ab-angle->ABCF C

Percentage Accurate: 80.0% → 80.1%

Time: 15.7s

Alternatives: 11

Speedup: 1.0×

• Could not converge on a ground truth

  1. a = 4.2827936845505934e+133
  2. b = 6.711507064688623e-51
  3. angle = 3.7651738337100724e+211

• 36.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 80.0% 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: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ {\left(a \cdot \cos \left(\frac{1}{\frac{1}{t\_0}}\right)\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI 0.005555555555555556))))
   (+ (pow (* a (cos (/ 1.0 (/ 1.0 t_0)))) 2.0) (pow (* b (sin t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * 0.005555555555555556);
	return pow((a * cos((1.0 / (1.0 / 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 * (Math.PI * 0.005555555555555556);
	return Math.pow((a * Math.cos((1.0 / (1.0 / t_0)))), 2.0) + Math.pow((b * Math.sin(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = angle * (math.pi * 0.005555555555555556)
	return math.pow((a * math.cos((1.0 / (1.0 / t_0)))), 2.0) + math.pow((b * math.sin(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi * 0.005555555555555556))
	return Float64((Float64(a * cos(Float64(1.0 / Float64(1.0 / t_0)))) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = angle * (pi * 0.005555555555555556);
	tmp = ((a * cos((1.0 / (1.0 / t_0)))) ^ 2.0) + ((b * sin(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 77.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*r/ N/A

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

   5. div-inv N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 77.5

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

4. Applied rewrites 77.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*r/ N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. *-rgt-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 77.5

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

6. Applied rewrites 77.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f64 N/A

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

   7. lower-/.f64 77.6

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. *-commutative N/A

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

   12. associate-*r/ N/A

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

   13. lift-/.f64 N/A

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

   14. lower-*.f64 77.7

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

   15. lift-/.f64 N/A

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

   16. div-inv N/A

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

   17. metadata-eval N/A

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

   18. lift-*.f64 77.7

       \[\leadsto {\left(a \cdot \cos \left(\frac{1}{\frac{1}{angle \cdot \color{blue}{\left(\pi \cdot 0.005555555555555556\right)}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{180} \cdot \frac{1}{\frac{1}{angle}}\right)\right)}^{2} \]

8. Applied rewrites 77.7%

   \[\leadsto {\left(a \cdot \cos \left(\frac{1}{\color{blue}{\frac{1}{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{180} \cdot \frac{1}{\frac{1}{angle}}\right)\right)}^{2} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. remove-double-div N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 77.8

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

10. Applied rewrites 77.8%

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

Alternative 2: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* angle (* PI 0.005555555555555556)))) 2.0)
  (pow (* b (sin (/ 1.0 (/ 180.0 (* angle PI))))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * cos((angle * (((double) M_PI) * 0.005555555555555556)))), 2.0) + pow((b * sin((1.0 / (180.0 / (angle * ((double) M_PI)))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.cos((angle * (Math.PI * 0.005555555555555556)))), 2.0) + Math.pow((b * Math.sin((1.0 / (180.0 / (angle * Math.PI))))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.cos((angle * (math.pi * 0.005555555555555556)))), 2.0) + math.pow((b * math.sin((1.0 / (180.0 / (angle * math.pi))))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * cos(Float64(angle * Float64(pi * 0.005555555555555556)))) ^ 2.0) + (Float64(b * sin(Float64(1.0 / Float64(180.0 / Float64(angle * pi))))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * cos((angle * (pi * 0.005555555555555556)))) ^ 2.0) + ((b * sin((1.0 / (180.0 / (angle * pi))))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(1.0 / N[(180.0 / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval 77.6

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

4. Applied rewrites 77.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 77.6

      \[\leadsto {\left(a \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{1}{\frac{180}{\color{blue}{angle \cdot \pi}}}\right)\right)}^{2} \]

6. Applied rewrites 77.6%

   \[\leadsto {\left(a \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)}\right)}^{2} \]

7. Final simplification 77.6%

   \[\leadsto {\left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 3: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval 77.6

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

4. Applied rewrites 77.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 77.6

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

6. Applied rewrites 77.6%

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

7. Final simplification 77.6%

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

Alternative 4: 79.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval 77.6

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

4. Applied rewrites 77.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*r/ N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. *-rgt-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 77.6

       \[\leadsto {\left(a \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{1}{\frac{\color{blue}{\frac{180}{angle}}}{\pi}}\right)\right)}^{2} \]

6. Applied rewrites 77.6%

   \[\leadsto {\left(a \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\frac{1}{\frac{\frac{180}{angle}}{\pi}}\right)}\right)}^{2} \]

7. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 77.4

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

9. Applied rewrites 77.4%

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

Alternative 5: 80.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 77.4

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

5. Applied rewrites 77.4%

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

Alternative 6: 55.6% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 6.4 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \left(\pi \cdot \pi\right)\right) \cdot \mathsf{fma}\left(b, b \cdot 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 6.4e+69)
   (fma
    (*
     (* angle (* PI PI))
     (fma b (* b 3.08641975308642e-5) (* (* a a) -3.08641975308642e-5)))
    angle
    (* a a))
   (* a a)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6.4e+69) {
		tmp = fma(((angle * (((double) M_PI) * ((double) M_PI))) * fma(b, (b * 3.08641975308642e-5), ((a * a) * -3.08641975308642e-5))), angle, (a * a));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 6.4e+69)
		tmp = fma(Float64(Float64(angle * Float64(pi * pi)) * fma(b, Float64(b * 3.08641975308642e-5), Float64(Float64(a * a) * -3.08641975308642e-5))), angle, Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 6.4e+69], N[(N[(N[(angle * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.3999999999999997e69

   1. Initial program 75.9%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.5%

      \[\leadsto \mathsf{fma}\left(\left(angle \cdot \left(\pi \cdot \pi\right)\right) \cdot \mathsf{fma}\left(b, b \cdot 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right), \color{blue}{angle}, a \cdot a\right) \]

   if 6.3999999999999997e69 < a

   1. Initial program 84.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.1

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

   5. Applied rewrites 82.1%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 64.1% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-36}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(angle \cdot b\right)\right) \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.7e-36)
   (* a a)
   (if (<= b 3e+183)
     (fma
      (* angle angle)
      (* PI (* PI (* b (* b 3.08641975308642e-5))))
      (* a a))
     (* 3.08641975308642e-5 (* (* angle (* angle b)) (* b (* PI PI)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.7e-36) {
		tmp = a * a;
	} else if (b <= 3e+183) {
		tmp = fma((angle * angle), (((double) M_PI) * (((double) M_PI) * (b * (b * 3.08641975308642e-5)))), (a * a));
	} else {
		tmp = 3.08641975308642e-5 * ((angle * (angle * b)) * (b * (((double) M_PI) * ((double) M_PI))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.7e-36)
		tmp = Float64(a * a);
	elseif (b <= 3e+183)
		tmp = fma(Float64(angle * angle), Float64(pi * Float64(pi * Float64(b * Float64(b * 3.08641975308642e-5)))), Float64(a * a));
	else
		tmp = Float64(3.08641975308642e-5 * Float64(Float64(angle * Float64(angle * b)) * Float64(b * Float64(pi * pi))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2.7e-36], N[(a * a), $MachinePrecision], If[LessEqual[b, 3e+183], N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(3.08641975308642e-5 * N[(N[(angle * N[(angle * b), $MachinePrecision]), $MachinePrecision] * N[(b * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.70000000000000007e-36

   1. Initial program 75.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.9

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

   5. Applied rewrites 61.9%

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

   if 2.70000000000000007e-36 < b < 2.99999999999999996e183

   1. Initial program 72.1%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 36.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 59.8%

      \[\leadsto \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \color{blue}{\left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)}\right)\right), a \cdot a\right) \]

   if 2.99999999999999996e183 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 46.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 63.2%

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

   10. Applied rewrites 83.8%

       \[\leadsto \left(\left(angle \cdot \left(angle \cdot b\right)\right) \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-36}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(angle \cdot b\right)\right) \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.9 \cdot 10^{+149}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(angle \cdot b\right)\right) \cdot \left(b \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.9e+149)
   (* a a)
   (* (* angle (* angle b)) (* b (* (* PI PI) 3.08641975308642e-5)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.9e+149) {
		tmp = a * a;
	} else {
		tmp = (angle * (angle * b)) * (b * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.9e+149) {
		tmp = a * a;
	} else {
		tmp = (angle * (angle * b)) * (b * ((Math.PI * Math.PI) * 3.08641975308642e-5));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 4.9e+149:
		tmp = a * a
	else:
		tmp = (angle * (angle * b)) * (b * ((math.pi * math.pi) * 3.08641975308642e-5))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 4.9e+149)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(angle * Float64(angle * b)) * Float64(b * Float64(Float64(pi * pi) * 3.08641975308642e-5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 4.9e+149)
		tmp = a * a;
	else
		tmp = (angle * (angle * b)) * (b * ((pi * pi) * 3.08641975308642e-5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 4.9e+149], N[(a * a), $MachinePrecision], N[(N[(angle * N[(angle * b), $MachinePrecision]), $MachinePrecision] * N[(b * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.9000000000000001e149

   1. Initial program 74.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if 4.9000000000000001e149 < b

   1. Initial program 99.8%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 67.3%

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

   10. Applied rewrites 87.0%

       \[\leadsto \left(b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(angle \cdot \color{blue}{\left(angle \cdot b\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.9 \cdot 10^{+149}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(angle \cdot b\right)\right) \cdot \left(b \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{+149}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.2e+149)
   (* a a)
   (* angle (* angle (* b (* b (* (* PI PI) 3.08641975308642e-5)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.2e+149) {
		tmp = a * a;
	} else {
		tmp = angle * (angle * (b * (b * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.2e+149) {
		tmp = a * a;
	} else {
		tmp = angle * (angle * (b * (b * ((Math.PI * Math.PI) * 3.08641975308642e-5))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2.2e+149:
		tmp = a * a
	else:
		tmp = angle * (angle * (b * (b * ((math.pi * math.pi) * 3.08641975308642e-5))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.2e+149)
		tmp = Float64(a * a);
	else
		tmp = Float64(angle * Float64(angle * Float64(b * Float64(b * Float64(Float64(pi * pi) * 3.08641975308642e-5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2.2e+149)
		tmp = a * a;
	else
		tmp = angle * (angle * (b * (b * ((pi * pi) * 3.08641975308642e-5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2.2e+149], N[(a * a), $MachinePrecision], N[(angle * N[(angle * N[(b * N[(b * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2e149

   1. Initial program 74.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if 2.2e149 < b

   1. Initial program 99.8%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 67.3%

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

   10. Applied rewrites 68.4%

       \[\leadsto \left(angle \cdot \left(b \cdot \left(b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \cdot angle \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{+149}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+149}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 5e+149)
   (* a a)
   (* (* 3.08641975308642e-5 (* angle angle)) (* (* PI PI) (* b b)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5e+149) {
		tmp = a * a;
	} else {
		tmp = (3.08641975308642e-5 * (angle * angle)) * ((((double) M_PI) * ((double) M_PI)) * (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5e+149) {
		tmp = a * a;
	} else {
		tmp = (3.08641975308642e-5 * (angle * angle)) * ((Math.PI * Math.PI) * (b * b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 5e+149:
		tmp = a * a
	else:
		tmp = (3.08641975308642e-5 * (angle * angle)) * ((math.pi * math.pi) * (b * b))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 5e+149)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(3.08641975308642e-5 * Float64(angle * angle)) * Float64(Float64(pi * pi) * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 5e+149)
		tmp = a * a;
	else
		tmp = (3.08641975308642e-5 * (angle * angle)) * ((pi * pi) * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 5e+149], N[(a * a), $MachinePrecision], N[(N[(3.08641975308642e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.9999999999999999e149

   1. Initial program 74.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if 4.9999999999999999e149 < b

   1. Initial program 99.8%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 67.3%

      \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)} \]

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 67.3%

       \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+149}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.0% accurate, 74.7× 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

real(8) function code(a, b, angle)
    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 77.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.2

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

5. Applied rewrites 56.2%

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

Reproduce

herbie shell --seed 2024234 
(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)))