ab-angle->ABCF A

Percentage Accurate: 80.3% → 80.3%

Time: 3.2s

Alternatives: 13

Speedup: 0.7×

• 63.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (/ angle 180) PI)))
  (+ (pow (* a (sin t_0)) 2) (pow (* b (cos t_0)) 2))))

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), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]]

Initial Program: 80.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (/ angle 180) PI)))
  (+ (pow (* a (sin t_0)) 2) (pow (* b (cos t_0)) 2))))

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), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 80.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (+
 (pow (* a (sin (* (/ angle 180) PI))) 2)
 (pow
  (* b (sin (* (pow PI 2/3) (* (cbrt PI) (- 1/2 (* 1/180 angle))))))
  2)))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * sin((pow(((double) M_PI), 0.6666666666666666) * (cbrt(((double) M_PI)) * (0.5 - (0.005555555555555556 * angle)))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.sin((Math.pow(Math.PI, 0.6666666666666666) * (Math.cbrt(Math.PI) * (0.5 - (0.005555555555555556 * angle)))))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * sin(Float64((pi ^ 0.6666666666666666) * Float64(cbrt(pi) * Float64(0.5 - Float64(0.005555555555555556 * angle)))))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[Power[Pi, 2/3], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[(1/2 - N[(1/180 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. lift-/.f64 N/A

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

   9. distribute-neg-frac N/A

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

   10. lift-PI.f64 N/A

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

   11. mult-flip N/A

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

   12. distribute-lft-out N/A

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

   13. lower-*.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. distribute-neg-frac N/A

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

   16. mult-flip N/A

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

   17. *-commutative N/A

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

   18. distribute-lft-neg-in N/A

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

   19. distribute-frac-neg2 N/A

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

   20. lower-*.f64 N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

3. Applied rewrites 80.3%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. pow1/3 N/A

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

   8. lift-PI.f64 N/A

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

   9. pow1/3 N/A

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

   10. pow-prod-up N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-cbrt.f64 80.2%

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

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

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

   20. metadata-eval N/A

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

   21. lift-*.f64 N/A

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

   22. lower--.f64 80.2%

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

5. Applied rewrites 80.2%

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

Alternative 2: 80.3% accurate, 1.2× speedup

Math

\[{\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{\left|angle\right|}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\pi \cdot \left(\frac{1}{2} - \frac{1}{90} \cdot \left|angle\right|\right)\right)\right) \cdot \left(b \cdot b\right) \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+
 (pow (* a (sin (/ PI (/ 180 (fabs angle))))) 2)
 (*
  (+ 1/2 (* 1/2 (sin (* PI (- 1/2 (* 1/90 (fabs angle)))))))
  (* b b))))

C

double code(double a, double b, double angle) {
	return pow((a * sin((((double) M_PI) / (180.0 / fabs(angle))))), 2.0) + ((0.5 + (0.5 * sin((((double) M_PI) * (0.5 - (0.011111111111111112 * fabs(angle))))))) * (b * b));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin((Math.PI / (180.0 / Math.abs(angle))))), 2.0) + ((0.5 + (0.5 * Math.sin((Math.PI * (0.5 - (0.011111111111111112 * Math.abs(angle))))))) * (b * b));
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin((math.pi / (180.0 / math.fabs(angle))))), 2.0) + ((0.5 + (0.5 * math.sin((math.pi * (0.5 - (0.011111111111111112 * math.fabs(angle))))))) * (b * b))

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(pi / Float64(180.0 / abs(angle))))) ^ 2.0) + Float64(Float64(0.5 + Float64(0.5 * sin(Float64(pi * Float64(0.5 - Float64(0.011111111111111112 * abs(angle))))))) * Float64(b * b)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin((pi / (180.0 / abs(angle))))) ^ 2.0) + ((0.5 + (0.5 * sin((pi * (0.5 - (0.011111111111111112 * abs(angle))))))) * (b * b));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(Pi / N[(180 / N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[(N[(1/2 + N[(1/2 * N[Sin[N[(Pi * N[(1/2 - N[(1/90 * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. swap-sqr N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.2%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \pi\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   2. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   4. div-flip-rev N/A

      \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   5. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \frac{1}{\color{blue}{\frac{180}{angle}}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   6. mult-flip-rev N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   7. lower-/.f64 80.2%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

5. Applied rewrites 80.2%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)}\right) \cdot \left(b \cdot b\right) \]

   2. cos-neg-rev N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right)}\right) \cdot \left(b \cdot b\right) \]

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

      \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \left(b \cdot b\right) \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(\pi \cdot angle\right) \cdot \frac{1}{90}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(b \cdot b\right) \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\color{blue}{\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{90}\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(b \cdot b\right) \]

   6. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \color{blue}{\frac{-1}{90}} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(b \cdot b\right) \]

   7. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\color{blue}{\frac{-1}{90} \cdot \left(\pi \cdot angle\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(b \cdot b\right) \]

   8. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\color{blue}{\frac{-1}{90} \cdot \left(\pi \cdot angle\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(b \cdot b\right) \]

   9. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\frac{-1}{90} \cdot \left(\pi \cdot angle\right) + \frac{\color{blue}{\pi}}{2}\right)\right) \cdot \left(b \cdot b\right) \]

   10. mult-flip N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\frac{-1}{90} \cdot \left(\pi \cdot angle\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)\right) \cdot \left(b \cdot b\right) \]

   11. metadata-eval N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\frac{-1}{90} \cdot \left(\pi \cdot angle\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \left(b \cdot b\right) \]

   12. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\frac{-1}{90} \cdot \left(\pi \cdot angle\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)\right) \cdot \left(b \cdot b\right) \]

   13. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\frac{-1}{90} \cdot \left(\pi \cdot angle\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)\right) \cdot \left(b \cdot b\right) \]

   14. +-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \pi + \frac{-1}{90} \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \left(b \cdot b\right) \]

   15. lift-+.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \pi + \frac{-1}{90} \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \left(b \cdot b\right) \]

   16. lift-sin.f64 80.2%

       \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \pi + \frac{-1}{90} \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \left(b \cdot b\right) \]

   17. lift-+.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \pi + \frac{-1}{90} \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \left(b \cdot b\right) \]

   18. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\frac{1}{2} \cdot \pi + \color{blue}{\frac{-1}{90} \cdot \left(\pi \cdot angle\right)}\right)\right) \cdot \left(b \cdot b\right) \]

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

       \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \pi - \left(\mathsf{neg}\left(\frac{-1}{90}\right)\right) \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \left(b \cdot b\right) \]

7. Applied rewrites 80.2%

   \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\sin \left(\pi \cdot \left(\frac{1}{2} - \frac{1}{90} \cdot angle\right)\right)}\right) \cdot \left(b \cdot b\right) \]
8. Add Preprocessing

Alternative 3: 80.2% accurate, 1.3× speedup

Math

\[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)\right) \cdot \left(b \cdot b\right) \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+
 (pow (* a (sin (* (/ angle 180) PI))) 2)
 (* (+ 1/2 (* 1/2 (cos (* (* 1/90 PI) angle)))) (* b b))))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + ((0.5 + (0.5 * cos(((0.011111111111111112 * ((double) M_PI)) * angle)))) * (b * b));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + ((0.5 + (0.5 * Math.cos(((0.011111111111111112 * Math.PI) * angle)))) * (b * b));
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin(((angle / 180.0) * math.pi))), 2.0) + ((0.5 + (0.5 * math.cos(((0.011111111111111112 * math.pi) * angle)))) * (b * b))

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + Float64(Float64(0.5 + Float64(0.5 * cos(Float64(Float64(0.011111111111111112 * pi) * angle)))) * Float64(b * b)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((angle / 180.0) * pi))) ^ 2.0) + ((0.5 + (0.5 * cos(((0.011111111111111112 * pi) * angle)))) * (b * b));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[(N[(1/2 + N[(1/2 * N[Cos[N[(N[(1/90 * Pi), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. swap-sqr N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.2%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)}\right) \cdot \left(b \cdot b\right) \]

   2. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\frac{1}{90} \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \left(b \cdot b\right) \]

   3. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \color{blue}{\left(\pi \cdot angle\right)}\right)\right) \cdot \left(b \cdot b\right) \]

   4. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)}\right) \cdot \left(b \cdot b\right) \]

   5. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)}\right) \cdot \left(b \cdot b\right) \]

   6. lower-*.f64 80.3%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\color{blue}{\left(\frac{1}{90} \cdot \pi\right)} \cdot angle\right)\right) \cdot \left(b \cdot b\right) \]

5. Applied rewrites 80.3%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)}\right) \cdot \left(b \cdot b\right) \]
6. Add Preprocessing

Alternative 4: 80.2% accurate, 1.3× speedup

Math

\[{\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+
 (pow (* a (sin (* (* PI angle) 1/180))) 2)
 (* (+ 1/2 (* 1/2 (cos (* (* PI angle) 1/90)))) (* b b))))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((((double) M_PI) * angle) * 0.005555555555555556))), 2.0) + ((0.5 + (0.5 * cos(((((double) M_PI) * angle) * 0.011111111111111112)))) * (b * b));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((Math.PI * angle) * 0.005555555555555556))), 2.0) + ((0.5 + (0.5 * Math.cos(((Math.PI * angle) * 0.011111111111111112)))) * (b * b));
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin(((math.pi * angle) * 0.005555555555555556))), 2.0) + ((0.5 + (0.5 * math.cos(((math.pi * angle) * 0.011111111111111112)))) * (b * b))

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(pi * angle) * 0.005555555555555556))) ^ 2.0) + Float64(Float64(0.5 + Float64(0.5 * cos(Float64(Float64(pi * angle) * 0.011111111111111112)))) * Float64(b * b)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((pi * angle) * 0.005555555555555556))) ^ 2.0) + ((0.5 + (0.5 * cos(((pi * angle) * 0.011111111111111112)))) * (b * b));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 1/180), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[(N[(1/2 + N[(1/2 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 1/90), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. swap-sqr N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.2%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \pi\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   2. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{angle}{180}} \cdot \pi\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   3. associate-*l/ N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   4. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{\color{blue}{\pi \cdot angle}}{180}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   5. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{\color{blue}{\pi \cdot angle}}{180}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   6. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   7. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \color{blue}{\frac{1}{180}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   8. lower-*.f64 80.3%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

5. Applied rewrites 80.3%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]
6. Add Preprocessing

Alternative 5: 80.1% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (+ (pow (* a (sin (/ PI (/ 180 angle)))) 2) (* (+ 1/2 1/2) (* b b))))

C

double code(double a, double b, double angle) {
	return pow((a * sin((((double) M_PI) / (180.0 / angle)))), 2.0) + ((0.5 + 0.5) * (b * b));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin((Math.PI / (180.0 / angle)))), 2.0) + ((0.5 + 0.5) * (b * b));
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin((math.pi / (180.0 / angle)))), 2.0) + ((0.5 + 0.5) * (b * b))

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(pi / Float64(180.0 / angle)))) ^ 2.0) + Float64(Float64(0.5 + 0.5) * Float64(b * b)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin((pi / (180.0 / angle)))) ^ 2.0) + ((0.5 + 0.5) * (b * b));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(Pi / N[(180 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[(N[(1/2 + 1/2), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. swap-sqr N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.2%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \pi\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   2. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   4. div-flip-rev N/A

      \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   5. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \frac{1}{\color{blue}{\frac{180}{angle}}}\right)\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   6. mult-flip-rev N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   7. lower-/.f64 80.2%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

5. Applied rewrites 80.2%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)}\right)}^{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

6. Taylor expanded in angle around 0

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

8. Applied rewrites 80.1%

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

Alternative 6: 80.1% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180) PI))) 2) (* (+ 1/2 1/2) (* b b))))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + ((0.5 + 0.5) * (b * b));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + ((0.5 + 0.5) * (b * b));
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin(((angle / 180.0) * math.pi))), 2.0) + ((0.5 + 0.5) * (b * b))

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + Float64(Float64(0.5 + 0.5) * Float64(b * b)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((angle / 180.0) * pi))) ^ 2.0) + ((0.5 + 0.5) * (b * b));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[(N[(1/2 + 1/2), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. swap-sqr N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.2%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right)} \]

4. Taylor expanded in angle around 0

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

6. Applied rewrites 80.1%

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

Alternative 7: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b\\ \mathbf{if}\;\left|angle\right| \leq \frac{5224175567749775}{18014398509481984}:\\ \;\;\;\;\left(\frac{1}{32400} \cdot \left(a \cdot \left({\left(\left|angle\right|\right)}^{2} \cdot {\pi}^{2}\right)\right)\right) \cdot a + t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(\left|angle\right| \cdot \pi\right)\right)\right)\right) \cdot a + t\_0\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (* (+ 1/2 1/2) b) b)))
  (if (<= (fabs angle) 5224175567749775/18014398509481984)
    (+
     (* (* 1/32400 (* a (* (pow (fabs angle) 2) (pow PI 2)))) a)
     t_0)
    (+
     (* (* a (- 1/2 (* 1/2 (cos (* 1/90 (* (fabs angle) PI)))))) a)
     t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((0.5 + 0.5) * b) * b;
	double tmp;
	if (fabs(angle) <= 0.29) {
		tmp = ((3.08641975308642e-5 * (a * (pow(fabs(angle), 2.0) * pow(((double) M_PI), 2.0)))) * a) + t_0;
	} else {
		tmp = ((a * (0.5 - (0.5 * cos((0.011111111111111112 * (fabs(angle) * ((double) M_PI))))))) * a) + t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = ((0.5 + 0.5) * b) * b;
	double tmp;
	if (Math.abs(angle) <= 0.29) {
		tmp = ((3.08641975308642e-5 * (a * (Math.pow(Math.abs(angle), 2.0) * Math.pow(Math.PI, 2.0)))) * a) + t_0;
	} else {
		tmp = ((a * (0.5 - (0.5 * Math.cos((0.011111111111111112 * (Math.abs(angle) * Math.PI)))))) * a) + t_0;
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = ((0.5 + 0.5) * b) * b
	tmp = 0
	if math.fabs(angle) <= 0.29:
		tmp = ((3.08641975308642e-5 * (a * (math.pow(math.fabs(angle), 2.0) * math.pow(math.pi, 2.0)))) * a) + t_0
	else:
		tmp = ((a * (0.5 - (0.5 * math.cos((0.011111111111111112 * (math.fabs(angle) * math.pi)))))) * a) + t_0
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(Float64(0.5 + 0.5) * b) * b)
	tmp = 0.0
	if (abs(angle) <= 0.29)
		tmp = Float64(Float64(Float64(3.08641975308642e-5 * Float64(a * Float64((abs(angle) ^ 2.0) * (pi ^ 2.0)))) * a) + t_0);
	else
		tmp = Float64(Float64(Float64(a * Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(abs(angle) * pi)))))) * a) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = ((0.5 + 0.5) * b) * b;
	tmp = 0.0;
	if (abs(angle) <= 0.29)
		tmp = ((3.08641975308642e-5 * (a * ((abs(angle) ^ 2.0) * (pi ^ 2.0)))) * a) + t_0;
	else
		tmp = ((a * (0.5 - (0.5 * cos((0.011111111111111112 * (abs(angle) * pi)))))) * a) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(N[(1/2 + 1/2), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[N[Abs[angle], $MachinePrecision], 5224175567749775/18014398509481984], N[(N[(N[(1/32400 * N[(a * N[(N[Power[N[Abs[angle], $MachinePrecision], 2], $MachinePrecision] * N[Power[Pi, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(a * N[(1/2 - N[(1/2 * N[Cos[N[(1/90 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 0.28999999999999998

   1. Initial program 80.3%

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

   3. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]
   5. Step-by-step derivation

   6. Applied rewrites 57.6%

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 57.7%

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

   10. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-PI.f64 71.0%

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

   12. Applied rewrites 71.0%

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

   if 0.28999999999999998 < angle

   1. Initial program 80.3%

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

   3. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]
   5. Step-by-step derivation

   6. Applied rewrites 57.6%

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 57.7%

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

   10. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\left(a \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-cos.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-PI.f64 68.4%

          \[\leadsto \left(a \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

   12. Applied rewrites 68.4%

       \[\leadsto \color{blue}{\left(a \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 71.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq \frac{7642277889662869}{463168356949264781694283940034751631413079938662562256157830336031652518559744}:\\ \;\;\;\;\left(\left(t\_0 \cdot \frac{-1}{32400} - \left(a \cdot a\right) \cdot \frac{-1}{32400}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot angle\right)\right) \cdot angle + t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot \left|b\right|\right) \cdot \left|b\right|\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b))))
  (if (<=
       (fabs b)
       7642277889662869/463168356949264781694283940034751631413079938662562256157830336031652518559744)
    (+
     (*
      (*
       (- (* t_0 -1/32400) (* (* a a) -1/32400))
       (* (* PI PI) angle))
      angle)
     t_0)
    (+
     (* (* a (- 1/2 (* 1/2 (cos (* 1/90 (* angle PI)))))) a)
     (* (* (+ 1/2 1/2) (fabs b)) (fabs b))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) * fabs(b);
	double tmp;
	if (fabs(b) <= 1.65e-62) {
		tmp = ((((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * ((((double) M_PI) * ((double) M_PI)) * angle)) * angle) + t_0;
	} else {
		tmp = ((a * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))))))) * a) + (((0.5 + 0.5) * fabs(b)) * fabs(b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) * Math.abs(b);
	double tmp;
	if (Math.abs(b) <= 1.65e-62) {
		tmp = ((((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * ((Math.PI * Math.PI) * angle)) * angle) + t_0;
	} else {
		tmp = ((a * (0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)))))) * a) + (((0.5 + 0.5) * Math.abs(b)) * Math.abs(b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) * math.fabs(b)
	tmp = 0
	if math.fabs(b) <= 1.65e-62:
		tmp = ((((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * ((math.pi * math.pi) * angle)) * angle) + t_0
	else:
		tmp = ((a * (0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))))) * a) + (((0.5 + 0.5) * math.fabs(b)) * math.fabs(b))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (abs(b) <= 1.65e-62)
		tmp = Float64(Float64(Float64(Float64(Float64(t_0 * -3.08641975308642e-5) - Float64(Float64(a * a) * -3.08641975308642e-5)) * Float64(Float64(pi * pi) * angle)) * angle) + t_0);
	else
		tmp = Float64(Float64(Float64(a * Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi)))))) * a) + Float64(Float64(Float64(0.5 + 0.5) * abs(b)) * abs(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) * abs(b);
	tmp = 0.0;
	if (abs(b) <= 1.65e-62)
		tmp = ((((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * ((pi * pi) * angle)) * angle) + t_0;
	else
		tmp = ((a * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi)))))) * a) + (((0.5 + 0.5) * abs(b)) * abs(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 7642277889662869/463168356949264781694283940034751631413079938662562256157830336031652518559744], N[(N[(N[(N[(N[(t$95$0 * -1/32400), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * -1/32400), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(a * N[(1/2 - N[(1/2 * N[Cos[N[(1/90 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[(1/2 + 1/2), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.65e-62

   1. Initial program 80.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 40.7%

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

   6. Applied rewrites 40.8%

      \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right) + \color{blue}{b \cdot b} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right) + b \cdot b \]

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 43.1%

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

   if 1.65e-62 < b

   1. Initial program 80.3%

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

   3. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]
   5. Step-by-step derivation

   6. Applied rewrites 57.6%

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 57.7%

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

   10. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\left(a \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-cos.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-PI.f64 68.4%

          \[\leadsto \left(a \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

   12. Applied rewrites 68.4%

       \[\leadsto \color{blue}{\left(a \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 2950000000000000059304930393302603223489185923540163102230528049495911188363352314194466435490646428401669785291287477296199626375736899804365586432:\\ \;\;\;\;\left(\left(t\_0 \cdot \frac{-1}{32400} - \left(a \cdot a\right) \cdot \frac{-1}{32400}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot angle\right)\right) \cdot angle + t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot \left|b\right|\right) \cdot \left|b\right|\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b))))
  (if (<=
       (fabs b)
       2950000000000000059304930393302603223489185923540163102230528049495911188363352314194466435490646428401669785291287477296199626375736899804365586432)
    (+
     (*
      (*
       (- (* t_0 -1/32400) (* (* a a) -1/32400))
       (* (* PI PI) angle))
      angle)
     t_0)
    (+
     (* (* (- 1/2 1/2) a) a)
     (* (* (+ 1/2 1/2) (fabs b)) (fabs b))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) * fabs(b);
	double tmp;
	if (fabs(b) <= 2.95e+147) {
		tmp = ((((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * ((((double) M_PI) * ((double) M_PI)) * angle)) * angle) + t_0;
	} else {
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * fabs(b)) * fabs(b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) * Math.abs(b);
	double tmp;
	if (Math.abs(b) <= 2.95e+147) {
		tmp = ((((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * ((Math.PI * Math.PI) * angle)) * angle) + t_0;
	} else {
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * Math.abs(b)) * Math.abs(b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) * math.fabs(b)
	tmp = 0
	if math.fabs(b) <= 2.95e+147:
		tmp = ((((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * ((math.pi * math.pi) * angle)) * angle) + t_0
	else:
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * math.fabs(b)) * math.fabs(b))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (abs(b) <= 2.95e+147)
		tmp = Float64(Float64(Float64(Float64(Float64(t_0 * -3.08641975308642e-5) - Float64(Float64(a * a) * -3.08641975308642e-5)) * Float64(Float64(pi * pi) * angle)) * angle) + t_0);
	else
		tmp = Float64(Float64(Float64(Float64(0.5 - 0.5) * a) * a) + Float64(Float64(Float64(0.5 + 0.5) * abs(b)) * abs(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) * abs(b);
	tmp = 0.0;
	if (abs(b) <= 2.95e+147)
		tmp = ((((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * ((pi * pi) * angle)) * angle) + t_0;
	else
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * abs(b)) * abs(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2950000000000000059304930393302603223489185923540163102230528049495911188363352314194466435490646428401669785291287477296199626375736899804365586432], N[(N[(N[(N[(N[(t$95$0 * -1/32400), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * -1/32400), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(N[(1/2 - 1/2), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[(1/2 + 1/2), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.9500000000000001e147

   1. Initial program 80.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 40.7%

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

   6. Applied rewrites 40.8%

      \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right) + \color{blue}{b \cdot b} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right) + b \cdot b \]

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 43.1%

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

   if 2.9500000000000001e147 < b

   1. Initial program 80.3%

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

   3. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]
   5. Step-by-step derivation

   6. Applied rewrites 57.6%

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 57.7%

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

Alternative 10: 66.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 2950000000000000059304930393302603223489185923540163102230528049495911188363352314194466435490646428401669785291287477296199626375736899804365586432:\\ \;\;\;\;\left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot t\_0 + \left(\frac{1}{32400} \cdot a\right) \cdot a\right)\right) \cdot \left(angle \cdot angle\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot \left|b\right|\right) \cdot \left|b\right|\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b))))
  (if (<=
       (fabs b)
       2950000000000000059304930393302603223489185923540163102230528049495911188363352314194466435490646428401669785291287477296199626375736899804365586432)
    (+
     (*
      (* (* PI PI) (+ (* -1/32400 t_0) (* (* 1/32400 a) a)))
      (* angle angle))
     t_0)
    (+
     (* (* (- 1/2 1/2) a) a)
     (* (* (+ 1/2 1/2) (fabs b)) (fabs b))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) * fabs(b);
	double tmp;
	if (fabs(b) <= 2.95e+147) {
		tmp = (((((double) M_PI) * ((double) M_PI)) * ((-3.08641975308642e-5 * t_0) + ((3.08641975308642e-5 * a) * a))) * (angle * angle)) + t_0;
	} else {
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * fabs(b)) * fabs(b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) * Math.abs(b);
	double tmp;
	if (Math.abs(b) <= 2.95e+147) {
		tmp = (((Math.PI * Math.PI) * ((-3.08641975308642e-5 * t_0) + ((3.08641975308642e-5 * a) * a))) * (angle * angle)) + t_0;
	} else {
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * Math.abs(b)) * Math.abs(b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) * math.fabs(b)
	tmp = 0
	if math.fabs(b) <= 2.95e+147:
		tmp = (((math.pi * math.pi) * ((-3.08641975308642e-5 * t_0) + ((3.08641975308642e-5 * a) * a))) * (angle * angle)) + t_0
	else:
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * math.fabs(b)) * math.fabs(b))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (abs(b) <= 2.95e+147)
		tmp = Float64(Float64(Float64(Float64(pi * pi) * Float64(Float64(-3.08641975308642e-5 * t_0) + Float64(Float64(3.08641975308642e-5 * a) * a))) * Float64(angle * angle)) + t_0);
	else
		tmp = Float64(Float64(Float64(Float64(0.5 - 0.5) * a) * a) + Float64(Float64(Float64(0.5 + 0.5) * abs(b)) * abs(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) * abs(b);
	tmp = 0.0;
	if (abs(b) <= 2.95e+147)
		tmp = (((pi * pi) * ((-3.08641975308642e-5 * t_0) + ((3.08641975308642e-5 * a) * a))) * (angle * angle)) + t_0;
	else
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * abs(b)) * abs(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2950000000000000059304930393302603223489185923540163102230528049495911188363352314194466435490646428401669785291287477296199626375736899804365586432], N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(-1/32400 * t$95$0), $MachinePrecision] + N[(N[(1/32400 * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle * angle), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(N[(1/2 - 1/2), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[(1/2 + 1/2), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.9500000000000001e147

   1. Initial program 80.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 40.7%

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

   6. Applied rewrites 40.8%

      \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right) + \color{blue}{b \cdot b} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right) + b \cdot b \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right) + b \cdot b \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + a \cdot \left(a \cdot \frac{1}{32400}\right)\right)\right) \cdot \left(angle \cdot angle\right) + b \cdot b \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot \frac{1}{32400}\right) \cdot a\right)\right) \cdot \left(angle \cdot angle\right) + b \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot \frac{1}{32400}\right) \cdot a\right)\right) \cdot \left(angle \cdot angle\right) + b \cdot b \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(\frac{1}{32400} \cdot a\right) \cdot a\right)\right) \cdot \left(angle \cdot angle\right) + b \cdot b \]

      7. lower-*.f64 40.9%

         \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(\frac{1}{32400} \cdot a\right) \cdot a\right)\right) \cdot \left(angle \cdot angle\right) + b \cdot b \]

   8. Applied rewrites 40.9%

      \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(\frac{1}{32400} \cdot a\right) \cdot a\right)\right) \cdot \left(angle \cdot angle\right) + b \cdot b \]

   if 2.9500000000000001e147 < b

   1. Initial program 80.3%

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

   3. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]
   5. Step-by-step derivation

   6. Applied rewrites 57.6%

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 57.7%

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

Alternative 11: 66.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 2950000000000000059304930393302603223489185923540163102230528049495911188363352314194466435490646428401669785291287477296199626375736899804365586432:\\ \;\;\;\;\left(\left(angle \cdot angle\right) \cdot \pi\right) \cdot \left(\left(t\_0 \cdot \frac{-1}{32400} - \left(a \cdot a\right) \cdot \frac{-1}{32400}\right) \cdot \pi\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot \left|b\right|\right) \cdot \left|b\right|\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b))))
  (if (<=
       (fabs b)
       2950000000000000059304930393302603223489185923540163102230528049495911188363352314194466435490646428401669785291287477296199626375736899804365586432)
    (+
     (*
      (* (* angle angle) PI)
      (* (- (* t_0 -1/32400) (* (* a a) -1/32400)) PI))
     t_0)
    (+
     (* (* (- 1/2 1/2) a) a)
     (* (* (+ 1/2 1/2) (fabs b)) (fabs b))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) * fabs(b);
	double tmp;
	if (fabs(b) <= 2.95e+147) {
		tmp = (((angle * angle) * ((double) M_PI)) * (((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * ((double) M_PI))) + t_0;
	} else {
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * fabs(b)) * fabs(b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) * Math.abs(b);
	double tmp;
	if (Math.abs(b) <= 2.95e+147) {
		tmp = (((angle * angle) * Math.PI) * (((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * Math.PI)) + t_0;
	} else {
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * Math.abs(b)) * Math.abs(b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) * math.fabs(b)
	tmp = 0
	if math.fabs(b) <= 2.95e+147:
		tmp = (((angle * angle) * math.pi) * (((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * math.pi)) + t_0
	else:
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * math.fabs(b)) * math.fabs(b))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (abs(b) <= 2.95e+147)
		tmp = Float64(Float64(Float64(Float64(angle * angle) * pi) * Float64(Float64(Float64(t_0 * -3.08641975308642e-5) - Float64(Float64(a * a) * -3.08641975308642e-5)) * pi)) + t_0);
	else
		tmp = Float64(Float64(Float64(Float64(0.5 - 0.5) * a) * a) + Float64(Float64(Float64(0.5 + 0.5) * abs(b)) * abs(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) * abs(b);
	tmp = 0.0;
	if (abs(b) <= 2.95e+147)
		tmp = (((angle * angle) * pi) * (((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * pi)) + t_0;
	else
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * abs(b)) * abs(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2950000000000000059304930393302603223489185923540163102230528049495911188363352314194466435490646428401669785291287477296199626375736899804365586432], N[(N[(N[(N[(angle * angle), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(N[(t$95$0 * -1/32400), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * -1/32400), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(N[(1/2 - 1/2), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[(1/2 + 1/2), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.9500000000000001e147

   1. Initial program 80.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 40.7%

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

   6. Applied rewrites 40.8%

      \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right) + \color{blue}{b \cdot b} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) + b \cdot b \]

      4. lift-*.f64 N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) + b \cdot b \]

      5. associate-*l* N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right)\right) + b \cdot b \]

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot angle\right) \cdot \pi\right) \cdot \left(\pi \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) + b \cdot b \]

      9. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right) \cdot \pi\right) + b \cdot b \]

      10. lower-*.f64 40.8%

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right) \cdot \pi\right) + b \cdot b \]

   8. Applied rewrites 40.8%

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

   if 2.9500000000000001e147 < b

   1. Initial program 80.3%

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

   3. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]
   5. Step-by-step derivation

   6. Applied rewrites 57.6%

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 57.7%

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

Alternative 12: 66.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 2950000000000000059304930393302603223489185923540163102230528049495911188363352314194466435490646428401669785291287477296199626375736899804365586432:\\ \;\;\;\;\pi \cdot \left(\pi \cdot \left(\left(t\_0 \cdot \frac{-1}{32400} - \left(a \cdot a\right) \cdot \frac{-1}{32400}\right) \cdot \left(angle \cdot angle\right)\right)\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot \left|b\right|\right) \cdot \left|b\right|\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b))))
  (if (<=
       (fabs b)
       2950000000000000059304930393302603223489185923540163102230528049495911188363352314194466435490646428401669785291287477296199626375736899804365586432)
    (+
     (*
      PI
      (*
       PI
       (* (- (* t_0 -1/32400) (* (* a a) -1/32400)) (* angle angle))))
     t_0)
    (+
     (* (* (- 1/2 1/2) a) a)
     (* (* (+ 1/2 1/2) (fabs b)) (fabs b))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) * fabs(b);
	double tmp;
	if (fabs(b) <= 2.95e+147) {
		tmp = (((double) M_PI) * (((double) M_PI) * (((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * (angle * angle)))) + t_0;
	} else {
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * fabs(b)) * fabs(b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) * Math.abs(b);
	double tmp;
	if (Math.abs(b) <= 2.95e+147) {
		tmp = (Math.PI * (Math.PI * (((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * (angle * angle)))) + t_0;
	} else {
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * Math.abs(b)) * Math.abs(b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) * math.fabs(b)
	tmp = 0
	if math.fabs(b) <= 2.95e+147:
		tmp = (math.pi * (math.pi * (((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * (angle * angle)))) + t_0
	else:
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * math.fabs(b)) * math.fabs(b))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (abs(b) <= 2.95e+147)
		tmp = Float64(Float64(pi * Float64(pi * Float64(Float64(Float64(t_0 * -3.08641975308642e-5) - Float64(Float64(a * a) * -3.08641975308642e-5)) * Float64(angle * angle)))) + t_0);
	else
		tmp = Float64(Float64(Float64(Float64(0.5 - 0.5) * a) * a) + Float64(Float64(Float64(0.5 + 0.5) * abs(b)) * abs(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) * abs(b);
	tmp = 0.0;
	if (abs(b) <= 2.95e+147)
		tmp = (pi * (pi * (((t_0 * -3.08641975308642e-5) - ((a * a) * -3.08641975308642e-5)) * (angle * angle)))) + t_0;
	else
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * abs(b)) * abs(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2950000000000000059304930393302603223489185923540163102230528049495911188363352314194466435490646428401669785291287477296199626375736899804365586432], N[(N[(Pi * N[(Pi * N[(N[(N[(t$95$0 * -1/32400), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * -1/32400), $MachinePrecision]), $MachinePrecision] * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(N[(1/2 - 1/2), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[(1/2 + 1/2), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.9500000000000001e147

   1. Initial program 80.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 40.7%

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

   6. Applied rewrites 40.8%

      \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right) + \color{blue}{b \cdot b} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right) + b \cdot b \]

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

         \[\leadsto \left(\pi \cdot \pi\right) \cdot \left(\left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right) \cdot \left(angle \cdot angle\right)\right) + b \cdot b \]

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

         \[\leadsto \pi \cdot \left(\pi \cdot \left(\left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right) \cdot \left(angle \cdot angle\right)\right)\right) + b \cdot b \]

      8. lower-*.f64 40.8%

         \[\leadsto \pi \cdot \left(\pi \cdot \left(\left(\frac{-1}{32400} \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \frac{1}{32400}\right) \cdot \left(angle \cdot angle\right)\right)\right) + b \cdot b \]

   8. Applied rewrites 40.8%

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

   if 2.9500000000000001e147 < b

   1. Initial program 80.3%

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

   3. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]
   5. Step-by-step derivation

   6. Applied rewrites 57.6%

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 57.7%

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

Alternative 13: 57.7% accurate, 14.9× speedup

Math

\[\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+ (* (* (- 1/2 1/2) a) a) (* (* (+ 1/2 1/2) b) b)))

C

double code(double a, double b, double angle) {
	return (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * b) * b);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = (((0.5d0 - 0.5d0) * a) * a) + (((0.5d0 + 0.5d0) * b) * b)
end function

Java

public static double code(double a, double b, double angle) {
	return (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * b) * b);
}

Python

def code(a, b, angle):
	return (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * b) * b)

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(Float64(0.5 - 0.5) * a) * a) + Float64(Float64(Float64(0.5 + 0.5) * b) * b))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * b) * b);
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(N[(1/2 - 1/2), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[(1/2 + 1/2), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

3. Applied rewrites 68.5%

   \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b} \]

4. Taylor expanded in angle around 0

   \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]
5. Step-by-step derivation

6. Applied rewrites 57.6%

   \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b \]

7. Taylor expanded in angle around 0

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

9. Applied rewrites 57.7%

   \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot a\right) \cdot a + \left(\left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180) PI))) 2) (pow (* b (cos (* (/ angle 180) PI))) 2)))