ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.6%

Time: 1.1min

Alternatives: 10

Speedup: 1.5×

• 36.6% 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: 79.6% 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: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Initial program 75.7%

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

2. Taylor expanded in angle around 0 75.8%

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

3. Taylor expanded in b around 0 75.8%

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

   1. *-commutative 75.8%

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

   2. metadata-eval 75.8%

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

   3. *-commutative 75.8%

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

   4. associate-/r/ 75.8%

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

   5. associate-/l* 75.8%

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

   6. *-commutative 75.8%

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

   7. associate-/l* 75.8%

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

   8. metadata-eval 75.8%

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

   9. *-commutative 75.8%

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

   10. associate-*r/ 75.9%

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

5. Simplified 75.9%

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

   1. expm1-log1p-u 60.4%

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

   2. div-inv 60.4%

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

   3. metadata-eval 60.4%

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

7. Applied egg-rr 60.4%

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

8. Final simplification 60.4%

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

Alternative 2: 79.6% accurate, 1.5× speedup

Math

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

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (+ (pow a 2.0) (pow (* b (sin (* 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((((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.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.pi * (angle * -0.005555555555555556)))), 2.0)

Julia

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

MATLAB

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

Derivation

1. Initial program 75.7%

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

   1. unpow2 75.7%

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

   2. swap-sqr 66.5%

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

   3. sqr-neg 66.5%

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

   4. swap-sqr 75.7%

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

   5. unpow2 75.7%

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

   6. distribute-lft-neg-out 75.7%

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

   7. distribute-rgt-neg-in 75.7%

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

   8. sin-neg 75.7%

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

   9. distribute-rgt-neg-out 75.7%

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

   10. distribute-frac-neg 75.7%

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

   11. unpow2 75.7%

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

   12. associate-*l* 74.3%

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

3. Simplified 75.7%

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

4. Taylor expanded in angle around 0 75.8%

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

5. Final simplification 75.8%

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

Alternative 3: 79.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

angle = abs(angle)
function tmp = code(a, b, angle)
	tmp = (a ^ 2.0) + ((b * sin((pi * (angle / 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[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

2. Taylor expanded in angle around 0 75.8%

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

3. Final simplification 75.8%

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

Alternative 4: 79.5% accurate, 1.5× speedup

Math

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

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(Float64(angle * 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[(N[(angle * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

   1. unpow2 75.7%

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

   2. swap-sqr 66.5%

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

   3. sqr-neg 66.5%

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

   4. swap-sqr 75.7%

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

   5. unpow2 75.7%

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

   6. distribute-lft-neg-out 75.7%

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

   7. distribute-rgt-neg-in 75.7%

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

   8. sin-neg 75.7%

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

   9. distribute-rgt-neg-out 75.7%

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

   10. distribute-frac-neg 75.7%

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

   11. unpow2 75.7%

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

   12. associate-*l* 74.3%

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

3. Simplified 75.7%

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

4. Taylor expanded in angle around 0 75.8%

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

   1. add-sqr-sqrt 41.5%

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

   2. sqrt-unprod 56.8%

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

   3. swap-sqr 56.8%

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

   4. metadata-eval 56.8%

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

   5. metadata-eval 56.8%

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

   6. metadata-eval 56.8%

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

   7. metadata-eval 56.8%

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

   8. swap-sqr 56.8%

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

   9. div-inv 56.8%

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

   10. div-inv 56.8%

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

   11. sqrt-unprod 34.3%

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

   12. add-sqr-sqrt 75.8%

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

   13. associate-*r/ 75.8%

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

6. Applied egg-rr 75.8%

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

7. Final simplification 75.8%

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

Alternative 5: 79.6% 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 75.7%

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

2. Taylor expanded in angle around 0 75.8%

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

3. Taylor expanded in b around 0 75.8%

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

   1. *-commutative 75.8%

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

   2. metadata-eval 75.8%

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

   3. *-commutative 75.8%

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

   4. associate-/r/ 75.8%

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

   5. associate-/l* 75.8%

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

   6. *-commutative 75.8%

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

   7. associate-/l* 75.8%

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

   8. metadata-eval 75.8%

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

   9. *-commutative 75.8%

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

   10. associate-*r/ 75.9%

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

5. Simplified 75.9%

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

6. Final simplification 75.9%

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

Alternative 6: 66.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle = |angle|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.2e-13)
   (* a a)
   (fma 3.08641975308642e-5 (pow (* PI (* angle b)) 2.0) (* a a))))

C

angle = abs(angle);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.2e-13) {
		tmp = a * a;
	} else {
		tmp = fma(3.08641975308642e-5, pow((((double) M_PI) * (angle * b)), 2.0), (a * a));
	}
	return tmp;
}

Julia

angle = abs(angle)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.2e-13)
		tmp = Float64(a * a);
	else
		tmp = fma(3.08641975308642e-5, (Float64(pi * Float64(angle * b)) ^ 2.0), Float64(a * a));
	end
	return tmp
end

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 2.2e-13], N[(a * a), $MachinePrecision], N[(3.08641975308642e-5 * N[Power[N[(Pi * N[(angle * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.19999999999999997e-13

   1. Initial program 74.0%

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

   2. Taylor expanded in angle around 0 74.3%

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

   3. Taylor expanded in angle around 0 67.6%

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

      1. *-commutative 67.6%

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

   5. Simplified 67.6%

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

      1. *-commutative 67.6%

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

      2. unpow-prod-down 67.5%

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

      3. *-commutative 67.5%

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

      4. associate-*l* 67.5%

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

      5. metadata-eval 67.5%

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

   7. Applied egg-rr 67.5%

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

   8. Taylor expanded in a around inf 58.9%

      \[\leadsto \color{blue}{{a}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 58.9%

         \[\leadsto \color{blue}{a \cdot a} \]

   10. Simplified 58.9%

       \[\leadsto \color{blue}{a \cdot a} \]

   if 2.19999999999999997e-13 < 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 79.8%

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

   3. Taylor expanded in angle around 0 75.9%

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

      1. *-commutative 75.9%

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

   5. Simplified 75.9%

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

      1. *-commutative 75.9%

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

      2. unpow-prod-down 76.0%

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

      3. *-commutative 76.0%

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

      4. associate-*l* 76.1%

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

      5. metadata-eval 76.1%

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

   7. Applied egg-rr 76.1%

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

   8. Taylor expanded in a around 0 56.1%

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

      1. fma-def 56.1%

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

      2. unpow2 56.1%

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

      3. *-commutative 56.1%

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

      4. unpow2 56.1%

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

      5. unpow2 56.1%

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

      6. swap-sqr 56.1%

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

      7. swap-sqr 76.0%

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

      8. unpow2 76.0%

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

      9. *-commutative 76.0%

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

      10. associate-*r* 76.2%

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

      11. unpow2 76.2%

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

   10. Simplified 76.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\left(\left(angle \cdot b\right) \cdot \pi\right)}^{2}, a \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}, a \cdot a\right)\\ \end{array} \]

Alternative 7: 66.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle = |angle|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, 0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.9e-13)
   (* a a)
   (pow (hypot a (* 0.005555555555555556 (* angle (* PI b)))) 2.0)))

C

angle = abs(angle);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.9e-13) {
		tmp = a * a;
	} else {
		tmp = pow(hypot(a, (0.005555555555555556 * (angle * (((double) M_PI) * b)))), 2.0);
	}
	return tmp;
}

Java

angle = Math.abs(angle);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.9e-13) {
		tmp = a * a;
	} else {
		tmp = Math.pow(Math.hypot(a, (0.005555555555555556 * (angle * (Math.PI * b)))), 2.0);
	}
	return tmp;
}

Python

angle = abs(angle)
def code(a, b, angle):
	tmp = 0
	if b <= 2.9e-13:
		tmp = a * a
	else:
		tmp = math.pow(math.hypot(a, (0.005555555555555556 * (angle * (math.pi * b)))), 2.0)
	return tmp

Julia

angle = abs(angle)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.9e-13)
		tmp = Float64(a * a);
	else
		tmp = hypot(a, Float64(0.005555555555555556 * Float64(angle * Float64(pi * b)))) ^ 2.0;
	end
	return tmp
end

MATLAB

angle = abs(angle)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2.9e-13)
		tmp = a * a;
	else
		tmp = hypot(a, (0.005555555555555556 * (angle * (pi * b)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 2.9e-13], N[(a * a), $MachinePrecision], N[Power[N[Sqrt[a ^ 2 + N[(0.005555555555555556 * N[(angle * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8999999999999998e-13

   1. Initial program 74.0%

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

   2. Taylor expanded in angle around 0 74.3%

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

   3. Taylor expanded in angle around 0 67.6%

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

      1. *-commutative 67.6%

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

   5. Simplified 67.6%

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

      1. *-commutative 67.6%

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

      2. unpow-prod-down 67.5%

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

      3. *-commutative 67.5%

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

      4. associate-*l* 67.5%

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

      5. metadata-eval 67.5%

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

   7. Applied egg-rr 67.5%

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

   8. Taylor expanded in a around inf 58.9%

      \[\leadsto \color{blue}{{a}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 58.9%

         \[\leadsto \color{blue}{a \cdot a} \]

   10. Simplified 58.9%

       \[\leadsto \color{blue}{a \cdot a} \]

   if 2.8999999999999998e-13 < 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 79.8%

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

   3. Taylor expanded in angle around 0 75.9%

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

      1. *-commutative 75.9%

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

   5. Simplified 75.9%

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

      1. *-commutative 75.9%

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

      2. unpow-prod-down 76.0%

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

      3. *-commutative 76.0%

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

      4. associate-*l* 76.1%

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

      5. metadata-eval 76.1%

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

   7. Applied egg-rr 76.1%

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

      1. add-sqr-sqrt 76.0%

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

      2. pow2 76.0%

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

   9. Applied egg-rr 75.9%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, 0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 8: 66.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle = |angle|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, 0.005555555555555556 \cdot \left(\pi \cdot \left(angle \cdot b\right)\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.2e-13)
   (* a a)
   (pow (hypot a (* 0.005555555555555556 (* PI (* angle b)))) 2.0)))

C

angle = abs(angle);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.2e-13) {
		tmp = a * a;
	} else {
		tmp = pow(hypot(a, (0.005555555555555556 * (((double) M_PI) * (angle * b)))), 2.0);
	}
	return tmp;
}

Java

angle = Math.abs(angle);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.2e-13) {
		tmp = a * a;
	} else {
		tmp = Math.pow(Math.hypot(a, (0.005555555555555556 * (Math.PI * (angle * b)))), 2.0);
	}
	return tmp;
}

Python

angle = abs(angle)
def code(a, b, angle):
	tmp = 0
	if b <= 2.2e-13:
		tmp = a * a
	else:
		tmp = math.pow(math.hypot(a, (0.005555555555555556 * (math.pi * (angle * b)))), 2.0)
	return tmp

Julia

angle = abs(angle)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.2e-13)
		tmp = Float64(a * a);
	else
		tmp = hypot(a, Float64(0.005555555555555556 * Float64(pi * Float64(angle * b)))) ^ 2.0;
	end
	return tmp
end

MATLAB

angle = abs(angle)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2.2e-13)
		tmp = a * a;
	else
		tmp = hypot(a, (0.005555555555555556 * (pi * (angle * b)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 2.2e-13], N[(a * a), $MachinePrecision], N[Power[N[Sqrt[a ^ 2 + N[(0.005555555555555556 * N[(Pi * N[(angle * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.19999999999999997e-13

   1. Initial program 74.0%

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

   2. Taylor expanded in angle around 0 74.3%

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

   3. Taylor expanded in angle around 0 67.6%

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

      1. *-commutative 67.6%

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

   5. Simplified 67.6%

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

      1. *-commutative 67.6%

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

      2. unpow-prod-down 67.5%

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

      3. *-commutative 67.5%

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

      4. associate-*l* 67.5%

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

      5. metadata-eval 67.5%

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

   7. Applied egg-rr 67.5%

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

   8. Taylor expanded in a around inf 58.9%

      \[\leadsto \color{blue}{{a}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 58.9%

         \[\leadsto \color{blue}{a \cdot a} \]

   10. Simplified 58.9%

       \[\leadsto \color{blue}{a \cdot a} \]

   if 2.19999999999999997e-13 < 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 79.8%

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

   3. Taylor expanded in angle around 0 75.9%

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

      1. *-commutative 75.9%

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

   5. Simplified 75.9%

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

      1. *-commutative 75.9%

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

      2. unpow-prod-down 76.0%

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

      3. *-commutative 76.0%

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

      4. associate-*l* 76.1%

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

      5. metadata-eval 76.1%

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

   7. Applied egg-rr 76.1%

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

      1. expm1-log1p-u 74.6%

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

      2. expm1-udef 57.6%

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

   9. Applied egg-rr 57.6%

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

       1. expm1-def 74.4%

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

       2. expm1-log1p 75.9%

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

       3. *-commutative 75.9%

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

       4. associate-*r* 75.9%

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

   11. Simplified 75.9%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, 0.005555555555555556 \cdot \left(\pi \cdot \left(angle \cdot b\right)\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 9: 66.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle = |angle|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-13}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, \left(angle \cdot 0.005555555555555556\right) \cdot \left(\pi \cdot b\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.6e-13)
   (* a a)
   (pow (hypot a (* (* angle 0.005555555555555556) (* PI b))) 2.0)))

C

angle = abs(angle);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.6e-13) {
		tmp = a * a;
	} else {
		tmp = pow(hypot(a, ((angle * 0.005555555555555556) * (((double) M_PI) * b))), 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-13) {
		tmp = a * a;
	} else {
		tmp = Math.pow(Math.hypot(a, ((angle * 0.005555555555555556) * (Math.PI * b))), 2.0);
	}
	return tmp;
}

Python

angle = abs(angle)
def code(a, b, angle):
	tmp = 0
	if b <= 2.6e-13:
		tmp = a * a
	else:
		tmp = math.pow(math.hypot(a, ((angle * 0.005555555555555556) * (math.pi * b))), 2.0)
	return tmp

Julia

angle = abs(angle)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.6e-13)
		tmp = Float64(a * a);
	else
		tmp = hypot(a, Float64(Float64(angle * 0.005555555555555556) * Float64(pi * b))) ^ 2.0;
	end
	return tmp
end

MATLAB

angle = abs(angle)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2.6e-13)
		tmp = a * a;
	else
		tmp = hypot(a, ((angle * 0.005555555555555556) * (pi * b))) ^ 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-13], N[(a * a), $MachinePrecision], N[Power[N[Sqrt[a ^ 2 + N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6e-13

   1. Initial program 74.0%

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

   2. Taylor expanded in angle around 0 74.3%

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

   3. Taylor expanded in angle around 0 67.6%

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

      1. *-commutative 67.6%

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

   5. Simplified 67.6%

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

      1. *-commutative 67.6%

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

      2. unpow-prod-down 67.5%

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

      3. *-commutative 67.5%

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

      4. associate-*l* 67.5%

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

      5. metadata-eval 67.5%

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

   7. Applied egg-rr 67.5%

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

   8. Taylor expanded in a around inf 58.9%

      \[\leadsto \color{blue}{{a}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 58.9%

         \[\leadsto \color{blue}{a \cdot a} \]

   10. Simplified 58.9%

       \[\leadsto \color{blue}{a \cdot a} \]

   if 2.6e-13 < 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 79.8%

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

   3. Taylor expanded in angle around 0 75.9%

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

      1. *-commutative 75.9%

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

   5. Simplified 75.9%

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

      1. *-commutative 75.9%

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

      2. unpow-prod-down 76.0%

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

      3. *-commutative 76.0%

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

      4. associate-*l* 76.1%

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

      5. metadata-eval 76.1%

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

   7. Applied egg-rr 76.1%

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

      1. expm1-log1p-u 74.6%

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

      2. expm1-udef 57.6%

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

   9. Applied egg-rr 57.6%

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

       1. expm1-def 74.4%

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

       2. expm1-log1p 75.9%

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

       3. associate-*r* 76.0%

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

   11. Simplified 76.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-13}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, \left(angle \cdot 0.005555555555555556\right) \cdot \left(\pi \cdot b\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 10: 56.5% accurate, 205.0× speedup

Math

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

FPCore

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

C

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

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 * a
end function

Java

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

Python

angle = abs(angle)
def code(a, b, angle):
	return a * a

Julia

angle = abs(angle)
function code(a, b, angle)
	return Float64(a * a)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

2. Taylor expanded in angle around 0 75.8%

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

3. Taylor expanded in angle around 0 69.9%

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

   1. *-commutative 69.9%

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

5. Simplified 69.9%

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

   1. *-commutative 69.9%

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

   2. unpow-prod-down 69.9%

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

   3. *-commutative 69.9%

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

   4. associate-*l* 70.0%

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

   5. metadata-eval 70.0%

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

7. Applied egg-rr 70.0%

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

8. Taylor expanded in a around inf 51.9%

   \[\leadsto \color{blue}{{a}^{2}} \]
9. Step-by-step derivation

   1. unpow2 51.9%

      \[\leadsto \color{blue}{a \cdot a} \]

10. Simplified 51.9%

    \[\leadsto \color{blue}{a \cdot a} \]

11. Final simplification 51.9%

    \[\leadsto a \cdot a \]

Reproduce

herbie shell --seed 2023287 
(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)))