ab-angle->ABCF C

Percentage Accurate: 79.8% → 79.8%

Time: 4.8s

Alternatives: 10

Speedup: 1.5×

• 36.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\mathsf{fma}\left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi \cdot \pi}, \sqrt[3]{\pi}, 0.5 \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} \end{array} \]

FPCore

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

C

double code(double a, double b, double angle) {
	return pow((a * sin(fma(((angle * 0.005555555555555556) * cbrt((((double) M_PI) * ((double) M_PI)))), cbrt(((double) M_PI)), (0.5 * ((double) M_PI))))), 2.0) + pow((b * sin((angle * (0.005555555555555556 * ((double) M_PI))))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(fma(Float64(Float64(angle * 0.005555555555555556) * cbrt(Float64(pi * pi))), cbrt(pi), Float64(0.5 * pi)))) ^ 2.0) + (Float64(b * sin(Float64(angle * Float64(0.005555555555555556 * pi)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[Power[N[(Pi * Pi), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.8

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

3. Applied rewrites 79.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.8

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

5. Applied rewrites 79.8%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lift-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. cos-neg-rev N/A

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

   12. sin-+PI/2-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. lift-/.f64 N/A

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

   18. mult-flip N/A

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

   19. metadata-eval N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 N/A

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

   22. lift-PI.f64 N/A

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

   23. mult-flip N/A

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

   24. metadata-eval N/A

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

   25. lower-*.f64 79.8

       \[\leadsto {\left(a \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \color{blue}{\pi \cdot 0.5}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} \]

7. Applied rewrites 79.8%

   \[\leadsto {\left(a \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} \]
8. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. mult-flip N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. mult-flip N/A

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

   16. lift-/.f64 N/A

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

   17. lift-PI.f64 N/A

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

   18. add-cube-cbrt N/A

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

   19. associate-*r* N/A

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

   20. lift-*.f64 N/A

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

   21. metadata-eval N/A

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

   22. mult-flip-rev N/A

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

   23. lift-PI.f64 N/A

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

9. Applied rewrites 79.8%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi \cdot \pi}, \sqrt[3]{\pi}, 0.5 \cdot \pi\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 2: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\pi \cdot \mathsf{fma}\left(angle, 0.005555555555555556, 0.5\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* PI (fma angle 0.005555555555555556 0.5)))) 2.0)
  (pow (* b (sin (* angle (* 0.005555555555555556 PI)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin((((double) M_PI) * fma(angle, 0.005555555555555556, 0.5)))), 2.0) + pow((b * sin((angle * (0.005555555555555556 * ((double) M_PI))))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(pi * fma(angle, 0.005555555555555556, 0.5)))) ^ 2.0) + (Float64(b * sin(Float64(angle * Float64(0.005555555555555556 * pi)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(Pi * N[(angle * 0.005555555555555556 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.8

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

3. Applied rewrites 79.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.8

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

5. Applied rewrites 79.8%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lift-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. cos-neg-rev N/A

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

   12. sin-+PI/2-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. lift-/.f64 N/A

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

   18. mult-flip N/A

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

   19. metadata-eval N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 N/A

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

   22. lift-PI.f64 N/A

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

   23. mult-flip N/A

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

   24. metadata-eval N/A

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

   25. lower-*.f64 79.8

       \[\leadsto {\left(a \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \color{blue}{\pi \cdot 0.5}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} \]

7. Applied rewrites 79.8%

   \[\leadsto {\left(a \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} \]
8. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. fp-cancel-sign-sub-inv N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. mult-flip-rev N/A

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

   18. lift-PI.f64 N/A

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

9. Applied rewrites 79.8%

   \[\leadsto {\left(a \cdot \color{blue}{\sin \left(\pi \cdot \mathsf{fma}\left(angle, 0.005555555555555556, 0.5\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 3: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* 0.005555555555555556 PI))))
   (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * ((double) M_PI));
	return pow((a * cos(t_0)), 2.0) + pow((b * sin(t_0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * Math.PI);
	return Math.pow((a * Math.cos(t_0)), 2.0) + Math.pow((b * Math.sin(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = angle * (0.005555555555555556 * math.pi)
	return math.pow((a * math.cos(t_0)), 2.0) + math.pow((b * math.sin(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(0.005555555555555556 * pi))
	return Float64((Float64(a * cos(t_0)) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = angle * (0.005555555555555556 * pi);
	tmp = ((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.8

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

3. Applied rewrites 79.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.8

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

5. Applied rewrites 79.8%

   \[\leadsto {\left(a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}^{2} \]
6. Add Preprocessing

Alternative 4: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ {\left(\sin t\_0 \cdot b\right)}^{2} + {\left(\cos t\_0 \cdot a\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) PI)))
   (+ (pow (* (sin t_0) b) 2.0) (pow (* (cos t_0) a) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 * angle) * ((double) M_PI);
	return pow((sin(t_0) * b), 2.0) + pow((cos(t_0) * a), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 * angle) * Math.PI;
	return Math.pow((Math.sin(t_0) * b), 2.0) + Math.pow((Math.cos(t_0) * a), 2.0);
}

Python

def code(a, b, angle):
	t_0 = (0.005555555555555556 * angle) * math.pi
	return math.pow((math.sin(t_0) * b), 2.0) + math.pow((math.cos(t_0) * a), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(0.005555555555555556 * angle) * pi)
	return Float64((Float64(sin(t_0) * b) ^ 2.0) + (Float64(cos(t_0) * a) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = (0.005555555555555556 * angle) * pi;
	tmp = ((sin(t_0) * b) ^ 2.0) + ((cos(t_0) * a) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(N[Sin[t$95$0], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Cos[t$95$0], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 79.8

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 79.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 79.8

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

   10. lift-/.f64 N/A

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

   11. mult-flip N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 79.8

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

3. Applied rewrites 79.8%

   \[\leadsto \color{blue}{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2} + {\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot a\right)}^{2}} \]
4. Add Preprocessing

Alternative 5: 79.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, b, angle):
	return math.pow((a * 1.0), 2.0) + math.pow((b * math.sin((angle * (0.005555555555555556 * math.pi)))), 2.0)

Julia

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

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * 1.0) ^ 2.0) + ((b * sin((angle * (0.005555555555555556 * pi)))) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.8

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

3. Applied rewrites 79.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.8

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

5. Applied rewrites 79.8%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 79.8%

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

Alternative 6: 60.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{+108}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.7e+108)
   (* a a)
   (* (pow b 2.0) (pow (sin (* 0.005555555555555556 (* angle PI))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.7e+108) {
		tmp = a * a;
	} else {
		tmp = pow(b, 2.0) * pow(sin((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.7e+108) {
		tmp = a * a;
	} else {
		tmp = Math.pow(b, 2.0) * Math.pow(Math.sin((0.005555555555555556 * (angle * Math.PI))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 4.7e+108:
		tmp = a * a
	else:
		tmp = math.pow(b, 2.0) * math.pow(math.sin((0.005555555555555556 * (angle * math.pi))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 4.7e+108)
		tmp = Float64(a * a);
	else
		tmp = Float64((b ^ 2.0) * (sin(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 4.7e+108)
		tmp = a * a;
	else
		tmp = (b ^ 2.0) * (sin((0.005555555555555556 * (angle * pi))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 4.7e+108], N[(a * a), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] * N[Power[N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.6999999999999996e108

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if 4.6999999999999996e108 < b

   1. Initial program 79.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 34.1

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

   4. Applied rewrites 34.1%

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

Alternative 7: 59.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{+158}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.6e+158)
   (* a a)
   (*
    (pow b 2.0)
    (- 0.5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.6e+158) {
		tmp = a * a;
	} else {
		tmp = pow(b, 2.0) * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.6e+158) {
		tmp = a * a;
	} else {
		tmp = Math.pow(b, 2.0) * (0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 3.6e+158:
		tmp = a * a
	else:
		tmp = math.pow(b, 2.0) * (0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.6e+158)
		tmp = Float64(a * a);
	else
		tmp = Float64((b ^ 2.0) * Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 3.6e+158)
		tmp = a * a;
	else
		tmp = (b ^ 2.0) * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.6e+158], N[(a * a), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.59999999999999988e158

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if 3.59999999999999988e158 < b

   1. Initial program 79.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 79.8

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

   3. Applied rewrites 79.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 79.8

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

   5. Applied rewrites 79.8%

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

   7. Applied rewrites 62.5%

      \[\leadsto {\left(a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} + \color{blue}{\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.5\right) \cdot \left(b \cdot b\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 26.2

         \[\leadsto {b}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   10. Applied rewrites 26.2%

       \[\leadsto \color{blue}{{b}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 59.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ t_1 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;{\left(a \cdot \cos t\_1\right)}^{2} + {\left(b \cdot \sin t\_1\right)}^{2} \leq 10^{+302}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{t\_0 \cdot t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* (* a a) a) a)) (t_1 (* PI (/ angle 180.0))))
   (if (<= (+ (pow (* a (cos t_1)) 2.0) (pow (* b (sin t_1)) 2.0)) 1e+302)
     (* a a)
     (sqrt (sqrt (* t_0 t_0))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((a * a) * a) * a;
	double t_1 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((pow((a * cos(t_1)), 2.0) + pow((b * sin(t_1)), 2.0)) <= 1e+302) {
		tmp = a * a;
	} else {
		tmp = sqrt(sqrt((t_0 * t_0)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = ((a * a) * a) * a;
	double t_1 = Math.PI * (angle / 180.0);
	double tmp;
	if ((Math.pow((a * Math.cos(t_1)), 2.0) + Math.pow((b * Math.sin(t_1)), 2.0)) <= 1e+302) {
		tmp = a * a;
	} else {
		tmp = Math.sqrt(Math.sqrt((t_0 * t_0)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = ((a * a) * a) * a
	t_1 = math.pi * (angle / 180.0)
	tmp = 0
	if (math.pow((a * math.cos(t_1)), 2.0) + math.pow((b * math.sin(t_1)), 2.0)) <= 1e+302:
		tmp = a * a
	else:
		tmp = math.sqrt(math.sqrt((t_0 * t_0)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(Float64(a * a) * a) * a)
	t_1 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64((Float64(a * cos(t_1)) ^ 2.0) + (Float64(b * sin(t_1)) ^ 2.0)) <= 1e+302)
		tmp = Float64(a * a);
	else
		tmp = sqrt(sqrt(Float64(t_0 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = ((a * a) * a) * a;
	t_1 = pi * (angle / 180.0);
	tmp = 0.0;
	if ((((a * cos(t_1)) ^ 2.0) + ((b * sin(t_1)) ^ 2.0)) <= 1e+302)
		tmp = a * a;
	else
		tmp = sqrt(sqrt((t_0 * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(a * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1e+302], N[(a * a), $MachinePrecision], N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < 1.0000000000000001e302

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if 1.0000000000000001e302 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)))

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{{a}^{2}} \cdot \color{blue}{\sqrt{{a}^{2}}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{{a}^{2} \cdot {a}^{2}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{{a}^{2} \cdot {a}^{2}} \]

      4. lower-*.f64 49.4

         \[\leadsto \sqrt{{a}^{2} \cdot {a}^{2}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \sqrt{{a}^{2} \cdot {a}^{2}} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\left(a \cdot a\right) \cdot {a}^{2}} \]

      7. lift-*.f64 49.4

         \[\leadsto \sqrt{\left(a \cdot a\right) \cdot {a}^{2}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(a \cdot a\right) \cdot {a}^{2}} \]

      9. pow2 N/A

         \[\leadsto \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]

      10. lift-*.f64 49.4

          \[\leadsto \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \cdot \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      4. lower-*.f64 45.4

         \[\leadsto \sqrt{\sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\sqrt{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\sqrt{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\sqrt{\left(\left(a \cdot \left(a \cdot a\right)\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\sqrt{\left(\left(a \cdot \left(a \cdot a\right)\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      11. lower-*.f64 45.4

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)}} \]

      14. associate-*l* N/A

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot a\right)}} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot a\right)}} \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right)}} \]

      18. lower-*.f64 45.4

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right)}} \]

   8. Applied rewrites 45.4%

      \[\leadsto \sqrt{\sqrt{\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 58.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;{\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \leq 10^{+302}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<= (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0)) 1e+302)
     (* a a)
     (sqrt (* (* a a) (* a a))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((pow((a * cos(t_0)), 2.0) + pow((b * sin(t_0)), 2.0)) <= 1e+302) {
		tmp = a * a;
	} else {
		tmp = sqrt(((a * a) * (a * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double tmp;
	if ((Math.pow((a * Math.cos(t_0)), 2.0) + Math.pow((b * Math.sin(t_0)), 2.0)) <= 1e+302) {
		tmp = a * a;
	} else {
		tmp = Math.sqrt(((a * a) * (a * a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	tmp = 0
	if (math.pow((a * math.cos(t_0)), 2.0) + math.pow((b * math.sin(t_0)), 2.0)) <= 1e+302:
		tmp = a * a
	else:
		tmp = math.sqrt(((a * a) * (a * a)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64((Float64(a * cos(t_0)) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0)) <= 1e+302)
		tmp = Float64(a * a);
	else
		tmp = sqrt(Float64(Float64(a * a) * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = 0.0;
	if ((((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0)) <= 1e+302)
		tmp = a * a;
	else
		tmp = sqrt(((a * a) * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1e+302], N[(a * a), $MachinePrecision], N[Sqrt[N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < 1.0000000000000001e302

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if 1.0000000000000001e302 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)))

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{{a}^{2}} \cdot \color{blue}{\sqrt{{a}^{2}}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{{a}^{2} \cdot {a}^{2}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{{a}^{2} \cdot {a}^{2}} \]

      4. lower-*.f64 49.4

         \[\leadsto \sqrt{{a}^{2} \cdot {a}^{2}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \sqrt{{a}^{2} \cdot {a}^{2}} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\left(a \cdot a\right) \cdot {a}^{2}} \]

      7. lift-*.f64 49.4

         \[\leadsto \sqrt{\left(a \cdot a\right) \cdot {a}^{2}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(a \cdot a\right) \cdot {a}^{2}} \]

      9. pow2 N/A

         \[\leadsto \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]

      10. lift-*.f64 49.4

          \[\leadsto \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.1% accurate, 29.7× speedup

Math

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

FPCore

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

C

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

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 = a * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

2. Taylor expanded in angle around 0

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

   1. lower-pow.f64 57.1

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

4. Applied rewrites 57.1%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 57.1

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

6. Applied rewrites 57.1%

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

Reproduce

herbie shell --seed 2025142 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))