ab-angle->ABCF C

Percentage Accurate: 79.4% → 79.4%

Time: 34.8s

Alternatives: 6

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: 79.4% 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: 79.4% accurate, 0.8× speedup

Math

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

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (+
  (pow a 2.0)
  (pow
   (*
    b
    (sin
     (expm1 (log1p (* PI (expm1 (log1p (* angle 0.005555555555555556))))))))
   2.0)))

C

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

Java

angle = Math.abs(angle);
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 * Math.expm1(Math.log1p((angle * 0.005555555555555556)))))))), 2.0);
}

Python

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

Julia

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

Wolfram

NOTE: angle should be positive before calling this function
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[(Exp[N[Log[1 + N[(angle * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

2. Taylor expanded in angle around 0 76.9%

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

   1. add-sqr-sqrt 39.9%

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

   2. sqrt-unprod 60.6%

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

   3. associate-*r/ 60.6%

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

   4. associate-*r/ 60.6%

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

   5. frac-times 60.5%

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

   6. metadata-eval 60.5%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)}^{2} \]

   7. metadata-eval 60.5%

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

   8. frac-times 60.6%

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

   9. associate-*l/ 60.6%

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

   10. associate-*l/ 60.6%

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

   11. sqrt-unprod 37.0%

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

   12. add-sqr-sqrt 77.0%

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

   13. expm1-log1p-u 63.9%

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

   14. add-sqr-sqrt 37.0%

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

   15. sqrt-unprod 60.7%

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

   16. associate-*l/ 60.7%

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

   17. associate-*l/ 60.7%

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

   18. frac-times 60.6%

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

4. Applied egg-rr 64.6%

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

   1. expm1-log1p-u 64.6%

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

6. Applied egg-rr 64.6%

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

7. Final simplification 64.6%

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

Alternative 2: 79.4% accurate, 1.0× speedup

Math

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

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (+
  (pow a 2.0)
  (pow (* b (sin (expm1 (log1p (* PI (* angle 0.005555555555555556)))))) 2.0)))

C

angle = abs(angle);
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

angle = Math.abs(angle);
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

angle = abs(angle)
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

angle = abs(angle)
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

NOTE: angle should be positive before calling this function
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 76.6%

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

2. Taylor expanded in angle around 0 76.9%

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

   1. add-sqr-sqrt 39.9%

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

   2. sqrt-unprod 60.6%

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

   3. associate-*r/ 60.6%

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

   4. associate-*r/ 60.6%

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

   5. frac-times 60.5%

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

   6. metadata-eval 60.5%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)}^{2} \]

   7. metadata-eval 60.5%

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

   8. frac-times 60.6%

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

   9. associate-*l/ 60.6%

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

   10. associate-*l/ 60.6%

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

   11. sqrt-unprod 37.0%

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

   12. add-sqr-sqrt 77.0%

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

   13. expm1-log1p-u 63.9%

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

   14. add-sqr-sqrt 37.0%

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

   15. sqrt-unprod 60.7%

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

   16. associate-*l/ 60.7%

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

   17. associate-*l/ 60.7%

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

   18. frac-times 60.6%

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

4. Applied egg-rr 64.6%

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

5. Final simplification 64.6%

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

Alternative 3: 79.4% accurate, 1.5× speedup

Math

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

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (+ (pow a 2.0) (pow (* b (sin (* angle (/ PI 180.0)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := N[(N[Power[a, 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 76.6%

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

2. Taylor expanded in angle around 0 76.9%

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

3. Taylor expanded in angle around inf 76.9%

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

   1. *-commutative 76.9%

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

   2. associate-*r* 77.0%

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

   3. metadata-eval 77.0%

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

   4. associate-/r/ 77.0%

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

   5. associate-*l/ 77.0%

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

   6. *-lft-identity 77.0%

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

   7. associate-/l* 77.0%

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

   8. associate-*r/ 77.0%

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

5. Simplified 77.0%

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

6. Final simplification 77.0%

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

Alternative 4: 66.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle = |angle|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-68}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 9.2e-68)
   (pow a 2.0)
   (+ (pow a 2.0) (pow (* b (* 0.005555555555555556 (* PI angle))) 2.0))))

C

angle = abs(angle);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9.2e-68) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(a, 2.0) + pow((b * (0.005555555555555556 * (((double) M_PI) * angle))), 2.0);
	}
	return tmp;
}

Java

angle = Math.abs(angle);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9.2e-68) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow((b * (0.005555555555555556 * (Math.PI * angle))), 2.0);
	}
	return tmp;
}

Python

angle = abs(angle)
def code(a, b, angle):
	tmp = 0
	if b <= 9.2e-68:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(a, 2.0) + math.pow((b * (0.005555555555555556 * (math.pi * angle))), 2.0)
	return tmp

Julia

angle = abs(angle)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 9.2e-68)
		tmp = a ^ 2.0;
	else
		tmp = Float64((a ^ 2.0) + (Float64(b * Float64(0.005555555555555556 * Float64(pi * angle))) ^ 2.0));
	end
	return tmp
end

MATLAB

angle = abs(angle)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 9.2e-68)
		tmp = a ^ 2.0;
	else
		tmp = (a ^ 2.0) + ((b * (0.005555555555555556 * (pi * angle))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 9.2e-68], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.19999999999999987e-68

   1. Initial program 75.2%

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

   3. Applied egg-rr 59.7%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{\frac{{b}^{2} \cdot \left(\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0\right)\right)}{2}} \]
   4. Step-by-step derivation

      1. *-commutative 59.7%

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \frac{\color{blue}{\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0\right)\right) \cdot {b}^{2}}}{2} \]

      2. associate-/l* 59.7%

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{\frac{\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0\right)}{\frac{2}{{b}^{2}}}} \]

      3. +-inverses 59.7%

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \frac{\color{blue}{0}}{\frac{2}{{b}^{2}}} \]

      4. div0 63.2%

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{0} \]

   5. Simplified 63.2%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{0} \]

   6. Taylor expanded in angle around 0 63.5%

      \[\leadsto {\color{blue}{a}}^{2} + 0 \]

   if 9.19999999999999987e-68 < b

   1. Initial program 80.0%

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

   2. Taylor expanded in angle around 0 80.3%

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

   3. Taylor expanded in angle around 0 77.2%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-68}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 5: 66.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle = |angle|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-67}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.6e-67)
   (pow a 2.0)
   (+ (pow a 2.0) (pow (* b (* PI (* angle 0.005555555555555556))) 2.0))))

C

angle = abs(angle);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.6e-67) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(a, 2.0) + pow((b * (((double) M_PI) * (angle * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Java

angle = Math.abs(angle);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.6e-67) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow((b * (Math.PI * (angle * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Python

angle = abs(angle)
def code(a, b, angle):
	tmp = 0
	if b <= 2.6e-67:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(a, 2.0) + math.pow((b * (math.pi * (angle * 0.005555555555555556))), 2.0)
	return tmp

Julia

angle = abs(angle)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.6e-67)
		tmp = a ^ 2.0;
	else
		tmp = Float64((a ^ 2.0) + (Float64(b * Float64(pi * Float64(angle * 0.005555555555555556))) ^ 2.0));
	end
	return tmp
end

MATLAB

angle = abs(angle)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2.6e-67)
		tmp = a ^ 2.0;
	else
		tmp = (a ^ 2.0) + ((b * (pi * (angle * 0.005555555555555556))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 2.6e-67], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5999999999999999e-67

   1. Initial program 75.2%

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

   3. Applied egg-rr 59.7%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{\frac{{b}^{2} \cdot \left(\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0\right)\right)}{2}} \]
   4. Step-by-step derivation

      1. *-commutative 59.7%

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \frac{\color{blue}{\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0\right)\right) \cdot {b}^{2}}}{2} \]

      2. associate-/l* 59.7%

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{\frac{\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0\right)}{\frac{2}{{b}^{2}}}} \]

      3. +-inverses 59.7%

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \frac{\color{blue}{0}}{\frac{2}{{b}^{2}}} \]

      4. div0 63.2%

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{0} \]

   5. Simplified 63.2%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{0} \]

   6. Taylor expanded in angle around 0 63.5%

      \[\leadsto {\color{blue}{a}}^{2} + 0 \]

   if 2.5999999999999999e-67 < b

   1. Initial program 80.0%

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

   2. Taylor expanded in angle around 0 80.3%

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

   3. Taylor expanded in angle around 0 77.2%

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

      1. associate-*r* 77.2%

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

      2. *-commutative 77.2%

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

      3. *-commutative 77.2%

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

      4. *-commutative 77.2%

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

   5. Simplified 77.2%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-67}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 6: 56.4% accurate, 6.0× speedup

Math

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

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle) :precision binary64 (pow a 2.0))

C

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

Fortran

NOTE: angle should be positive before calling this function
real(8) function code(a, b, angle)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = a ** 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := N[Power[a, 2.0], $MachinePrecision]

Derivation

1. Initial program 76.6%

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

3. Applied egg-rr 51.3%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{\frac{{b}^{2} \cdot \left(\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0\right)\right)}{2}} \]
4. Step-by-step derivation

   1. *-commutative 51.3%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \frac{\color{blue}{\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0\right)\right) \cdot {b}^{2}}}{2} \]

   2. associate-/l* 51.3%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{\frac{\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0\right)}{\frac{2}{{b}^{2}}}} \]

   3. +-inverses 51.3%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \frac{\color{blue}{0}}{\frac{2}{{b}^{2}}} \]

   4. div0 58.7%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{0} \]

5. Simplified 58.7%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{0} \]

6. Taylor expanded in angle around 0 58.9%

   \[\leadsto {\color{blue}{a}}^{2} + 0 \]

7. Final simplification 58.9%

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

Reproduce

herbie shell --seed 2023301 
(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)))