ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.5%

Time: 18.3s

Alternatives: 14

Speedup: 1.0×

• 35.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \cos \left(\frac{\sqrt{\pi}}{180} \cdot e^{\log \left(\sqrt{\pi} \cdot angle\_m\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow
   (* a (cos (* (/ (sqrt PI) 180.0) (exp (log (* (sqrt PI) angle_m))))))
   2.0)
  (pow (* b (sin (* PI (/ angle_m 180.0)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * cos(((sqrt(((double) M_PI)) / 180.0) * exp(log((sqrt(((double) M_PI)) * angle_m)))))), 2.0) + pow((b * sin((((double) M_PI) * (angle_m / 180.0)))), 2.0);
}

Java

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

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.cos(((math.sqrt(math.pi) / 180.0) * math.exp(math.log((math.sqrt(math.pi) * angle_m)))))), 2.0) + math.pow((b * math.sin((math.pi * (angle_m / 180.0)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * cos(Float64(Float64(sqrt(pi) / 180.0) * exp(log(Float64(sqrt(pi) * angle_m)))))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle_m / 180.0)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * cos(((sqrt(pi) / 180.0) * exp(log((sqrt(pi) * angle_m)))))) ^ 2.0) + ((b * sin((pi * (angle_m / 180.0)))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Cos[N[(N[(N[Sqrt[Pi], $MachinePrecision] / 180.0), $MachinePrecision] * N[Exp[N[Log[N[(N[Sqrt[Pi], $MachinePrecision] * angle$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

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

   1. lift-PI.f64 N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. lift-PI.f64 N/A

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

   5. add-sqr-sqrt N/A

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

   6. div-inv N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-/.f64 82.3

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

4. Applied egg-rr 82.3%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. inv-pow N/A

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

   6. pow-to-exp N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. clear-num N/A

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

   11. lift-/.f64 N/A

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

   12. lower-/.f64 82.4

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. associate-/r/ N/A

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

   16. /-rgt-identity N/A

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

   17. lower-*.f64 82.4

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

6. Applied egg-rr 82.4%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. unpow1 N/A

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

   5. pow-flip N/A

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

   6. metadata-eval N/A

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

   7. pow-to-exp N/A

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

   8. pow-exp N/A

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

   9. inv-pow N/A

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

   10. log-div N/A

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

   11. rem-log-exp N/A

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

   12. log-div N/A

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

   13. inv-pow N/A

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

   14. pow-to-exp N/A

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

   15. rem-log-exp N/A

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

8. Applied egg-rr 82.4%

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

Alternative 2: 79.5% accurate, 0.9× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (* PI (/ angle_m 180.0)))) 2.0)
  (pow (* a (cos (* (/ (sqrt PI) 180.0) (/ (sqrt PI) (/ 1.0 angle_m))))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(N[(N[Sqrt[Pi], $MachinePrecision] / 180.0), $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] / N[(1.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.6%

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

   1. lift-PI.f64 N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. lift-PI.f64 N/A

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

   5. add-sqr-sqrt N/A

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

   6. div-inv N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-/.f64 79.5

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

4. Applied egg-rr 79.5%

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

5. Final simplification 79.5%

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

Reproduce

herbie shell --seed 2024218 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))