ((a + fabs(b)) * fabs(b))));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (Math.PI * ((a + Math.abs(b)) * Math.abs(b))));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * (angle * (math.pi * ((a + math.fabs(b)) * math.fabs(b))))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(a + abs(b)) * abs(b)))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (angle * (pi * ((a + abs(b)) * abs(b))));
end

Wolfram

code[a_, b_, angle_] := N[(1/90 * N[(angle * N[(Pi * N[(N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

3. Applied rewrites 57.9%

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

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

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

9. Applied rewrites 37.2%

   \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot b\right)\right)\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -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)))))