ab-angle->ABCF A

Percentage Accurate: 79.8% → 79.6%

Time: 37.1s

Alternatives: 9

Speedup: N/A×

• 35.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(Exp[N[Log[1 + N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

   1. add-sqr-sqrt 38.3%

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

   2. pow2 38.3%

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

   3. associate-*l/ 38.3%

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

   4. associate-*r/ 38.3%

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

   5. div-inv 38.3%

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

   6. metadata-eval 38.3%

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

3. Applied egg-rr 38.3%

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

   1. unpow2 38.3%

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

   2. add-sqr-sqrt 79.0%

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

   3. associate-*r* 78.9%

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

   4. metadata-eval 78.9%

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

   5. div-inv 78.9%

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

   6. clear-num 78.9%

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

   7. *-commutative 78.9%

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

5. Applied egg-rr 78.9%

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

6. Taylor expanded in angle around 0 79.3%

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

   1. expm1-log1p-u 65.6%

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

   2. associate-*l/ 65.6%

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

   3. associate-*r/ 65.6%

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

   4. div-inv 65.6%

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

   5. metadata-eval 65.6%

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

8. Applied egg-rr 65.6%

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

9. Final simplification 65.6%

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

Alternative 2: 79.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. unpow2 78.9%

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

   2. swap-sqr 78.9%

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

   3. associate-*l/ 78.9%

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

   4. associate-*r/ 79.0%

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

   5. swap-sqr 79.0%

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

   6. unpow2 79.0%

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

   7. associate-*l/ 79.0%

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

   8. associate-*r/ 79.0%

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

3. Simplified 79.0%

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

4. Taylor expanded in angle around 0 79.3%

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

5. Taylor expanded in angle around inf 79.3%

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

6. Final simplification 79.3%

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

Alternative 3: 79.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.9%

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

   1. unpow2 78.9%

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

   2. swap-sqr 78.9%

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

   3. associate-*l/ 78.9%

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

   4. associate-*r/ 79.0%

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

   5. swap-sqr 79.0%

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

   6. unpow2 79.0%

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

   7. associate-*l/ 79.0%

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

   8. associate-*r/ 79.0%

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

3. Simplified 79.0%

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

4. Taylor expanded in angle around 0 79.3%

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

5. Final simplification 79.3%

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

Alternative 4: 79.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

   1. add-sqr-sqrt 38.3%

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

   2. pow2 38.3%

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

   3. associate-*l/ 38.3%

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

   4. associate-*r/ 38.3%

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

   5. div-inv 38.3%

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

   6. metadata-eval 38.3%

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

3. Applied egg-rr 38.3%

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

   1. unpow2 38.3%

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

   2. add-sqr-sqrt 79.0%

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

   3. associate-*r* 78.9%

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

   4. metadata-eval 78.9%

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

   5. div-inv 78.9%

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

   6. clear-num 78.9%

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

   7. *-commutative 78.9%

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

5. Applied egg-rr 78.9%

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

6. Taylor expanded in angle around 0 79.3%

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

   1. *-commutative 79.3%

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

   2. clear-num 79.3%

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

   3. un-div-inv 79.3%

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

8. Applied egg-rr 79.3%

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

9. Final simplification 79.3%

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

Alternative 5: 74.4% accurate, 2.0× speedup

Math

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

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (+
  (pow b 2.0)
  (* 3.08641975308642e-5 (* (* a angle) (* PI (* a (* angle PI)))))))

C

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

Java

angle = Math.abs(angle);
public static double code(double a, double b, double angle) {
	return Math.pow(b, 2.0) + (3.08641975308642e-5 * ((a * angle) * (Math.PI * (a * (angle * Math.PI)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. unpow2 78.9%

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

   2. swap-sqr 78.9%

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

   3. associate-*l/ 78.9%

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

   4. associate-*r/ 79.0%

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

   5. swap-sqr 79.0%

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

   6. unpow2 79.0%

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

   7. associate-*l/ 79.0%

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

   8. associate-*r/ 79.0%

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

3. Simplified 79.0%

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

4. Taylor expanded in angle around 0 79.3%

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

   1. associate-*r/ 79.2%

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

   2. associate-*l/ 79.3%

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

   3. expm1-log1p-u 79.3%

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

   4. associate-*l/ 79.2%

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

   5. associate-*r/ 79.3%

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

   6. div-inv 79.3%

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

   7. metadata-eval 79.3%

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

6. Applied egg-rr 79.3%

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

7. Taylor expanded in angle around 0 63.9%

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

   1. *-commutative 63.9%

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

   2. unpow2 63.9%

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

   3. unpow2 63.9%

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

   4. swap-sqr 63.9%

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

   5. unpow2 63.9%

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

   6. swap-sqr 74.3%

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

   7. associate-*r* 74.3%

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

   8. associate-*r* 74.3%

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

   9. unpow2 74.3%

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

   10. associate-*r* 74.3%

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

   11. *-commutative 74.3%

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

   12. associate-*r* 74.3%

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

   13. *-commutative 74.3%

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

   14. *-commutative 74.3%

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

9. Simplified 74.3%

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

    1. unpow2 74.3%

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

    2. associate-*r* 74.3%

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

    3. *-commutative 74.3%

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

    4. *-commutative 74.3%

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

    5. associate-*l* 74.3%

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

11. Applied egg-rr 74.3%

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

12. Final simplification 74.3%

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

Alternative 6: 74.4% accurate, 2.0× speedup

Math

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

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (+
  (pow b 2.0)
  (* 3.08641975308642e-5 (* PI (* (* a (* angle PI)) (* a angle))))))

C

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

Java

angle = Math.abs(angle);
public static double code(double a, double b, double angle) {
	return Math.pow(b, 2.0) + (3.08641975308642e-5 * (Math.PI * ((a * (angle * Math.PI)) * (a * angle))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. unpow2 78.9%

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

   2. swap-sqr 78.9%

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

   3. associate-*l/ 78.9%

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

   4. associate-*r/ 79.0%

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

   5. swap-sqr 79.0%

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

   6. unpow2 79.0%

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

   7. associate-*l/ 79.0%

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

   8. associate-*r/ 79.0%

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

3. Simplified 79.0%

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

4. Taylor expanded in angle around 0 79.3%

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

   1. associate-*r/ 79.2%

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

   2. associate-*l/ 79.3%

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

   3. expm1-log1p-u 79.3%

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

   4. associate-*l/ 79.2%

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

   5. associate-*r/ 79.3%

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

   6. div-inv 79.3%

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

   7. metadata-eval 79.3%

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

6. Applied egg-rr 79.3%

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

7. Taylor expanded in angle around 0 63.9%

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

   1. *-commutative 63.9%

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

   2. unpow2 63.9%

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

   3. unpow2 63.9%

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

   4. swap-sqr 63.9%

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

   5. unpow2 63.9%

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

   6. swap-sqr 74.3%

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

   7. associate-*r* 74.3%

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

   8. associate-*r* 74.3%

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

   9. unpow2 74.3%

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

   10. associate-*r* 74.3%

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

   11. *-commutative 74.3%

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

   12. associate-*r* 74.3%

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

   13. *-commutative 74.3%

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

   14. *-commutative 74.3%

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

9. Simplified 74.3%

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

    1. unpow2 74.3%

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

    2. *-commutative 74.3%

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

    3. associate-*r* 74.3%

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

    4. *-commutative 74.3%

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

    5. *-commutative 74.3%

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

    6. associate-*l* 74.3%

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

11. Applied egg-rr 74.3%

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

12. Final simplification 74.3%

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

Alternative 7: 74.4% accurate, 2.0× speedup

Math

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

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (+ (pow b 2.0) (* 3.08641975308642e-5 (pow (* PI (* a angle)) 2.0))))

C

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

Java

angle = Math.abs(angle);
public static double code(double a, double b, double angle) {
	return Math.pow(b, 2.0) + (3.08641975308642e-5 * Math.pow((Math.PI * (a * angle)), 2.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. unpow2 78.9%

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

   2. swap-sqr 78.9%

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

   3. associate-*l/ 78.9%

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

   4. associate-*r/ 79.0%

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

   5. swap-sqr 79.0%

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

   6. unpow2 79.0%

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

   7. associate-*l/ 79.0%

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

   8. associate-*r/ 79.0%

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

3. Simplified 79.0%

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

4. Taylor expanded in angle around 0 79.3%

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

   1. associate-*r/ 79.2%

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

   2. associate-*l/ 79.3%

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

   3. expm1-log1p-u 79.3%

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

   4. associate-*l/ 79.2%

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

   5. associate-*r/ 79.3%

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

   6. div-inv 79.3%

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

   7. metadata-eval 79.3%

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

6. Applied egg-rr 79.3%

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

7. Taylor expanded in angle around 0 63.9%

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

   1. *-commutative 63.9%

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

   2. unpow2 63.9%

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

   3. unpow2 63.9%

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

   4. swap-sqr 63.9%

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

   5. unpow2 63.9%

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

   6. swap-sqr 74.3%

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

   7. associate-*r* 74.3%

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

   8. associate-*r* 74.3%

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

   9. unpow2 74.3%

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

   10. associate-*r* 74.3%

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

   11. *-commutative 74.3%

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

   12. associate-*r* 74.3%

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

   13. *-commutative 74.3%

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

   14. *-commutative 74.3%

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

9. Simplified 74.3%

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

10. Final simplification 74.3%

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

Alternative 8: 74.4% accurate, 2.0× speedup

Math

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

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (+ (pow b 2.0) (* 3.08641975308642e-5 (pow (* angle (* a PI)) 2.0))))

C

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

Java

angle = Math.abs(angle);
public static double code(double a, double b, double angle) {
	return Math.pow(b, 2.0) + (3.08641975308642e-5 * Math.pow((angle * (a * Math.PI)), 2.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. unpow2 78.9%

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

   2. swap-sqr 78.9%

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

   3. associate-*l/ 78.9%

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

   4. associate-*r/ 79.0%

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

   5. swap-sqr 79.0%

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

   6. unpow2 79.0%

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

   7. associate-*l/ 79.0%

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

   8. associate-*r/ 79.0%

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

3. Simplified 79.0%

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

4. Taylor expanded in angle around 0 79.3%

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

5. Taylor expanded in angle around 0 74.2%

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

   1. *-commutative 74.2%

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

7. Simplified 74.2%

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

   1. *-commutative 74.2%

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

   2. unpow-prod-down 74.3%

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

   3. associate-*l* 74.3%

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

   4. metadata-eval 74.3%

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

9. Applied egg-rr 74.3%

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

10. Final simplification 74.3%

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

Alternative 9: 74.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. unpow2 78.9%

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

   2. swap-sqr 78.9%

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

   3. associate-*l/ 78.9%

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

   4. associate-*r/ 79.0%

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

   5. swap-sqr 79.0%

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

   6. unpow2 79.0%

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

   7. associate-*l/ 79.0%

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

   8. associate-*r/ 79.0%

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

3. Simplified 79.0%

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

4. Taylor expanded in angle around 0 79.3%

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

5. Taylor expanded in angle around 0 74.2%

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

   1. *-commutative 74.2%

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

   2. associate-*r* 74.3%

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

   3. associate-*l* 74.3%

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

   4. *-commutative 74.3%

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

7. Simplified 74.3%

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

8. Final simplification 74.3%

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

Reproduce

herbie shell --seed 2023314 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))