ab-angle->ABCF C

Percentage Accurate: 80.1% → 80.0%

Time: 39.8s

Alternatives: 9

Speedup: N/A×

• 37.0% 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: 80.1% 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, 0.8× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (pow
   (* b (sin (expm1 (log1p (* 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(expm1(log1p((((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.expm1(Math.log1p((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.expm1(math.log1p((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(expm1(log1p(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[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Exp[N[Log[1 + N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in angle around 0 80.6%

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

   1. expm1-log1p-u 66.8%

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

   2. div-inv 66.8%

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

   3. metadata-eval 66.8%

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

4. Applied egg-rr 66.8%

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

5. Final simplification 66.8%

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

Alternative 2: 80.1% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow a 2.0) (pow (* b (sin (* 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 * 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.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.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 * sin(Float64(angle_m * Float64(pi * 0.005555555555555556)))) ^ 2.0))
end

MATLAB

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

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in angle around 0 80.6%

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

3. Taylor expanded in angle around inf 80.6%

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

   1. associate-*r* 80.6%

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

   2. *-commutative 80.6%

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

   3. associate-*r* 80.6%

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

   4. *-commutative 80.6%

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

5. Simplified 80.6%

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

6. Final simplification 80.6%

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

Alternative 3: 80.1% accurate, 1.3× 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 80.5%

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

2. Taylor expanded in angle around 0 80.6%

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

3. Final simplification 80.6%

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

Alternative 4: 80.0% 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 80.5%

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

3. Simplified 80.6%

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

   1. add-sqr-sqrt 38.4%

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

   2. sqrt-unprod 63.3%

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

   3. associate-/r/ 63.3%

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

   4. associate-/r/ 63.3%

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

   5. associate-*l/ 63.3%

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

   6. associate-*l/ 63.3%

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

   7. frac-times 63.2%

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

   8. metadata-eval 63.2%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)}^{2} \]

   9. metadata-eval 63.2%

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

   10. frac-times 63.3%

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

   11. associate-*r/ 63.3%

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

   12. associate-*r/ 63.3%

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

   13. sqrt-unprod 42.5%

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

   14. add-sqr-sqrt 80.6%

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

   15. associate-*r/ 80.6%

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

   16. div-inv 80.6%

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

   17. metadata-eval 80.6%

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

5. Applied egg-rr 80.6%

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

6. Taylor expanded in angle around 0 80.6%

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

7. Final simplification 80.6%

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

Alternative 5: 74.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {a}^{2} + {\left(angle_m \cdot \left(b \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow a 2.0) (* (pow (* angle_m (* b PI)) 2.0) 3.08641975308642e-5)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow(a, 2.0) + (pow((angle_m * (b * ((double) M_PI))), 2.0) * 3.08641975308642e-5);
}

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((angle_m * (b * Math.PI)), 2.0) * 3.08641975308642e-5);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow(a, 2.0) + (math.pow((angle_m * (b * math.pi)), 2.0) * 3.08641975308642e-5)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((a ^ 2.0) + Float64((Float64(angle_m * Float64(b * pi)) ^ 2.0) * 3.08641975308642e-5))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = (a ^ 2.0) + (((angle_m * (b * pi)) ^ 2.0) * 3.08641975308642e-5);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[Power[N[(angle$95$m * N[(b * Pi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in angle around 0 80.6%

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

3. Taylor expanded in angle around 0 76.9%

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

   1. *-commutative 76.9%

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

5. Simplified 76.9%

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

   1. unpow-prod-down 76.9%

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

   2. *-commutative 76.9%

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

   3. metadata-eval 76.9%

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

7. Applied egg-rr 76.9%

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

8. Final simplification 76.9%

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

Alternative 6: 73.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in angle around 0 80.6%

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

3. Taylor expanded in angle around 0 76.9%

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

   1. *-commutative 76.9%

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

5. Simplified 76.9%

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

   1. unpow2 76.9%

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

   2. associate-*r* 76.9%

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

   3. *-commutative 76.9%

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

   4. associate-*l* 74.4%

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

   5. associate-*r* 74.4%

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

   6. *-commutative 74.4%

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

   7. associate-*l* 74.4%

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

7. Applied egg-rr 74.4%

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

8. Final simplification 74.4%

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

Alternative 7: 73.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in angle around 0 80.6%

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

3. Taylor expanded in angle around 0 76.9%

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

   1. *-commutative 76.9%

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

5. Simplified 76.9%

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

   1. unpow2 76.9%

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

   2. associate-*r* 76.9%

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

   3. *-commutative 76.9%

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

   4. associate-*l* 74.4%

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

   5. associate-*r* 74.4%

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

   6. *-commutative 74.4%

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

   7. associate-*l* 74.4%

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

7. Applied egg-rr 74.4%

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

8. Taylor expanded in b around 0 74.4%

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

   1. *-commutative 74.4%

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

   2. *-commutative 74.4%

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

   3. associate-*r* 74.4%

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

   4. *-commutative 74.4%

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

10. Simplified 74.4%

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

11. Final simplification 74.4%

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

Alternative 8: 74.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in angle around 0 80.6%

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

3. Taylor expanded in angle around 0 76.9%

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

   1. *-commutative 76.9%

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

5. Simplified 76.9%

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

   1. unpow2 76.9%

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

   2. associate-*r* 76.9%

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

   3. *-commutative 76.9%

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

   4. associate-*r* 76.9%

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

   5. *-commutative 76.9%

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

   6. associate-*l* 76.9%

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

   7. associate-*l* 76.9%

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

7. Applied egg-rr 76.9%

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

8. Final simplification 76.9%

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

Alternative 9: 74.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in angle around 0 80.6%

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

3. Taylor expanded in angle around 0 76.9%

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

   1. *-commutative 76.9%

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

5. Simplified 76.9%

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

   1. unpow2 76.9%

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

   2. associate-*r* 76.9%

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

   3. associate-*r* 76.9%

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

   4. *-commutative 76.9%

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

   5. associate-*l* 76.9%

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

7. Applied egg-rr 76.9%

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

8. Final simplification 76.9%

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

Reproduce

herbie shell --seed 2023336 
(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)))