ab-angle->ABCF A

Percentage Accurate: 79.6% → 79.1%

Time: 14.0s

Alternatives: 11

Speedup: N/A×

• 37.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;{\left(\left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\_m\right) \cdot a\right)}^{2} + b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right), 0.5\right) \cdot a, a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 5e+14)
   (+ (pow (* (* (* 0.005555555555555556 PI) angle_m) a) 2.0) (* b b))
   (fma
    (* (fma -0.5 (cos (* 0.011111111111111112 (* angle_m PI))) 0.5) a)
    a
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e+14) {
		tmp = pow((((0.005555555555555556 * ((double) M_PI)) * angle_m) * a), 2.0) + (b * b);
	} else {
		tmp = fma((fma(-0.5, cos((0.011111111111111112 * (angle_m * ((double) M_PI)))), 0.5) * a), a, (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+14)
		tmp = Float64((Float64(Float64(Float64(0.005555555555555556 * pi) * angle_m) * a) ^ 2.0) + Float64(b * b));
	else
		tmp = fma(Float64(fma(-0.5, cos(Float64(0.011111111111111112 * Float64(angle_m * pi))), 0.5) * a), a, Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+14], N[(N[Power[N[(N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 * N[Cos[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * a), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5e14

   1. Initial program 85.5%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.5

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

   8. Applied rewrites 85.5%

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

   9. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 82.3

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

   11. Applied rewrites 82.3%

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

   if 5e14 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 64.0%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.1

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

   8. Applied rewrites 64.1%

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

   10. Applied rewrites 64.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right) \cdot a, a, b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;{\left(\left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right) \cdot a\right)}^{2} + b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right) \cdot a, a, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ b \cdot b + {\left(\sin \left(\left(0.005555555555555556 \cdot \sqrt{\pi}\right) \cdot \left(angle\_m \cdot \sqrt{\pi}\right)\right) \cdot a\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (* b b)
  (pow
   (* (sin (* (* 0.005555555555555556 (sqrt PI)) (* angle_m (sqrt PI)))) a)
   2.0)))

C

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

Java

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

Python

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

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(Float64(b * b) + (Float64(sin(Float64(Float64(0.005555555555555556 * sqrt(pi)) * Float64(angle_m * sqrt(pi)))) * a) ^ 2.0))
end

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(b * b), $MachinePrecision] + N[Power[N[(N[Sin[N[(N[(0.005555555555555556 * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 80.1

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

8. Applied rewrites 80.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. div-inv N/A

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

   10. lift-/.f64 N/A

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

   11. *-commutative N/A

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

   12. times-frac N/A

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

   13. lift-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. associate-/r/ N/A

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

   18. /-rgt-identity N/A

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

   19. lower-*.f64 N/A

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

   20. div-inv N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 80.1

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

10. Applied rewrites 80.1%

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

11. Final simplification 80.1%

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

Alternative 3: 79.6% accurate, 2.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* (sin (* (* 0.005555555555555556 PI) angle_m)) a) 2.0) (* b b)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(N[Sin[N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 80.1

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

8. Applied rewrites 80.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 80.1

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

10. Applied rewrites 80.1%

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

11. Final simplification 80.1%

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

Alternative 4: 66.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{-65}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\_m\right) \cdot a\right)}^{2} + b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 4.2e-65)
   (* b b)
   (+ (pow (* (* (* 0.005555555555555556 PI) angle_m) a) 2.0) (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.2e-65) {
		tmp = b * b;
	} else {
		tmp = pow((((0.005555555555555556 * ((double) M_PI)) * angle_m) * a), 2.0) + (b * b);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.2e-65) {
		tmp = b * b;
	} else {
		tmp = Math.pow((((0.005555555555555556 * Math.PI) * angle_m) * a), 2.0) + (b * b);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 4.2e-65:
		tmp = b * b
	else:
		tmp = math.pow((((0.005555555555555556 * math.pi) * angle_m) * a), 2.0) + (b * b)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 4.2e-65)
		tmp = Float64(b * b);
	else
		tmp = Float64((Float64(Float64(Float64(0.005555555555555556 * pi) * angle_m) * a) ^ 2.0) + Float64(b * b));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 4.2e-65)
		tmp = b * b;
	else
		tmp = ((((0.005555555555555556 * pi) * angle_m) * a) ^ 2.0) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 4.2e-65], N[(b * b), $MachinePrecision], N[(N[Power[N[(N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.20000000000000006e-65

   1. Initial program 79.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   if 4.20000000000000006e-65 < a

   1. Initial program 81.2%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

   9. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 78.6

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

   11. Applied rewrites 78.6%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{-65}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right) \cdot a\right)}^{2} + b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{-65}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\pi \cdot a\right) \cdot 0.005555555555555556\right) \cdot angle\_m\right)}^{2} + b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 4.2e-65)
   (* b b)
   (+ (pow (* (* (* PI a) 0.005555555555555556) angle_m) 2.0) (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.2e-65) {
		tmp = b * b;
	} else {
		tmp = pow((((((double) M_PI) * a) * 0.005555555555555556) * angle_m), 2.0) + (b * b);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.2e-65) {
		tmp = b * b;
	} else {
		tmp = Math.pow((((Math.PI * a) * 0.005555555555555556) * angle_m), 2.0) + (b * b);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 4.2e-65:
		tmp = b * b
	else:
		tmp = math.pow((((math.pi * a) * 0.005555555555555556) * angle_m), 2.0) + (b * b)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 4.2e-65)
		tmp = Float64(b * b);
	else
		tmp = Float64((Float64(Float64(Float64(pi * a) * 0.005555555555555556) * angle_m) ^ 2.0) + Float64(b * b));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 4.2e-65)
		tmp = b * b;
	else
		tmp = ((((pi * a) * 0.005555555555555556) * angle_m) ^ 2.0) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 4.2e-65], N[(b * b), $MachinePrecision], N[(N[Power[N[(N[(N[(Pi * a), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.20000000000000006e-65

   1. Initial program 79.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   if 4.20000000000000006e-65 < a

   1. Initial program 81.2%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/r/ N/A

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

      18. /-rgt-identity N/A

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

      19. lower-*.f64 N/A

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

      20. div-inv N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 81.6

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

   10. Applied rewrites 81.6%

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

   11. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lower-PI.f64 78.6

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

   13. Applied rewrites 78.6%

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

Alternative 6: 64.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{-65}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \pi\right) \cdot \pi, angle\_m \cdot angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \pi\right) \cdot \left(\left(angle\_m \cdot a\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 4.2e-65)
   (* b b)
   (if (<= a 9.5e+139)
     (fma
      (* (* (* (* a a) 3.08641975308642e-5) PI) PI)
      (* angle_m angle_m)
      (* b b))
     (* (* PI PI) (* (* angle_m a) (* (* 3.08641975308642e-5 a) angle_m))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.2e-65) {
		tmp = b * b;
	} else if (a <= 9.5e+139) {
		tmp = fma(((((a * a) * 3.08641975308642e-5) * ((double) M_PI)) * ((double) M_PI)), (angle_m * angle_m), (b * b));
	} else {
		tmp = (((double) M_PI) * ((double) M_PI)) * ((angle_m * a) * ((3.08641975308642e-5 * a) * angle_m));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 4.2e-65)
		tmp = Float64(b * b);
	elseif (a <= 9.5e+139)
		tmp = fma(Float64(Float64(Float64(Float64(a * a) * 3.08641975308642e-5) * pi) * pi), Float64(angle_m * angle_m), Float64(b * b));
	else
		tmp = Float64(Float64(pi * pi) * Float64(Float64(angle_m * a) * Float64(Float64(3.08641975308642e-5 * a) * angle_m)));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 4.2e-65], N[(b * b), $MachinePrecision], If[LessEqual[a, 9.5e+139], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * Pi), $MachinePrecision] * Pi), $MachinePrecision] * N[(angle$95$m * angle$95$m), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(angle$95$m * a), $MachinePrecision] * N[(N[(3.08641975308642e-5 * a), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 4.20000000000000006e-65

   1. Initial program 79.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   if 4.20000000000000006e-65 < a < 9.5000000000000002e139

   1. Initial program 68.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 35.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 64.3%

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

   if 9.5000000000000002e139 < a

   1. Initial program 95.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 78.5%

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

   10. Applied rewrites 92.9%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{-65}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \pi\right) \cdot \pi, angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \pi\right) \cdot \left(\left(angle \cdot a\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 9.2 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \pi\right) \cdot \left(\left(angle\_m \cdot a\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 9.2e+139)
   (* b b)
   (* (* PI PI) (* (* angle_m a) (* (* 3.08641975308642e-5 a) angle_m)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 9.2e+139) {
		tmp = b * b;
	} else {
		tmp = (((double) M_PI) * ((double) M_PI)) * ((angle_m * a) * ((3.08641975308642e-5 * a) * angle_m));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 9.2e+139) {
		tmp = b * b;
	} else {
		tmp = (Math.PI * Math.PI) * ((angle_m * a) * ((3.08641975308642e-5 * a) * angle_m));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 9.2e+139:
		tmp = b * b
	else:
		tmp = (math.pi * math.pi) * ((angle_m * a) * ((3.08641975308642e-5 * a) * angle_m))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 9.2e+139)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(pi * pi) * Float64(Float64(angle_m * a) * Float64(Float64(3.08641975308642e-5 * a) * angle_m)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 9.2e+139)
		tmp = b * b;
	else
		tmp = (pi * pi) * ((angle_m * a) * ((3.08641975308642e-5 * a) * angle_m));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 9.2e+139], N[(b * b), $MachinePrecision], N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(angle$95$m * a), $MachinePrecision] * N[(N[(3.08641975308642e-5 * a), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.2e139

   1. Initial program 77.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 9.2e139 < a

   1. Initial program 95.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 78.5%

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

   10. Applied rewrites 92.9%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.2 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \pi\right) \cdot \left(\left(angle \cdot a\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle\_m \cdot a\right) \cdot angle\_m\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 9.5e+139)
   (* b b)
   (* (* (* (* (* angle_m a) angle_m) a) 3.08641975308642e-5) (* PI PI))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 9.5e+139) {
		tmp = b * b;
	} else {
		tmp = ((((angle_m * a) * angle_m) * a) * 3.08641975308642e-5) * (((double) M_PI) * ((double) M_PI));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 9.5e+139) {
		tmp = b * b;
	} else {
		tmp = ((((angle_m * a) * angle_m) * a) * 3.08641975308642e-5) * (Math.PI * Math.PI);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 9.5e+139:
		tmp = b * b
	else:
		tmp = ((((angle_m * a) * angle_m) * a) * 3.08641975308642e-5) * (math.pi * math.pi)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 9.5e+139)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(angle_m * a) * angle_m) * a) * 3.08641975308642e-5) * Float64(pi * pi));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 9.5e+139)
		tmp = b * b;
	else
		tmp = ((((angle_m * a) * angle_m) * a) * 3.08641975308642e-5) * (pi * pi);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 9.5e+139], N[(b * b), $MachinePrecision], N[(N[(N[(N[(N[(angle$95$m * a), $MachinePrecision] * angle$95$m), $MachinePrecision] * a), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.5000000000000002e139

   1. Initial program 77.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 9.5000000000000002e139 < a

   1. Initial program 95.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 78.5%

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

   10. Applied rewrites 88.2%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot a\right) \cdot angle\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot angle\_m\right) \cdot a\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 9.5e+139)
   (* b b)
   (* (* (* (* angle_m angle_m) a) (* PI PI)) (* 3.08641975308642e-5 a))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 9.5e+139) {
		tmp = b * b;
	} else {
		tmp = (((angle_m * angle_m) * a) * (((double) M_PI) * ((double) M_PI))) * (3.08641975308642e-5 * a);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 9.5e+139) {
		tmp = b * b;
	} else {
		tmp = (((angle_m * angle_m) * a) * (Math.PI * Math.PI)) * (3.08641975308642e-5 * a);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 9.5e+139:
		tmp = b * b
	else:
		tmp = (((angle_m * angle_m) * a) * (math.pi * math.pi)) * (3.08641975308642e-5 * a)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 9.5e+139)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(Float64(Float64(angle_m * angle_m) * a) * Float64(pi * pi)) * Float64(3.08641975308642e-5 * a));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 9.5e+139)
		tmp = b * b;
	else
		tmp = (((angle_m * angle_m) * a) * (pi * pi)) * (3.08641975308642e-5 * a);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 9.5e+139], N[(b * b), $MachinePrecision], N[(N[(N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * a), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(3.08641975308642e-5 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.5000000000000002e139

   1. Initial program 77.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 9.5000000000000002e139 < a

   1. Initial program 95.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 78.5%

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

   10. Applied rewrites 78.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 9.2 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot angle\_m\right) \cdot angle\_m\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 9.2e+139)
   (* b b)
   (* (* (* (* (* a a) 3.08641975308642e-5) angle_m) angle_m) (* PI PI))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 9.2e+139) {
		tmp = b * b;
	} else {
		tmp = ((((a * a) * 3.08641975308642e-5) * angle_m) * angle_m) * (((double) M_PI) * ((double) M_PI));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 9.2e+139) {
		tmp = b * b;
	} else {
		tmp = ((((a * a) * 3.08641975308642e-5) * angle_m) * angle_m) * (Math.PI * Math.PI);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 9.2e+139:
		tmp = b * b
	else:
		tmp = ((((a * a) * 3.08641975308642e-5) * angle_m) * angle_m) * (math.pi * math.pi)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 9.2e+139)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * 3.08641975308642e-5) * angle_m) * angle_m) * Float64(pi * pi));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 9.2e+139)
		tmp = b * b;
	else
		tmp = ((((a * a) * 3.08641975308642e-5) * angle_m) * angle_m) * (pi * pi);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 9.2e+139], N[(b * b), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * angle$95$m), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.2e139

   1. Initial program 77.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 9.2e139 < a

   1. Initial program 95.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 78.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 72.1%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.2 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot angle\right) \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.8% accurate, 74.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ b \cdot b \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m) :precision binary64 (* b b))

C

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

Fortran

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

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return b * b;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return b * b

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.7

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

5. Applied rewrites 56.7%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))