ab-angle->ABCF C

Percentage Accurate: 79.3% → 79.2%

Time: 33.1s

Alternatives: 8

Speedup: 1.5×

• 36.0% 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: 79.3% 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: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow(a, 2.0) + pow((b * sin(pow(sqrt((((double) M_PI) * (angle_m * 0.005555555555555556))), 2.0))), 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(Math.pow(Math.sqrt((Math.PI * (angle_m * 0.005555555555555556))), 2.0))), 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(math.pow(math.sqrt((math.pi * (angle_m * 0.005555555555555556))), 2.0))), 2.0)

Julia

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

MATLAB

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

Derivation

1. Initial program 76.4%

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

2. Taylor expanded in angle around 0 76.5%

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

   1. add-sqr-sqrt 39.0%

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

   2. pow2 39.0%

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

   3. div-inv 39.0%

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

   4. metadata-eval 39.0%

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

4. Applied egg-rr 39.0%

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

5. Final simplification 39.0%

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

Alternative 2: 79.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (pow
   (* b (expm1 (log1p (sin (* angle_m (* PI -0.005555555555555556))))))
   2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow(a, 2.0) + pow((b * expm1(log1p(sin((angle_m * (((double) M_PI) * -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, 2.0) + Math.pow((b * Math.expm1(Math.log1p(Math.sin((angle_m * (Math.PI * -0.005555555555555556)))))), 2.0);
}

Python

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

Julia

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

Derivation

1. Initial program 76.4%

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

2. Taylor expanded in angle around 0 76.5%

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

   1. add-sqr-sqrt 39.0%

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

   2. pow2 39.0%

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

   3. div-inv 39.0%

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

   4. metadata-eval 39.0%

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

4. Applied egg-rr 39.0%

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

   1. add-sqr-sqrt 39.0%

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

   2. sqrt-unprod 30.2%

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

   3. sqrt-pow2 30.2%

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

   4. metadata-eval 30.2%

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

   5. pow1 30.2%

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

   6. associate-*r* 30.2%

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

   7. sqrt-pow2 61.7%

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

   8. metadata-eval 61.7%

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

   9. pow1 61.7%

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

   10. associate-*r* 61.7%

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

   11. swap-sqr 61.3%

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

   12. *-commutative 61.3%

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

   13. *-commutative 61.3%

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

   14. metadata-eval 61.3%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\sqrt{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}}}\right)\right)}^{2} \]

   15. metadata-eval 61.3%

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

   16. swap-sqr 61.7%

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

   17. associate-*r* 61.7%

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

6. Applied egg-rr 76.5%

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

7. Final simplification 76.5%

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

Alternative 3: 79.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (pow
   (* b (log1p (expm1 (sin (* angle_m (* PI -0.005555555555555556))))))
   2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow(a, 2.0) + pow((b * log1p(expm1(sin((angle_m * (((double) M_PI) * -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, 2.0) + Math.pow((b * Math.log1p(Math.expm1(Math.sin((angle_m * (Math.PI * -0.005555555555555556)))))), 2.0);
}

Python

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

Julia

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

Derivation

1. Initial program 76.4%

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

2. Taylor expanded in angle around 0 76.5%

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

   1. add-sqr-sqrt 39.0%

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

   2. pow2 39.0%

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

   3. div-inv 39.0%

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

   4. metadata-eval 39.0%

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

4. Applied egg-rr 39.0%

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

   1. add-sqr-sqrt 39.0%

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

   2. sqrt-unprod 30.2%

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

   3. sqrt-pow2 30.2%

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

   4. metadata-eval 30.2%

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

   5. pow1 30.2%

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

   6. associate-*r* 30.2%

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

   7. sqrt-pow2 61.7%

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

   8. metadata-eval 61.7%

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

   9. pow1 61.7%

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

   10. associate-*r* 61.7%

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

   11. swap-sqr 61.3%

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

   12. *-commutative 61.3%

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

   13. *-commutative 61.3%

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

   14. metadata-eval 61.3%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\sqrt{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}}}\right)\right)}^{2} \]

   15. metadata-eval 61.3%

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

   16. swap-sqr 61.7%

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

   17. associate-*r* 61.7%

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

6. Applied egg-rr 76.5%

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

7. Final simplification 76.5%

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

Alternative 4: 79.2% accurate, 1.5× 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 76.4%

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

2. Taylor expanded in angle around 0 76.5%

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

3. Taylor expanded in b around 0 76.0%

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

4. Final simplification 76.0%

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

Alternative 5: 79.3% accurate, 1.5× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Derivation

1. Initial program 76.4%

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

2. Taylor expanded in angle around 0 76.5%

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

3. Taylor expanded in angle around inf 76.0%

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

   1. associate-*r* 76.4%

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

   2. *-commutative 76.4%

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

   3. *-commutative 76.4%

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

   4. *-commutative 76.4%

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

5. Simplified 76.4%

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

6. Final simplification 76.4%

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

Alternative 6: 79.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

2. Taylor expanded in angle around 0 76.5%

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

3. Final simplification 76.5%

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

Alternative 7: 66.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.9 \cdot 10^{-21}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b \cdot \left(angle_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.9e-21)
   (pow a 2.0)
   (pow (hypot a (* b (* angle_m (* PI 0.005555555555555556)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.9e-21) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(hypot(a, (b * (angle_m * (((double) M_PI) * 0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.9e-21) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(Math.hypot(a, (b * (angle_m * (Math.PI * 0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 5.9e-21:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(math.hypot(a, (b * (angle_m * (math.pi * 0.005555555555555556)))), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 5.9e-21)
		tmp = a ^ 2.0;
	else
		tmp = hypot(a, Float64(b * Float64(angle_m * Float64(pi * 0.005555555555555556)))) ^ 2.0;
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 5.9e-21)
		tmp = a ^ 2.0;
	else
		tmp = hypot(a, (b * (angle_m * (pi * 0.005555555555555556)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 5.9e-21], N[Power[a, 2.0], $MachinePrecision], N[Power[N[Sqrt[a ^ 2 + N[(b * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.9000000000000003e-21

   1. Initial program 75.2%

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

   2. Taylor expanded in angle around 0 75.4%

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

   3. Taylor expanded in angle around 0 70.8%

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

      1. *-commutative 70.8%

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

      2. unpow-prod-down 70.4%

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

      3. associate-*r* 70.4%

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

      4. *-commutative 70.4%

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

      5. metadata-eval 70.4%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} \]

   5. Applied egg-rr 70.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{{\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} \]

   6. Taylor expanded in a around inf 63.1%

      \[\leadsto \color{blue}{{a}^{2}} \]

   if 5.9000000000000003e-21 < b

   1. Initial program 80.2%

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

   2. Taylor expanded in angle around 0 80.1%

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

   3. Taylor expanded in angle around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. unpow-prod-down 75.0%

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

      3. associate-*r* 75.0%

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

      4. *-commutative 75.0%

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

      5. metadata-eval 75.0%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} \]

   5. Applied egg-rr 75.0%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{{\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 73.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(a \cdot 1\right)}^{2} + {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)} \]

      2. expm1-udef 61.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(a \cdot 1\right)}^{2} + {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)} - 1} \]

   7. Applied egg-rr 61.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, \left(b \cdot \left(angle \cdot \pi\right)\right) \cdot 0.005555555555555556\right)\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 73.2%

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

      2. expm1-log1p 75.1%

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

      3. associate-*l* 75.0%

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

      4. remove-double-div 75.0%

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

      5. associate-*l/ 75.0%

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

      6. *-lft-identity 75.0%

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

      7. associate-/r/ 75.0%

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

      8. associate-/l* 74.9%

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

      9. associate-/r/ 75.2%

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

      10. /-rgt-identity 75.2%

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

      11. *-commutative 75.2%

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

   9. Simplified 75.2%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.9 \cdot 10^{-21}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 8: 56.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {a}^{2} \end{array} \]

FPCore

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

C

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

Fortran

angle_m = abs(angle)
real(8) function code(a, b, angle_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = a ** 2.0d0
end function

Java

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

Python

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

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return a ^ 2.0
end

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[Power[a, 2.0], $MachinePrecision]

Derivation

1. Initial program 76.4%

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

2. Taylor expanded in angle around 0 76.5%

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

3. Taylor expanded in angle around 0 71.8%

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

   1. *-commutative 71.8%

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

   2. unpow-prod-down 71.4%

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

   3. associate-*r* 71.4%

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

   4. *-commutative 71.4%

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

   5. metadata-eval 71.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} \]

5. Applied egg-rr 71.4%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{{\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} \]

6. Taylor expanded in a around inf 58.9%

   \[\leadsto \color{blue}{{a}^{2}} \]

7. Final simplification 58.9%

   \[\leadsto {a}^{2} \]

Reproduce

herbie shell --seed 2023332 
(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)))