ab-angle->ABCF A

Percentage Accurate: 80.5% → 80.5%

Time: 4.8s

Alternatives: 8

Speedup: N/A×

• 37.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\left({\pi}^{0.6666666666666666} \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow
   (*
    a
    (sin
     (*
      (* (pow PI 0.6666666666666666) (* angle 0.005555555555555556))
      (cbrt PI))))
   2.0)
  (* (* (* 1.0 b) 1.0) b)))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((pow(((double) M_PI), 0.6666666666666666) * (angle * 0.005555555555555556)) * cbrt(((double) M_PI))))), 2.0) + (((1.0 * b) * 1.0) * b);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((Math.pow(Math.PI, 0.6666666666666666) * (angle * 0.005555555555555556)) * Math.cbrt(Math.PI)))), 2.0) + (((1.0 * b) * 1.0) * b);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64((pi ^ 0.6666666666666666) * Float64(angle * 0.005555555555555556)) * cbrt(pi)))) ^ 2.0) + Float64(Float64(Float64(1.0 * b) * 1.0) * b))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(N[Power[Pi, 0.6666666666666666], $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(1.0 * b), $MachinePrecision] * 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 80.5%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   4. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\frac{angle}{180} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   5. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\frac{angle}{180} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{angle}{180} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\frac{angle}{180}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   8. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   11. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   12. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   13. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\color{blue}{\pi}}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   14. cbrt-unprod N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt[3]{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   15. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)} \cdot \pi}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   16. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   17. lower-cbrt.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   18. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\color{blue}{\pi} \cdot \mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   19. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \color{blue}{\pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   20. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\color{blue}{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

6. Applied rewrites 80.4%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{{\left(b \cdot 1\right)}^{2}} \]

   2. unpow2 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(b \cdot 1\right) \cdot \left(b \cdot 1\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(b \cdot 1\right) \cdot \color{blue}{\left(b \cdot 1\right)} \]

   4. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(b \cdot 1\right) \cdot \color{blue}{\left(1 \cdot b\right)} \]

   5. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(\left(b \cdot 1\right) \cdot 1\right) \cdot b} \]

   6. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(\left(b \cdot 1\right) \cdot 1\right) \cdot b} \]

   7. lower-*.f64 80.4

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(\left(b \cdot 1\right) \cdot 1\right)} \cdot b \]

   8. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\color{blue}{\left(b \cdot 1\right)} \cdot 1\right) \cdot b \]

   9. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\color{blue}{\left(1 \cdot b\right)} \cdot 1\right) \cdot b \]

   10. lower-*.f64 80.4

       \[\leadsto {\left(a \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\color{blue}{\left(1 \cdot b\right)} \cdot 1\right) \cdot b \]

8. Applied rewrites 80.4%

   \[\leadsto {\left(a \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(\left(1 \cdot b\right) \cdot 1\right) \cdot b} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right)} \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   3. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   4. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   5. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\frac{angle}{180}} \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   6. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\frac{angle}{180}} \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   7. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\sqrt[3]{\pi \cdot \pi} \cdot \frac{angle}{180}\right)} \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   8. lower-*.f64 80.4

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\sqrt[3]{\pi \cdot \pi} \cdot \frac{angle}{180}\right)} \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   9. lift-cbrt.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\sqrt[3]{\pi \cdot \pi}} \cdot \frac{angle}{180}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   10. pow1/3 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{{\left(\pi \cdot \pi\right)}^{\frac{1}{3}}} \cdot \frac{angle}{180}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   11. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left({\color{blue}{\left(\pi \cdot \pi\right)}}^{\frac{1}{3}} \cdot \frac{angle}{180}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   12. unpow-prod-down N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left({\pi}^{\frac{1}{3}} \cdot {\pi}^{\frac{1}{3}}\right)} \cdot \frac{angle}{180}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   13. pow-prod-up N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \frac{angle}{180}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   14. lower-pow.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \frac{angle}{180}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   15. metadata-eval 80.4

       \[\leadsto {\left(a \cdot \sin \left(\left({\pi}^{\color{blue}{0.6666666666666666}} \cdot \frac{angle}{180}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   16. lift-/.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   17. mult-flip N/A

       \[\leadsto {\left(a \cdot \sin \left(\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   18. metadata-eval N/A

       \[\leadsto {\left(a \cdot \sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   19. lift-*.f64 80.4

       \[\leadsto {\left(a \cdot \sin \left(\left({\pi}^{0.6666666666666666} \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

10. Applied rewrites 80.4%

    \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left({\pi}^{0.6666666666666666} \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]
11. Add Preprocessing

Alternative 2: 80.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\frac{1}{\frac{180}{angle}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 80.5%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 80.4

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

6. Applied rewrites 80.4%

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

Alternative 3: 80.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 80.5%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 80.5

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 80.5

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

6. Applied rewrites 80.5%

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

Alternative 4: 80.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, {\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (* (* 1.0 b) 1.0)
  b
  (pow (* (sin (* PI (* 0.005555555555555556 angle))) a) 2.0)))

C

double code(double a, double b, double angle) {
	return fma(((1.0 * b) * 1.0), b, pow((sin((((double) M_PI) * (0.005555555555555556 * angle))) * a), 2.0));
}

Julia

function code(a, b, angle)
	return fma(Float64(Float64(1.0 * b) * 1.0), b, (Float64(sin(Float64(pi * Float64(0.005555555555555556 * angle))) * a) ^ 2.0))
end

Wolfram

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

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 80.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

6. Applied rewrites 80.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, {\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}\right)} \]
7. Add Preprocessing

Alternative 5: 68.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{-106}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.4e-106)
   (* b b)
   (+
    (pow (* a (* 0.005555555555555556 (* angle PI))) 2.0)
    (* (* (* 1.0 b) 1.0) b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.4e-106) {
		tmp = b * b;
	} else {
		tmp = pow((a * (0.005555555555555556 * (angle * ((double) M_PI)))), 2.0) + (((1.0 * b) * 1.0) * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.4e-106) {
		tmp = b * b;
	} else {
		tmp = Math.pow((a * (0.005555555555555556 * (angle * Math.PI))), 2.0) + (((1.0 * b) * 1.0) * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.4e-106:
		tmp = b * b
	else:
		tmp = math.pow((a * (0.005555555555555556 * (angle * math.pi))), 2.0) + (((1.0 * b) * 1.0) * b)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.4e-106)
		tmp = Float64(b * b);
	else
		tmp = Float64((Float64(a * Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0) + Float64(Float64(Float64(1.0 * b) * 1.0) * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.4e-106)
		tmp = b * b;
	else
		tmp = ((a * (0.005555555555555556 * (angle * pi))) ^ 2.0) + (((1.0 * b) * 1.0) * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.4e-106], N[(b * b), $MachinePrecision], N[(N[Power[N[(a * N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(1.0 * b), $MachinePrecision] * 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.39999999999999994e-106

   1. Initial program 80.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 58.5

         \[\leadsto {b}^{\color{blue}{2}} \]

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{{b}^{2}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {b}^{\color{blue}{2}} \]

      2. unpow2 N/A

         \[\leadsto b \cdot \color{blue}{b} \]

      3. lower-*.f64 58.5

         \[\leadsto b \cdot \color{blue}{b} \]

   6. Applied rewrites 58.5%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 1.39999999999999994e-106 < a

   1. Initial program 80.5%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      4. associate-*r* N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\frac{angle}{180} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      5. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\frac{angle}{180} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      6. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{angle}{180} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      7. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\frac{angle}{180}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      8. mult-flip N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      9. metadata-eval N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\color{blue}{\pi}}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. cbrt-unprod N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt[3]{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      15. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)} \cdot \pi}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      16. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      17. lower-cbrt.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      18. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\color{blue}{\pi} \cdot \mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      19. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \color{blue}{\pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      20. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\color{blue}{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied rewrites 80.4%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{{\left(b \cdot 1\right)}^{2}} \]

      2. unpow2 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(b \cdot 1\right) \cdot \left(b \cdot 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(b \cdot 1\right) \cdot \color{blue}{\left(b \cdot 1\right)} \]

      4. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(b \cdot 1\right) \cdot \color{blue}{\left(1 \cdot b\right)} \]

      5. associate-*r* N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(\left(b \cdot 1\right) \cdot 1\right) \cdot b} \]

      6. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(\left(b \cdot 1\right) \cdot 1\right) \cdot b} \]

      7. lower-*.f64 80.4

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(\left(b \cdot 1\right) \cdot 1\right)} \cdot b \]

      8. lift-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\color{blue}{\left(b \cdot 1\right)} \cdot 1\right) \cdot b \]

      9. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\color{blue}{\left(1 \cdot b\right)} \cdot 1\right) \cdot b \]

      10. lower-*.f64 80.4

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\color{blue}{\left(1 \cdot b\right)} \cdot 1\right) \cdot b \]

   8. Applied rewrites 80.4%

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(\left(1 \cdot b\right) \cdot 1\right) \cdot b} \]

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 75.5

          \[\leadsto {\left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   11. Applied rewrites 75.5%

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

Alternative 6: 68.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{-106}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(0.005555555555555556 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.4e-106)
   (* b b)
   (+
    (pow (* 0.005555555555555556 (* a (* angle PI))) 2.0)
    (* (* (* 1.0 b) 1.0) b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.4e-106) {
		tmp = b * b;
	} else {
		tmp = pow((0.005555555555555556 * (a * (angle * ((double) M_PI)))), 2.0) + (((1.0 * b) * 1.0) * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.4e-106) {
		tmp = b * b;
	} else {
		tmp = Math.pow((0.005555555555555556 * (a * (angle * Math.PI))), 2.0) + (((1.0 * b) * 1.0) * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.4e-106:
		tmp = b * b
	else:
		tmp = math.pow((0.005555555555555556 * (a * (angle * math.pi))), 2.0) + (((1.0 * b) * 1.0) * b)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.4e-106)
		tmp = Float64(b * b);
	else
		tmp = Float64((Float64(0.005555555555555556 * Float64(a * Float64(angle * pi))) ^ 2.0) + Float64(Float64(Float64(1.0 * b) * 1.0) * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.4e-106)
		tmp = b * b;
	else
		tmp = ((0.005555555555555556 * (a * (angle * pi))) ^ 2.0) + (((1.0 * b) * 1.0) * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.4e-106], N[(b * b), $MachinePrecision], N[(N[Power[N[(0.005555555555555556 * N[(a * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(1.0 * b), $MachinePrecision] * 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.39999999999999994e-106

   1. Initial program 80.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 58.5

         \[\leadsto {b}^{\color{blue}{2}} \]

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{{b}^{2}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {b}^{\color{blue}{2}} \]

      2. unpow2 N/A

         \[\leadsto b \cdot \color{blue}{b} \]

      3. lower-*.f64 58.5

         \[\leadsto b \cdot \color{blue}{b} \]

   6. Applied rewrites 58.5%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 1.39999999999999994e-106 < a

   1. Initial program 80.5%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      4. associate-*r* N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\frac{angle}{180} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      5. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\frac{angle}{180} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      6. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{angle}{180} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      7. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\frac{angle}{180}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      8. mult-flip N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      9. metadata-eval N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\color{blue}{\pi}}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. cbrt-unprod N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt[3]{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      15. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)} \cdot \pi}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      16. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      17. lower-cbrt.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      18. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\color{blue}{\pi} \cdot \mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      19. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \color{blue}{\pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      20. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\color{blue}{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied rewrites 80.4%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{{\left(b \cdot 1\right)}^{2}} \]

      2. unpow2 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(b \cdot 1\right) \cdot \left(b \cdot 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(b \cdot 1\right) \cdot \color{blue}{\left(b \cdot 1\right)} \]

      4. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(b \cdot 1\right) \cdot \color{blue}{\left(1 \cdot b\right)} \]

      5. associate-*r* N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(\left(b \cdot 1\right) \cdot 1\right) \cdot b} \]

      6. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(\left(b \cdot 1\right) \cdot 1\right) \cdot b} \]

      7. lower-*.f64 80.4

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(\left(b \cdot 1\right) \cdot 1\right)} \cdot b \]

      8. lift-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\color{blue}{\left(b \cdot 1\right)} \cdot 1\right) \cdot b \]

      9. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\color{blue}{\left(1 \cdot b\right)} \cdot 1\right) \cdot b \]

      10. lower-*.f64 80.4

          \[\leadsto {\left(a \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \left(\color{blue}{\left(1 \cdot b\right)} \cdot 1\right) \cdot b \]

   8. Applied rewrites 80.4%

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right)}^{2} + \color{blue}{\left(\left(1 \cdot b\right) \cdot 1\right) \cdot b} \]

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-PI.f64 75.5

          \[\leadsto {\left(0.005555555555555556 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + \left(\left(1 \cdot b\right) \cdot 1\right) \cdot b \]

   11. Applied rewrites 75.5%

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

Alternative 7: 66.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{-106}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\left(angle \cdot angle\right) \cdot a\right)\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.4e-106)
   (* b b)
   (fma
    (* (* 1.0 b) 1.0)
    b
    (* (* 3.08641975308642e-5 (* a (* (* angle angle) a))) (* PI PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.4e-106) {
		tmp = b * b;
	} else {
		tmp = fma(((1.0 * b) * 1.0), b, ((3.08641975308642e-5 * (a * ((angle * angle) * a))) * (((double) M_PI) * ((double) M_PI))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.4e-106)
		tmp = Float64(b * b);
	else
		tmp = fma(Float64(Float64(1.0 * b) * 1.0), b, Float64(Float64(3.08641975308642e-5 * Float64(a * Float64(Float64(angle * angle) * a))) * Float64(pi * pi)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.4e-106], N[(b * b), $MachinePrecision], N[(N[(N[(1.0 * b), $MachinePrecision] * 1.0), $MachinePrecision] * b + N[(N[(3.08641975308642e-5 * N[(a * N[(N[(angle * angle), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.39999999999999994e-106

   1. Initial program 80.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 58.5

         \[\leadsto {b}^{\color{blue}{2}} \]

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{{b}^{2}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {b}^{\color{blue}{2}} \]

      2. unpow2 N/A

         \[\leadsto b \cdot \color{blue}{b} \]

      3. lower-*.f64 58.5

         \[\leadsto b \cdot \color{blue}{b} \]

   6. Applied rewrites 58.5%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 1.39999999999999994e-106 < a

   1. Initial program 80.5%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 80.5%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{32400} \cdot \left({a}^{2} \cdot \color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{1}{32400} \cdot \left({a}^{2} \cdot \left(\color{blue}{{angle}^{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      7. lower-PI.f64 64.5

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) + {\left(b \cdot 1\right)}^{2} \]

   7. Applied rewrites 64.5%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) + {\left(b \cdot 1\right)}^{2}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{{\left(b \cdot 1\right)}^{2} + \frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(b \cdot 1\right)}^{2}} + \frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \color{blue}{\left(b \cdot 1\right) \cdot \left(b \cdot 1\right)} + \frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(b \cdot 1\right) \cdot \color{blue}{\left(b \cdot 1\right)} + \frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(b \cdot 1\right) \cdot \color{blue}{\left(1 \cdot b\right)} + \frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(b \cdot 1\right) \cdot 1\right) \cdot b} + \frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(b \cdot 1\right) \cdot 1, b, \frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)\right)} \]

   9. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\left(angle \cdot angle\right) \cdot a\right)\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 58.5% accurate, 29.7× speedup

Math

\[\begin{array}{l} \\ b \cdot b \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (* b b))

C

double code(double a, double b, double angle) {
	return b * b;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = b * b
end function

Java

public static double code(double a, double b, double angle) {
	return b * b;
}

Python

def code(a, b, angle):
	return b * b

Julia

function code(a, b, angle)
	return Float64(b * b)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = b * b;
end

Wolfram

code[a_, b_, angle_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{{b}^{2}} \]
3. Step-by-step derivation

   1. lower-pow.f64 58.5

      \[\leadsto {b}^{\color{blue}{2}} \]

4. Applied rewrites 58.5%

   \[\leadsto \color{blue}{{b}^{2}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto {b}^{\color{blue}{2}} \]

   2. unpow2 N/A

      \[\leadsto b \cdot \color{blue}{b} \]

   3. lower-*.f64 58.5

      \[\leadsto b \cdot \color{blue}{b} \]

6. Applied rewrites 58.5%

   \[\leadsto \color{blue}{b \cdot b} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025148 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))