ab-angle->ABCF B

Percentage Accurate: 54.0% → 53.7%

Time: 1.1min

Alternatives: 7

Speedup: 1.0×

• 31.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 53.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := \cos t\_0\\ t_2 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ t_3 := \sqrt[3]{\log \pi}\\ t_4 := \frac{angle\_m}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(t\_2 \cdot \sin t\_0\right) \cdot t\_1 \leq 2 \cdot 10^{+174}:\\ \;\;\;\;\left(t\_2 \cdot \sin t\_4\right) \cdot \cos t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \sin \left(\frac{angle\_m}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{{\left(e^{{t\_3}^{2}}\right)}^{t\_3}}\right)}^{2}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1 (cos t_0))
        (t_2 (* 2.0 (- (pow b 2.0) (pow a 2.0))))
        (t_3 (cbrt (log PI)))
        (t_4 (* (/ angle_m 180.0) (* (cbrt PI) (pow (cbrt PI) 2.0)))))
   (*
    angle_s
    (if (<= (* (* t_2 (sin t_0)) t_1) 2e+174)
      (* (* t_2 (sin t_4)) (cos t_4))
      (*
       t_1
       (*
        t_2
        (sin
         (*
          (/ angle_m 180.0)
          (* (cbrt PI) (pow (cbrt (pow (exp (pow t_3 2.0)) t_3)) 2.0))))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = cos(t_0);
	double t_2 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double t_3 = cbrt(log(((double) M_PI)));
	double t_4 = (angle_m / 180.0) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0));
	double tmp;
	if (((t_2 * sin(t_0)) * t_1) <= 2e+174) {
		tmp = (t_2 * sin(t_4)) * cos(t_4);
	} else {
		tmp = t_1 * (t_2 * sin(((angle_m / 180.0) * (cbrt(((double) M_PI)) * pow(cbrt(pow(exp(pow(t_3, 2.0)), t_3)), 2.0)))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m / 180.0);
	double t_1 = Math.cos(t_0);
	double t_2 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	double t_3 = Math.cbrt(Math.log(Math.PI));
	double t_4 = (angle_m / 180.0) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0));
	double tmp;
	if (((t_2 * Math.sin(t_0)) * t_1) <= 2e+174) {
		tmp = (t_2 * Math.sin(t_4)) * Math.cos(t_4);
	} else {
		tmp = t_1 * (t_2 * Math.sin(((angle_m / 180.0) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.pow(Math.exp(Math.pow(t_3, 2.0)), t_3)), 2.0)))));
	}
	return angle_s * tmp;
}

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = cos(t_0)
	t_2 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	t_3 = cbrt(log(pi))
	t_4 = Float64(Float64(angle_m / 180.0) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)))
	tmp = 0.0
	if (Float64(Float64(t_2 * sin(t_0)) * t_1) <= 2e+174)
		tmp = Float64(Float64(t_2 * sin(t_4)) * cos(t_4));
	else
		tmp = Float64(t_1 * Float64(t_2 * sin(Float64(Float64(angle_m / 180.0) * Float64(cbrt(pi) * (cbrt((exp((t_3 ^ 2.0)) ^ t_3)) ^ 2.0))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Log[Pi], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[(t$95$2 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 2e+174], N[(N[(t$95$2 * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$2 * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[N[Power[N[Exp[N[Power[t$95$3, 2.0], $MachinePrecision]], $MachinePrecision], t$95$3], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) < 2.00000000000000014e174

   1. Initial program 58.6%

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

      1. add-cube-cbrt 60.0%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right) \]

      2. pow2 60.0%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 60.0%

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

      1. add-cube-cbrt 60.0%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right) \]

      2. pow2 60.0%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 58.9%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \]

   if 2.00000000000000014e174 < (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))

   1. Initial program 37.1%

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

      1. add-cube-cbrt 39.2%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right) \]

      2. pow2 39.2%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 33.5%

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

      1. add-exp-log 33.5%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left({\left(\sqrt[3]{\color{blue}{e^{\log \pi}}}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. add-cube-cbrt 37.0%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left({\left(\sqrt[3]{e^{\color{blue}{\left(\sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}\right) \cdot \sqrt[3]{\log \pi}}}}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. exp-prod 38.3%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left({\left(\sqrt[3]{\color{blue}{{\left(e^{\sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}}\right)}^{\left(\sqrt[3]{\log \pi}\right)}}}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. pow2 38.3%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left({\left(\sqrt[3]{{\left(e^{\color{blue}{{\left(\sqrt[3]{\log \pi}\right)}^{2}}}\right)}^{\left(\sqrt[3]{\log \pi}\right)}}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 38.3%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left({\left(\sqrt[3]{\color{blue}{{\left(e^{{\left(\sqrt[3]{\log \pi}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \pi}\right)}}}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq 2 \cdot 10^{+174}:\\ \;\;\;\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{{\left(e^{{\left(\sqrt[3]{\log \pi}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \pi}\right)}}\right)}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 53.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := \frac{angle\_m}{180} \cdot \sqrt[3]{{\pi}^{3}}\\ t_2 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ t_3 := \frac{angle\_m}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(t\_2 \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -2 \cdot 10^{-42}:\\ \;\;\;\;\left(t\_2 \cdot \sin t\_3\right) \cdot \cos t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot \sin t\_1\right) \cdot \cos t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1 (* (/ angle_m 180.0) (cbrt (pow PI 3.0))))
        (t_2 (* 2.0 (- (pow b 2.0) (pow a 2.0))))
        (t_3 (* (/ angle_m 180.0) (* (cbrt PI) (pow (cbrt PI) 2.0)))))
   (*
    angle_s
    (if (<= (* (* t_2 (sin t_0)) (cos t_0)) -2e-42)
      (* (* t_2 (sin t_3)) (cos t_3))
      (* (* t_2 (sin t_1)) (cos t_1))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = (angle_m / 180.0) * cbrt(pow(((double) M_PI), 3.0));
	double t_2 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double t_3 = (angle_m / 180.0) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0));
	double tmp;
	if (((t_2 * sin(t_0)) * cos(t_0)) <= -2e-42) {
		tmp = (t_2 * sin(t_3)) * cos(t_3);
	} else {
		tmp = (t_2 * sin(t_1)) * cos(t_1);
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m / 180.0);
	double t_1 = (angle_m / 180.0) * Math.cbrt(Math.pow(Math.PI, 3.0));
	double t_2 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	double t_3 = (angle_m / 180.0) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0));
	double tmp;
	if (((t_2 * Math.sin(t_0)) * Math.cos(t_0)) <= -2e-42) {
		tmp = (t_2 * Math.sin(t_3)) * Math.cos(t_3);
	} else {
		tmp = (t_2 * Math.sin(t_1)) * Math.cos(t_1);
	}
	return angle_s * tmp;
}

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = Float64(Float64(angle_m / 180.0) * cbrt((pi ^ 3.0)))
	t_2 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	t_3 = Float64(Float64(angle_m / 180.0) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)))
	tmp = 0.0
	if (Float64(Float64(t_2 * sin(t_0)) * cos(t_0)) <= -2e-42)
		tmp = Float64(Float64(t_2 * sin(t_3)) * cos(t_3));
	else
		tmp = Float64(Float64(t_2 * sin(t_1)) * cos(t_1));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[(t$95$2 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], -2e-42], N[(N[(t$95$2 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) < -2.00000000000000008e-42

   1. Initial program 46.2%

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

      1. add-cube-cbrt 48.3%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right) \]

      2. pow2 48.3%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 48.3%

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

      1. add-cube-cbrt 48.3%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right) \]

      2. pow2 48.3%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 47.0%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \]

   if -2.00000000000000008e-42 < (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))

   1. Initial program 56.7%

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

      1. rem-cbrt-cube 54.8%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 54.8%

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

      1. rem-cbrt-cube 54.8%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 58.0%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq -2 \cdot 10^{-42}:\\ \;\;\;\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sqrt[3]{\log \pi}\\ t_1 := \frac{angle\_m}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\\ t_2 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 1.4 \cdot 10^{+180}:\\ \;\;\;\;\left(t\_2 \cdot \sin t\_1\right) \cdot \cos t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot \sin \left(\frac{angle\_m}{180} \cdot {\left(e^{{t\_0}^{2}}\right)}^{t\_0}\right)\right) \cdot \cos \left(\frac{angle\_m}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (cbrt (log PI)))
        (t_1 (* (/ angle_m 180.0) (* (cbrt PI) (pow (cbrt PI) 2.0))))
        (t_2 (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1.4e+180)
      (* (* t_2 (sin t_1)) (cos t_1))
      (*
       (* t_2 (sin (* (/ angle_m 180.0) (pow (exp (pow t_0 2.0)) t_0))))
       (cos (* (/ angle_m 180.0) (pow (sqrt PI) 2.0))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = cbrt(log(((double) M_PI)));
	double t_1 = (angle_m / 180.0) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0));
	double t_2 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double tmp;
	if ((angle_m / 180.0) <= 1.4e+180) {
		tmp = (t_2 * sin(t_1)) * cos(t_1);
	} else {
		tmp = (t_2 * sin(((angle_m / 180.0) * pow(exp(pow(t_0, 2.0)), t_0)))) * cos(((angle_m / 180.0) * pow(sqrt(((double) M_PI)), 2.0)));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.cbrt(Math.log(Math.PI));
	double t_1 = (angle_m / 180.0) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0));
	double t_2 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	double tmp;
	if ((angle_m / 180.0) <= 1.4e+180) {
		tmp = (t_2 * Math.sin(t_1)) * Math.cos(t_1);
	} else {
		tmp = (t_2 * Math.sin(((angle_m / 180.0) * Math.pow(Math.exp(Math.pow(t_0, 2.0)), t_0)))) * Math.cos(((angle_m / 180.0) * Math.pow(Math.sqrt(Math.PI), 2.0)));
	}
	return angle_s * tmp;
}

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = cbrt(log(pi))
	t_1 = Float64(Float64(angle_m / 180.0) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)))
	t_2 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1.4e+180)
		tmp = Float64(Float64(t_2 * sin(t_1)) * cos(t_1));
	else
		tmp = Float64(Float64(t_2 * sin(Float64(Float64(angle_m / 180.0) * (exp((t_0 ^ 2.0)) ^ t_0)))) * cos(Float64(Float64(angle_m / 180.0) * (sqrt(pi) ^ 2.0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Power[N[Log[Pi], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1.4e+180], N[(N[(t$95$2 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Exp[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 1.40000000000000006e180

   1. Initial program 55.9%

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

      1. add-cube-cbrt 56.1%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right) \]

      2. pow2 56.1%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 56.1%

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

      1. add-cube-cbrt 56.1%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right) \]

      2. pow2 56.1%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 56.9%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \]

   if 1.40000000000000006e180 < (/.f64 angle 180)

   1. Initial program 22.6%

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

      1. add-exp-log 37.7%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left({\left(\sqrt[3]{\color{blue}{e^{\log \pi}}}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. add-cube-cbrt 27.7%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left({\left(\sqrt[3]{e^{\color{blue}{\left(\sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}\right) \cdot \sqrt[3]{\log \pi}}}}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. exp-prod 40.0%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left({\left(\sqrt[3]{\color{blue}{{\left(e^{\sqrt[3]{\log \pi} \cdot \sqrt[3]{\log \pi}}\right)}^{\left(\sqrt[3]{\log \pi}\right)}}}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. pow2 40.0%

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left({\left(\sqrt[3]{{\left(e^{\color{blue}{{\left(\sqrt[3]{\log \pi}\right)}^{2}}}\right)}^{\left(\sqrt[3]{\log \pi}\right)}}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 32.2%

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

      1. add-sqr-sqrt 33.6%

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

      2. pow2 33.6%

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

   6. Applied egg-rr 41.2%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left({\left(e^{{\left(\sqrt[3]{\log \pi}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \pi}\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{{\left(\sqrt{\pi}\right)}^{2}} \cdot \frac{angle}{180}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 1.4 \cdot 10^{+180}:\\ \;\;\;\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot {\left(e^{{\left(\sqrt[3]{\log \pi}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \pi}\right)}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle\_m}{180} \cdot \sqrt[3]{{\pi}^{3}}\\ angle\_s \cdot \left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (/ angle_m 180.0) (cbrt (pow PI 3.0)))))
   (*
    angle_s
    (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0)))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * cbrt(pow(((double) M_PI), 3.0));
	return angle_s * (((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * Math.cbrt(Math.pow(Math.PI, 3.0));
	return angle_s * (((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0));
}

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(angle_m / 180.0) * cbrt((pi ^ 3.0)))
	return Float64(angle_s * Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.0%

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

   1. rem-cbrt-cube 51.9%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \frac{angle}{180}\right) \]

4. Applied egg-rr 51.9%

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

   1. rem-cbrt-cube 51.9%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \frac{angle}{180}\right) \]

6. Applied egg-rr 55.4%

   \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\sqrt[3]{{\pi}^{3}} \cdot \frac{angle}{180}\right) \]

7. Final simplification 55.4%

   \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right) \]
8. Add Preprocessing

Alternative 5: 54.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right) \cdot \cos \left(\frac{angle\_m}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle_m 180.0))))
   (cos (* (/ angle_m 180.0) (* (cbrt PI) (pow (cbrt PI) 2.0)))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin((((double) M_PI) * (angle_m / 180.0)))) * cos(((angle_m / 180.0) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0)))));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin((Math.PI * (angle_m / 180.0)))) * Math.cos(((angle_m / 180.0) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0)))));
}

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(Float64(pi * Float64(angle_m / 180.0)))) * cos(Float64(Float64(angle_m / 180.0) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0))))))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

   1. add-cube-cbrt 54.6%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right) \]

   2. pow2 54.6%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \]

4. Applied egg-rr 54.6%

   \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right) \]

5. Final simplification 54.6%

   \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right) \]
6. Add Preprocessing

Alternative 6: 53.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right) \cdot \cos \left(\frac{angle\_m}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle_m 180.0))))
   (cos (* (/ angle_m 180.0) (pow (sqrt PI) 2.0))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin((((double) M_PI) * (angle_m / 180.0)))) * cos(((angle_m / 180.0) * pow(sqrt(((double) M_PI)), 2.0))));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin((Math.PI * (angle_m / 180.0)))) * Math.cos(((angle_m / 180.0) * Math.pow(Math.sqrt(Math.PI), 2.0))));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin((math.pi * (angle_m / 180.0)))) * math.cos(((angle_m / 180.0) * math.pow(math.sqrt(math.pi), 2.0))))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(Float64(pi * Float64(angle_m / 180.0)))) * cos(Float64(Float64(angle_m / 180.0) * (sqrt(pi) ^ 2.0)))))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin((pi * (angle_m / 180.0)))) * cos(((angle_m / 180.0) * (sqrt(pi) ^ 2.0))));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

   1. add-sqr-sqrt 54.3%

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

   2. pow2 54.3%

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

4. Applied egg-rr 54.3%

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

5. Final simplification 54.3%

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

Alternative 7: 54.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ angle\_s \cdot \left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0))))
   (*
    angle_s
    (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0)))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	return angle_s * (((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m / 180.0);
	return angle_s * (((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (angle_m / 180.0)
	return angle_s * (((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	return Float64(angle_s * Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	t_0 = pi * (angle_m / 180.0);
	tmp = angle_s * (((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.0%

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

3. Final simplification 53.0%

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

Reproduce

herbie shell --seed 2024034 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))