ab-angle->ABCF C

Percentage Accurate: 80.3% → 80.3%

Time: 41.8s

Alternatives: 14

Speedup: 1.0×

• 38.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \cos \left(\pi \cdot \left({\left({\left(angle\_m \cdot 0.005555555555555556\right)}^{0.3333333333333333}\right)}^{2} \cdot \sqrt[3]{angle\_m \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow
   (*
    a
    (cos
     (*
      PI
      (*
       (pow (pow (* angle_m 0.005555555555555556) 0.3333333333333333) 2.0)
       (cbrt (* angle_m 0.005555555555555556))))))
   2.0)
  (pow (* b (sin (* PI (* angle_m 0.005555555555555556)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * cos((((double) M_PI) * (pow(pow((angle_m * 0.005555555555555556), 0.3333333333333333), 2.0) * cbrt((angle_m * 0.005555555555555556)))))), 2.0) + pow((b * sin((((double) M_PI) * (angle_m * 0.005555555555555556)))), 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.PI * (Math.pow(Math.pow((angle_m * 0.005555555555555556), 0.3333333333333333), 2.0) * Math.cbrt((angle_m * 0.005555555555555556)))))), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle_m * 0.005555555555555556)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * cos(Float64(pi * Float64(((Float64(angle_m * 0.005555555555555556) ^ 0.3333333333333333) ^ 2.0) * cbrt(Float64(angle_m * 0.005555555555555556)))))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Cos[N[(Pi * N[(N[Power[N[Power[N[(angle$95$m * 0.005555555555555556), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(angle$95$m * 0.005555555555555556), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.0%

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

   1. metadata-eval 80.0%

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

   2. div-inv 80.0%

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

   3. add-cbrt-cube 65.9%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{\sqrt[3]{\left(\frac{angle}{180} \cdot \frac{angle}{180}\right) \cdot \frac{angle}{180}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. pow1/3 51.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{{\left(\left(\frac{angle}{180} \cdot \frac{angle}{180}\right) \cdot \frac{angle}{180}\right)}^{0.3333333333333333}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. pow-to-exp 51.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{e^{\log \left(\left(\frac{angle}{180} \cdot \frac{angle}{180}\right) \cdot \frac{angle}{180}\right) \cdot 0.3333333333333333}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. pow3 51.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\log \color{blue}{\left({\left(\frac{angle}{180}\right)}^{3}\right)} \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. log-pow 44.0%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\color{blue}{\left(3 \cdot \log \left(\frac{angle}{180}\right)\right)} \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. div-inv 44.0%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\left(3 \cdot \log \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. metadata-eval 44.0%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\left(3 \cdot \log \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 44.0%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{e^{\left(3 \cdot \log \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.3333333333333333}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
7. Step-by-step derivation

   1. *-commutative 44.0%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(angle \cdot 0.005555555555555556\right)\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. associate-*r* 43.9%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \log \left(angle \cdot 0.005555555555555556\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   3. metadata-eval 43.9%

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

   4. *-un-lft-identity 43.9%

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

   5. add-exp-log 80.0%

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

   6. add-cube-cbrt 80.1%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. metadata-eval 80.1%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot \color{blue}{\frac{1}{180}}} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. div-inv 80.1%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(\left(\sqrt[3]{\color{blue}{\frac{angle}{180}}} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. metadata-eval 80.1%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(\left(\sqrt[3]{\frac{angle}{180}} \cdot \sqrt[3]{angle \cdot \color{blue}{\frac{1}{180}}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   10. div-inv 80.1%

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(\left(\sqrt[3]{\frac{angle}{180}} \cdot \sqrt[3]{\color{blue}{\frac{angle}{180}}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   11. pow2 80.1%

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(\color{blue}{{\left(\sqrt[3]{\frac{angle}{180}}\right)}^{2}} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   12. div-inv 80.1%

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left({\left(\sqrt[3]{\color{blue}{angle \cdot \frac{1}{180}}}\right)}^{2} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   13. metadata-eval 80.1%

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left({\left(\sqrt[3]{angle \cdot \color{blue}{0.005555555555555556}}\right)}^{2} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

8. Applied egg-rr 80.1%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{\left({\left(\sqrt[3]{angle \cdot 0.005555555555555556}\right)}^{2} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
9. Step-by-step derivation

   1. pow1/3 43.9%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left({\color{blue}{\left({\left(angle \cdot 0.005555555555555556\right)}^{0.3333333333333333}\right)}}^{2} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

10. Applied egg-rr 43.9%

    \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left({\color{blue}{\left({\left(angle \cdot 0.005555555555555556\right)}^{0.3333333333333333}\right)}}^{2} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

11. Final simplification 43.9%

    \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left({\left({\left(angle \cdot 0.005555555555555556\right)}^{0.3333333333333333}\right)}^{2} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 2: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt[3]{angle\_m \cdot 0.005555555555555556}\\ {\left(b \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \left(t\_0 \cdot {t\_0}^{2}\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (cbrt (* angle_m 0.005555555555555556))))
   (+
    (pow (* b (sin (* PI (* angle_m 0.005555555555555556)))) 2.0)
    (pow (* a (cos (* PI (* t_0 (pow t_0 2.0))))) 2.0))))

C

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

Java

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

Julia

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[Power[N[(angle$95$m * 0.005555555555555556), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(Pi * N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.0%

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

   1. metadata-eval 80.0%

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

   2. div-inv 80.0%

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

   3. add-cbrt-cube 65.9%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{\sqrt[3]{\left(\frac{angle}{180} \cdot \frac{angle}{180}\right) \cdot \frac{angle}{180}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. pow1/3 51.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{{\left(\left(\frac{angle}{180} \cdot \frac{angle}{180}\right) \cdot \frac{angle}{180}\right)}^{0.3333333333333333}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. pow-to-exp 51.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{e^{\log \left(\left(\frac{angle}{180} \cdot \frac{angle}{180}\right) \cdot \frac{angle}{180}\right) \cdot 0.3333333333333333}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. pow3 51.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\log \color{blue}{\left({\left(\frac{angle}{180}\right)}^{3}\right)} \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. log-pow 44.0%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\color{blue}{\left(3 \cdot \log \left(\frac{angle}{180}\right)\right)} \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. div-inv 44.0%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\left(3 \cdot \log \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. metadata-eval 44.0%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\left(3 \cdot \log \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 44.0%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{e^{\left(3 \cdot \log \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.3333333333333333}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
7. Step-by-step derivation

   1. *-commutative 44.0%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(angle \cdot 0.005555555555555556\right)\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. associate-*r* 43.9%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \log \left(angle \cdot 0.005555555555555556\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   3. metadata-eval 43.9%

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

   4. *-un-lft-identity 43.9%

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

   5. add-exp-log 80.0%

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

   6. add-cube-cbrt 80.1%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. metadata-eval 80.1%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot \color{blue}{\frac{1}{180}}} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. div-inv 80.1%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(\left(\sqrt[3]{\color{blue}{\frac{angle}{180}}} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. metadata-eval 80.1%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(\left(\sqrt[3]{\frac{angle}{180}} \cdot \sqrt[3]{angle \cdot \color{blue}{\frac{1}{180}}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   10. div-inv 80.1%

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(\left(\sqrt[3]{\frac{angle}{180}} \cdot \sqrt[3]{\color{blue}{\frac{angle}{180}}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   11. pow2 80.1%

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(\color{blue}{{\left(\sqrt[3]{\frac{angle}{180}}\right)}^{2}} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   12. div-inv 80.1%

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left({\left(\sqrt[3]{\color{blue}{angle \cdot \frac{1}{180}}}\right)}^{2} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   13. metadata-eval 80.1%

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left({\left(\sqrt[3]{angle \cdot \color{blue}{0.005555555555555556}}\right)}^{2} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

8. Applied egg-rr 80.1%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{\left({\left(\sqrt[3]{angle \cdot 0.005555555555555556}\right)}^{2} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

9. Final simplification 80.1%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot {\left(\sqrt[3]{angle \cdot 0.005555555555555556}\right)}^{2}\right)\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 3: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(b \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot e^{0.3333333333333333 \cdot \left(3 \cdot \log \left(angle\_m \cdot 0.005555555555555556\right)\right)}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (* PI (* angle_m 0.005555555555555556)))) 2.0)
  (pow
   (*
    a
    (cos
     (*
      PI
      (exp
       (*
        0.3333333333333333
        (* 3.0 (log (* angle_m 0.005555555555555556))))))))
   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 * 0.005555555555555556)))), 2.0) + pow((a * cos((((double) M_PI) * exp((0.3333333333333333 * (3.0 * log((angle_m * 0.005555555555555556)))))))), 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 * 0.005555555555555556)))), 2.0) + Math.pow((a * Math.cos((Math.PI * Math.exp((0.3333333333333333 * (3.0 * Math.log((angle_m * 0.005555555555555556)))))))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((b * math.sin((math.pi * (angle_m * 0.005555555555555556)))), 2.0) + math.pow((a * math.cos((math.pi * math.exp((0.3333333333333333 * (3.0 * math.log((angle_m * 0.005555555555555556)))))))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 2.0) + (Float64(a * cos(Float64(pi * exp(Float64(0.3333333333333333 * Float64(3.0 * log(Float64(angle_m * 0.005555555555555556)))))))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin((pi * (angle_m * 0.005555555555555556)))) ^ 2.0) + ((a * cos((pi * exp((0.3333333333333333 * (3.0 * log((angle_m * 0.005555555555555556)))))))) ^ 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 * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(Pi * N[Exp[N[(0.3333333333333333 * N[(3.0 * N[Log[N[(angle$95$m * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.0%

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

   1. metadata-eval 80.0%

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

   2. div-inv 80.0%

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

   3. add-cbrt-cube 65.9%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{\sqrt[3]{\left(\frac{angle}{180} \cdot \frac{angle}{180}\right) \cdot \frac{angle}{180}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. pow1/3 51.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{{\left(\left(\frac{angle}{180} \cdot \frac{angle}{180}\right) \cdot \frac{angle}{180}\right)}^{0.3333333333333333}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. pow-to-exp 51.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{e^{\log \left(\left(\frac{angle}{180} \cdot \frac{angle}{180}\right) \cdot \frac{angle}{180}\right) \cdot 0.3333333333333333}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. pow3 51.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\log \color{blue}{\left({\left(\frac{angle}{180}\right)}^{3}\right)} \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. log-pow 44.0%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\color{blue}{\left(3 \cdot \log \left(\frac{angle}{180}\right)\right)} \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. div-inv 44.0%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\left(3 \cdot \log \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. metadata-eval 44.0%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot e^{\left(3 \cdot \log \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 44.0%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{e^{\left(3 \cdot \log \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.3333333333333333}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

7. Final simplification 44.0%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot e^{0.3333333333333333 \cdot \left(3 \cdot \log \left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 4: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(b \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot {\left(\sqrt[3]{angle\_m \cdot 0.005555555555555556}\right)}^{3}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (* PI (* angle_m 0.005555555555555556)))) 2.0)
  (pow
   (* a (cos (* PI (pow (cbrt (* angle_m 0.005555555555555556)) 3.0))))
   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 * 0.005555555555555556)))), 2.0) + pow((a * cos((((double) M_PI) * pow(cbrt((angle_m * 0.005555555555555556)), 3.0)))), 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 * 0.005555555555555556)))), 2.0) + Math.pow((a * Math.cos((Math.PI * Math.pow(Math.cbrt((angle_m * 0.005555555555555556)), 3.0)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 2.0) + (Float64(a * cos(Float64(pi * (cbrt(Float64(angle_m * 0.005555555555555556)) ^ 3.0)))) ^ 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 * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(Pi * N[Power[N[Power[N[(angle$95$m * 0.005555555555555556), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.0%

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

   1. metadata-eval 80.0%

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

   2. div-inv 80.0%

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

   3. add-cube-cbrt 80.1%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{angle}{180}} \cdot \sqrt[3]{\frac{angle}{180}}\right) \cdot \sqrt[3]{\frac{angle}{180}}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. pow3 80.1%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{{\left(\sqrt[3]{\frac{angle}{180}}\right)}^{3}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. div-inv 80.1%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot {\left(\sqrt[3]{\color{blue}{angle \cdot \frac{1}{180}}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. metadata-eval 80.1%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot {\left(\sqrt[3]{angle \cdot \color{blue}{0.005555555555555556}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 80.1%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{{\left(\sqrt[3]{angle \cdot 0.005555555555555556}\right)}^{3}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

7. Final simplification 80.1%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot {\left(\sqrt[3]{angle \cdot 0.005555555555555556}\right)}^{3}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 5: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(b \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (* PI (* angle_m 0.005555555555555556)))) 2.0)
  (pow
   (* a (cos (* PI (expm1 (log1p (* angle_m 0.005555555555555556))))))
   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 * 0.005555555555555556)))), 2.0) + pow((a * cos((((double) M_PI) * expm1(log1p((angle_m * 0.005555555555555556)))))), 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 * 0.005555555555555556)))), 2.0) + Math.pow((a * Math.cos((Math.PI * Math.expm1(Math.log1p((angle_m * 0.005555555555555556)))))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((b * math.sin((math.pi * (angle_m * 0.005555555555555556)))), 2.0) + math.pow((a * math.cos((math.pi * math.expm1(math.log1p((angle_m * 0.005555555555555556)))))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 2.0) + (Float64(a * cos(Float64(pi * expm1(log1p(Float64(angle_m * 0.005555555555555556)))))) ^ 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 * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(Pi * N[(Exp[N[Log[1 + N[(angle$95$m * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.0%

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

   1. expm1-log1p-u 67.7%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 67.7%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

7. Final simplification 67.7%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 6: 80.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))))
   (+ (pow (* b (sin t_0)) 2.0) (pow (* a (cos t_0)) 2.0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

3. Simplified 80.0%

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

5. Final simplification 80.0%

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

Alternative 7: 80.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0))))
   (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 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]]

Derivation

1. Initial program 80.0%

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

3. Final simplification 80.0%

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

Alternative 8: 80.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (* PI (* angle_m 0.005555555555555556)))) 2.0)
  (pow (* a (cos (/ PI (/ 180.0 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 * 0.005555555555555556)))), 2.0) + pow((a * cos((((double) M_PI) / (180.0 / 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 * 0.005555555555555556)))), 2.0) + Math.pow((a * Math.cos((Math.PI / (180.0 / 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 * 0.005555555555555556)))), 2.0) + math.pow((a * math.cos((math.pi / (180.0 / 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 * 0.005555555555555556)))) ^ 2.0) + (Float64(a * cos(Float64(pi / Float64(180.0 / angle_m)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin((pi * (angle_m * 0.005555555555555556)))) ^ 2.0) + ((a * cos((pi / (180.0 / 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 * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.0%

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

   1. metadata-eval 80.0%

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

   2. div-inv 80.0%

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

   3. clear-num 80.0%

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

   4. un-div-inv 80.1%

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

6. Applied egg-rr 80.1%

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

7. Final simplification 80.1%

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

Alternative 9: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in angle around 0 79.9%

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

6. Taylor expanded in angle around inf 79.5%

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

7. Final simplification 79.5%

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

Alternative 10: 80.2% accurate, 1.3× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin((pi * (angle_m * 0.005555555555555556)))) ^ 2.0) + (a ^ 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 * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in angle around 0 79.9%

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

6. Final simplification 79.9%

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

Alternative 11: 80.2% accurate, 1.3× speedup

Math

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

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 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, 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, 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, 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) + (a ^ 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 ^ 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[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in angle around 0 79.9%

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

   1. metadata-eval 79.9%

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

   2. div-inv 79.9%

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

7. Applied egg-rr 79.9%

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

8. Final simplification 79.9%

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

Alternative 12: 74.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in angle around 0 79.9%

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

6. Taylor expanded in angle around 0 75.7%

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

   1. unpow2 75.7%

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

   2. associate-*r* 75.7%

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

   3. associate-*l* 74.4%

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

   4. *-commutative 74.4%

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

   5. *-commutative 74.4%

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

   6. *-commutative 74.4%

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

   7. associate-*r* 74.4%

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

   8. *-commutative 74.4%

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

   9. *-commutative 74.4%

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

   10. associate-*l* 74.4%

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

8. Applied egg-rr 74.4%

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

   1. associate-*l* 74.4%

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

   2. associate-*r* 74.4%

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

   3. *-commutative 74.4%

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

   4. associate-*l* 74.4%

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

   5. *-commutative 74.4%

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

   6. *-commutative 74.4%

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

10. Simplified 74.4%

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

11. Taylor expanded in angle around 0 74.5%

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

12. Final simplification 74.5%

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

Alternative 13: 74.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in angle around 0 79.9%

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

6. Taylor expanded in angle around 0 75.7%

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

   1. unpow2 75.7%

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

   2. associate-*r* 75.7%

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

   3. associate-*l* 74.4%

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

   4. *-commutative 74.4%

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

   5. *-commutative 74.4%

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

   6. *-commutative 74.4%

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

   7. associate-*r* 74.4%

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

   8. *-commutative 74.4%

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

   9. *-commutative 74.4%

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

   10. associate-*l* 74.4%

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

8. Applied egg-rr 74.4%

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

   1. associate-*l* 74.4%

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

   2. *-commutative 74.4%

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

   3. associate-*r* 74.5%

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

   4. *-commutative 74.5%

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

   5. associate-*l* 74.5%

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

   6. *-commutative 74.5%

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

   7. *-commutative 74.5%

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

10. Simplified 74.5%

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

11. Final simplification 74.5%

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

Alternative 14: 75.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in angle around 0 79.9%

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

6. Taylor expanded in angle around 0 75.7%

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

   1. unpow2 75.7%

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

   2. associate-*r* 75.7%

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

   3. associate-*l* 74.4%

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

   4. *-commutative 74.4%

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

   5. *-commutative 74.4%

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

   6. *-commutative 74.4%

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

   7. associate-*r* 74.4%

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

   8. *-commutative 74.4%

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

   9. *-commutative 74.4%

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

   10. associate-*l* 74.4%

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

8. Applied egg-rr 74.4%

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

   1. associate-*r* 75.7%

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

   2. *-commutative 75.7%

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

   3. associate-*l* 75.7%

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

   4. *-commutative 75.7%

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

10. Simplified 75.7%

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

11. Final simplification 75.7%

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

Reproduce

herbie shell --seed 2024069 
(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)))