ab-angle->ABCF C

Percentage Accurate: 80.0% → 80.0%

Time: 28.6s

Alternatives: 11

Speedup: 1.0×

• 35.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{\frac{180}{angle}}{\pi}}\\ {\left(a \cdot \cos \left(\frac{\frac{1}{{t_0}^{2}}}{t_0}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (/ (/ 180.0 angle) PI))))
   (+
    (pow (* a (cos (/ (/ 1.0 (pow t_0 2.0)) t_0))) 2.0)
    (pow (* b (sin (* angle (/ PI -180.0)))) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt(((180.0 / angle) / ((double) M_PI)));
	return pow((a * cos(((1.0 / pow(t_0, 2.0)) / t_0))), 2.0) + pow((b * sin((angle * (((double) M_PI) / -180.0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt(((180.0 / angle) / Math.PI));
	return Math.pow((a * Math.cos(((1.0 / Math.pow(t_0, 2.0)) / t_0))), 2.0) + Math.pow((b * Math.sin((angle * (Math.PI / -180.0)))), 2.0);
}

Julia

function code(a, b, angle)
	t_0 = cbrt(Float64(Float64(180.0 / angle) / pi))
	return Float64((Float64(a * cos(Float64(Float64(1.0 / (t_0 ^ 2.0)) / t_0))) ^ 2.0) + (Float64(b * sin(Float64(angle * Float64(pi / -180.0)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[(N[(180.0 / angle), $MachinePrecision] / Pi), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[N[(N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.1%

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

3. Simplified 79.2%

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

4. Taylor expanded in angle around inf 79.2%

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

   1. add-sqr-sqrt 32.9%

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

   2. sqrt-unprod 66.8%

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

   3. *-commutative 66.8%

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

   4. *-commutative 66.8%

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

   5. swap-sqr 66.8%

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

   6. metadata-eval 66.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}} \cdot \left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{-180} \cdot angle\right)\right)}^{2} \]

   7. metadata-eval 66.8%

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

   8. swap-sqr 66.8%

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

   9. metadata-eval 66.8%

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

   10. associate-/r/ 66.8%

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

   11. metadata-eval 66.8%

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

   12. associate-/r/ 66.9%

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

   13. sqrt-unprod 46.3%

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

   14. add-sqr-sqrt 79.3%

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

   15. add-cube-cbrt 79.3%

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

6. Applied egg-rr 79.4%

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

7. Final simplification 79.4%

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

Alternative 2: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + {\left(a \cdot \cos \left({\left(\sqrt[3]{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* angle (/ PI -180.0)))) 2.0)
  (pow
   (* a (cos (pow (cbrt (* angle (* PI 0.005555555555555556))) 3.0)))
   2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin((angle * (((double) M_PI) / -180.0)))), 2.0) + pow((a * cos(pow(cbrt((angle * (((double) M_PI) * 0.005555555555555556))), 3.0))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin((angle * (Math.PI / -180.0)))), 2.0) + Math.pow((a * Math.cos(Math.pow(Math.cbrt((angle * (Math.PI * 0.005555555555555556))), 3.0))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(angle * Float64(pi / -180.0)))) ^ 2.0) + (Float64(a * cos((cbrt(Float64(angle * Float64(pi * 0.005555555555555556))) ^ 3.0))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[Power[N[Power[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.1%

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

3. Simplified 79.2%

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

4. Taylor expanded in angle around inf 79.2%

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

   1. add-sqr-sqrt 32.9%

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

   2. sqrt-unprod 66.8%

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

   3. *-commutative 66.8%

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

   4. *-commutative 66.8%

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

   5. swap-sqr 66.8%

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

   6. metadata-eval 66.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}} \cdot \left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{-180} \cdot angle\right)\right)}^{2} \]

   7. metadata-eval 66.8%

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

   8. swap-sqr 66.8%

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

   9. metadata-eval 66.8%

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

   10. associate-/r/ 66.8%

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

   11. metadata-eval 66.8%

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

   12. associate-/r/ 66.9%

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

   13. sqrt-unprod 46.3%

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

   14. add-sqr-sqrt 79.3%

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

   15. add-cube-cbrt 79.2%

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

   16. pow3 79.3%

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

6. Applied egg-rr 79.3%

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

7. Final simplification 79.3%

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

Alternative 3: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Simplified 79.2%

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

4. Taylor expanded in angle around inf 79.2%

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

   1. add-sqr-sqrt 32.9%

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

   2. sqrt-unprod 66.8%

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

   3. *-commutative 66.8%

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

   4. *-commutative 66.8%

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

   5. swap-sqr 66.8%

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

   6. metadata-eval 66.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}} \cdot \left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{-180} \cdot angle\right)\right)}^{2} \]

   7. metadata-eval 66.8%

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

   8. swap-sqr 66.8%

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

   9. metadata-eval 66.8%

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

   10. associate-/r/ 66.8%

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

   11. metadata-eval 66.8%

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

   12. associate-/r/ 66.9%

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

   13. sqrt-unprod 46.3%

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

   14. add-sqr-sqrt 79.3%

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

   15. *-commutative 79.3%

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

   16. associate-/r* 79.3%

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

6. Applied egg-rr 79.3%

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

7. Final simplification 79.3%

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

Alternative 4: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Simplified 79.2%

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

4. Taylor expanded in angle around inf 79.2%

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

5. Final simplification 79.2%

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

Alternative 5: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi / -180.0), $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 79.1%

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

3. Simplified 79.2%

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

4. Final simplification 79.2%

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

Alternative 6: 65.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow a 2.0)
  (pow (* b (sin (expm1 (log1p (* PI (* angle 0.005555555555555556)))))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + pow((b * sin(expm1(log1p((((double) M_PI) * (angle * 0.005555555555555556)))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0) + Math.pow((b * Math.sin(Math.expm1(Math.log1p((Math.PI * (angle * 0.005555555555555556)))))), 2.0);
}

Python

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

Julia

function code(a, b, angle)
	return Float64((a ^ 2.0) + (Float64(b * sin(expm1(log1p(Float64(pi * Float64(angle * 0.005555555555555556)))))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Exp[N[Log[1 + N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0 79.1%

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

   1. expm1-log1p-u 67.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} \]

   2. div-inv 67.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right)}^{2} \]

   3. metadata-eval 67.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)\right)}^{2} \]

4. Applied egg-rr 67.4%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} \]

5. Final simplification 67.4%

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

Alternative 7: 79.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0 79.1%

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

3. Taylor expanded in b around 0 79.1%

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

4. Final simplification 79.1%

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

Alternative 8: 79.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Simplified 79.2%

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

4. Taylor expanded in angle around inf 79.2%

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

5. Taylor expanded in angle around 0 79.2%

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

6. Final simplification 79.2%

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

Alternative 9: 74.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0 79.1%

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

3. Taylor expanded in angle around 0 73.2%

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

   1. associate-*r* 73.2%

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

   2. *-commutative 73.2%

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

   3. associate-*l* 73.2%

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

   4. *-commutative 73.2%

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

5. Simplified 73.2%

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

6. Final simplification 73.2%

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

Alternative 10: 74.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0 79.1%

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

3. Taylor expanded in angle around 0 73.5%

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

4. Final simplification 73.5%

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

Alternative 11: 74.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0 79.1%

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

3. Taylor expanded in angle around 0 73.5%

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

   1. associate-*r* 73.5%

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

   2. *-commutative 73.5%

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

   3. associate-*l* 73.6%

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

5. Simplified 73.6%

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

6. Final simplification 73.6%

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

Reproduce

herbie shell --seed 2023319 
(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)))