ab-angle->ABCF A

Percentage Accurate: 79.3% → 79.2%

Time: 36.9s

Alternatives: 13

Speedup: 1.0×

• 36.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 79.3% 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: 79.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (/ (* angle PI) 180.0)))
   (+ (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 * ((double) M_PI)) / 180.0;
	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 * Math.PI) / 180.0;
	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 * math.pi) / 180.0
	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 * pi) / 180.0)
	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 * pi) / 180.0;
	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 * Pi), $MachinePrecision] / 180.0), $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]]

Derivation

1. Initial program 80.2%

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

   1. associate-*l/ 80.6%

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

3. Applied egg-rr 80.6%

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

   1. associate-*l/ 80.6%

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

5. Applied egg-rr 80.6%

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

6. Final simplification 80.6%

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

Alternative 2: 79.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (/ PI 180.0))))
   (+ (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 * (((double) M_PI) / 180.0);
	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 * (Math.PI / 180.0);
	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 * (math.pi / 180.0)
	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(angle * Float64(pi / 180.0))
	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 * (pi / 180.0);
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi / 180.0), $MachinePrecision]), $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]]

Derivation

1. Initial program 80.2%

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

   1. associate-*l/ 80.6%

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

   2. associate-*r/ 80.6%

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

   3. associate-*l/ 80.6%

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

   4. associate-*r/ 80.6%

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

3. Simplified 80.6%

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

4. Final simplification 80.6%

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

Alternative 3: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l/ 80.6%

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

3. Applied egg-rr 80.6%

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

4. Final simplification 80.6%

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

Alternative 4: 79.2% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow b 2.0) (pow (* a (sin (* (* angle PI) 0.005555555555555556))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. associate-*l/ 80.6%

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

   2. associate-*r/ 80.6%

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

   3. associate-*l/ 80.6%

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

   4. associate-*r/ 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in angle around 0 79.9%

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

5. Taylor expanded in angle around inf 79.5%

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

6. Final simplification 79.5%

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

Alternative 5: 79.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l/ 80.6%

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

   2. associate-*r/ 80.6%

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

   3. associate-*l/ 80.6%

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

   4. associate-*r/ 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in angle around 0 79.9%

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

5. Final simplification 79.9%

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

Alternative 6: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 8 \cdot 10^{-5}:\\ \;\;\;\;{b}^{2} + \left(a \cdot angle\right) \cdot \left(\pi \cdot \left(\left(\pi \cdot \left(a \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(a \cdot a\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 8e-5)
   (+
    (pow b 2.0)
    (* (* a angle) (* PI (* (* PI (* a angle)) 3.08641975308642e-5))))
   (+
    (pow b 2.0)
    (* (* a a) (- 0.5 (* 0.5 (cos (* (* angle PI) 0.011111111111111112))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 8e-5) {
		tmp = pow(b, 2.0) + ((a * angle) * (((double) M_PI) * ((((double) M_PI) * (a * angle)) * 3.08641975308642e-5)));
	} else {
		tmp = pow(b, 2.0) + ((a * a) * (0.5 - (0.5 * cos(((angle * ((double) M_PI)) * 0.011111111111111112)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 8e-5) {
		tmp = Math.pow(b, 2.0) + ((a * angle) * (Math.PI * ((Math.PI * (a * angle)) * 3.08641975308642e-5)));
	} else {
		tmp = Math.pow(b, 2.0) + ((a * a) * (0.5 - (0.5 * Math.cos(((angle * Math.PI) * 0.011111111111111112)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 8e-5:
		tmp = math.pow(b, 2.0) + ((a * angle) * (math.pi * ((math.pi * (a * angle)) * 3.08641975308642e-5)))
	else:
		tmp = math.pow(b, 2.0) + ((a * a) * (0.5 - (0.5 * math.cos(((angle * math.pi) * 0.011111111111111112)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 8e-5)
		tmp = Float64((b ^ 2.0) + Float64(Float64(a * angle) * Float64(pi * Float64(Float64(pi * Float64(a * angle)) * 3.08641975308642e-5))));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(a * a) * Float64(0.5 - Float64(0.5 * cos(Float64(Float64(angle * pi) * 0.011111111111111112))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 8e-5)
		tmp = (b ^ 2.0) + ((a * angle) * (pi * ((pi * (a * angle)) * 3.08641975308642e-5)));
	else
		tmp = (b ^ 2.0) + ((a * a) * (0.5 - (0.5 * cos(((angle * pi) * 0.011111111111111112)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 8e-5], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * angle), $MachinePrecision] * N[(Pi * N[(N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 8.00000000000000065e-5

   1. Initial program 86.7%

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

      1. associate-*l/ 86.6%

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

      2. associate-*r/ 86.7%

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

      3. associate-*l/ 86.7%

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

      4. associate-*r/ 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in angle around 0 86.4%

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

   5. Taylor expanded in angle around 0 83.2%

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

      1. *-commutative 83.2%

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

      2. associate-*l* 83.2%

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

   7. Simplified 83.2%

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

      1. unpow2 83.2%

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

      2. associate-*r* 83.3%

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

      3. *-commutative 83.3%

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

      4. associate-*r* 83.3%

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

      5. *-commutative 83.3%

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

      6. associate-*l* 83.3%

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

      7. associate-*r* 83.3%

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

      8. *-commutative 83.3%

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

      9. *-commutative 83.3%

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

      10. associate-*l* 83.3%

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

      11. associate-*r* 83.4%

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

      12. *-commutative 83.4%

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

      13. associate-*l* 83.3%

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

      14. metadata-eval 83.3%

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

   9. Applied egg-rr 83.3%

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

   if 8.00000000000000065e-5 < angle

   1. Initial program 61.7%

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

      1. associate-*l/ 63.0%

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

      2. associate-*r/ 63.1%

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

      3. associate-*l/ 63.1%

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

      4. associate-*r/ 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in angle around 0 61.4%

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

      1. unpow-prod-down 61.4%

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

      2. *-commutative 61.4%

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

      3. associate-/r/ 61.4%

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

      4. associate-/l* 61.4%

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

      5. associate-*r/ 61.3%

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

      6. *-commutative 61.3%

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

      7. *-commutative 61.3%

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

      8. associate-*l/ 61.4%

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

      9. associate-*r/ 61.4%

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

      10. div-inv 61.4%

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

      11. metadata-eval 61.4%

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

      12. unpow2 61.4%

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

   6. Applied egg-rr 61.4%

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

      1. unpow2 61.4%

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

      2. *-commutative 61.4%

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

      3. associate-*l* 54.4%

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

      4. *-commutative 54.4%

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

      5. associate-*l* 61.3%

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

      6. sqr-sin-a 61.3%

         \[\leadsto \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)} \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

      7. count-2 61.3%

         \[\leadsto \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}\right) \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

      8. associate-*l* 61.5%

         \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(\color{blue}{angle \cdot \left(0.005555555555555556 \cdot \pi\right)} + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

      9. *-commutative 61.5%

         \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(angle \cdot \color{blue}{\left(\pi \cdot 0.005555555555555556\right)} + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

      10. associate-*r* 61.4%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(\color{blue}{\left(angle \cdot \pi\right) \cdot 0.005555555555555556} + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

      11. associate-*l* 61.5%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \color{blue}{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

      12. *-commutative 61.5%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + angle \cdot \color{blue}{\left(\pi \cdot 0.005555555555555556\right)}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

      13. associate-*r* 61.4%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \color{blue}{\left(angle \cdot \pi\right) \cdot 0.005555555555555556}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

      14. distribute-lft-out 61.4%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.005555555555555556 + 0.005555555555555556\right)\right)}\right) \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

      15. metadata-eval 61.4%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \color{blue}{0.011111111111111112}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

   8. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 8 \cdot 10^{-5}:\\ \;\;\;\;{b}^{2} + \left(a \cdot angle\right) \cdot \left(\pi \cdot \left(\left(\pi \cdot \left(a \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(a \cdot a\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 7: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 8 \cdot 10^{-5}:\\ \;\;\;\;{b}^{2} + \left(a \cdot angle\right) \cdot \left(\pi \cdot \left(\left(\pi \cdot \left(a \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{-1 + \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}{-2} \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 8e-5)
   (+
    (pow b 2.0)
    (* (* a angle) (* PI (* (* PI (* a angle)) 3.08641975308642e-5))))
   (+
    (pow b 2.0)
    (*
     (/ (+ -1.0 (cos (* angle (* PI 0.011111111111111112)))) -2.0)
     (* a a)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 8e-5) {
		tmp = pow(b, 2.0) + ((a * angle) * (((double) M_PI) * ((((double) M_PI) * (a * angle)) * 3.08641975308642e-5)));
	} else {
		tmp = pow(b, 2.0) + (((-1.0 + cos((angle * (((double) M_PI) * 0.011111111111111112)))) / -2.0) * (a * a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 8e-5) {
		tmp = Math.pow(b, 2.0) + ((a * angle) * (Math.PI * ((Math.PI * (a * angle)) * 3.08641975308642e-5)));
	} else {
		tmp = Math.pow(b, 2.0) + (((-1.0 + Math.cos((angle * (Math.PI * 0.011111111111111112)))) / -2.0) * (a * a));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 8e-5:
		tmp = math.pow(b, 2.0) + ((a * angle) * (math.pi * ((math.pi * (a * angle)) * 3.08641975308642e-5)))
	else:
		tmp = math.pow(b, 2.0) + (((-1.0 + math.cos((angle * (math.pi * 0.011111111111111112)))) / -2.0) * (a * a))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 8e-5)
		tmp = Float64((b ^ 2.0) + Float64(Float64(a * angle) * Float64(pi * Float64(Float64(pi * Float64(a * angle)) * 3.08641975308642e-5))));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(Float64(-1.0 + cos(Float64(angle * Float64(pi * 0.011111111111111112)))) / -2.0) * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 8e-5)
		tmp = (b ^ 2.0) + ((a * angle) * (pi * ((pi * (a * angle)) * 3.08641975308642e-5)));
	else
		tmp = (b ^ 2.0) + (((-1.0 + cos((angle * (pi * 0.011111111111111112)))) / -2.0) * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 8e-5], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * angle), $MachinePrecision] * N[(Pi * N[(N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(N[(-1.0 + N[Cos[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 8.00000000000000065e-5

   1. Initial program 86.7%

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

      1. associate-*l/ 86.6%

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

      2. associate-*r/ 86.7%

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

      3. associate-*l/ 86.7%

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

      4. associate-*r/ 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in angle around 0 86.4%

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

   5. Taylor expanded in angle around 0 83.2%

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

      1. *-commutative 83.2%

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

      2. associate-*l* 83.2%

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

   7. Simplified 83.2%

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

      1. unpow2 83.2%

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

      2. associate-*r* 83.3%

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

      3. *-commutative 83.3%

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

      4. associate-*r* 83.3%

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

      5. *-commutative 83.3%

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

      6. associate-*l* 83.3%

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

      7. associate-*r* 83.3%

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

      8. *-commutative 83.3%

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

      9. *-commutative 83.3%

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

      10. associate-*l* 83.3%

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

      11. associate-*r* 83.4%

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

      12. *-commutative 83.4%

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

      13. associate-*l* 83.3%

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

      14. metadata-eval 83.3%

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

   9. Applied egg-rr 83.3%

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

   if 8.00000000000000065e-5 < angle

   1. Initial program 61.7%

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

      1. associate-*l/ 63.0%

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

      2. associate-*r/ 63.1%

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

      3. associate-*l/ 63.1%

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

      4. associate-*r/ 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in angle around 0 61.4%

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

      1. unpow-prod-down 61.4%

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

      2. *-commutative 61.4%

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

      3. associate-/r/ 61.4%

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

      4. associate-/l* 61.4%

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

      5. associate-*r/ 61.3%

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

      6. *-commutative 61.3%

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

      7. *-commutative 61.3%

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

      8. associate-*l/ 61.4%

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

      9. associate-*r/ 61.4%

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

      10. div-inv 61.4%

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

      11. metadata-eval 61.4%

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

      12. unpow2 61.4%

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

   6. Applied egg-rr 61.4%

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

      1. unpow2 61.4%

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

      2. *-commutative 61.4%

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

      3. associate-*l* 54.4%

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

      4. *-commutative 54.4%

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

      5. associate-*l* 61.3%

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

      6. sin-mult 61.3%

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

      7. frac-2neg 61.3%

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

   8. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\frac{-\left(\cos 0 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{-2}} \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]
   9. Step-by-step derivation

      1. neg-sub0 61.4%

         \[\leadsto \frac{\color{blue}{0 - \left(\cos 0 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}}{-2} \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

      2. cos-0 61.4%

         \[\leadsto \frac{0 - \left(\color{blue}{1} - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{-2} \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

      3. associate--r- 61.4%

         \[\leadsto \frac{\color{blue}{\left(0 - 1\right) + \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}}{-2} \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

      4. metadata-eval 61.4%

         \[\leadsto \frac{\color{blue}{-1} + \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{-2} \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

      5. associate-*l* 61.4%

         \[\leadsto \frac{-1 + \cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}}{-2} \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]

   10. Simplified 61.4%

       \[\leadsto \color{blue}{\frac{-1 + \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}{-2}} \cdot \left(a \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 8 \cdot 10^{-5}:\\ \;\;\;\;{b}^{2} + \left(a \cdot angle\right) \cdot \left(\pi \cdot \left(\left(\pi \cdot \left(a \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{-1 + \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}{-2} \cdot \left(a \cdot a\right)\\ \end{array} \]

Alternative 8: 67.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{-154}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(a \cdot angle\right) \cdot \left(\pi \cdot \left(\left(\pi \cdot \left(a \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.25e-154)
   (* b b)
   (+
    (pow b 2.0)
    (* (* a angle) (* PI (* (* PI (* a angle)) 3.08641975308642e-5))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.25e-154) {
		tmp = b * b;
	} else {
		tmp = pow(b, 2.0) + ((a * angle) * (((double) M_PI) * ((((double) M_PI) * (a * angle)) * 3.08641975308642e-5)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.25e-154) {
		tmp = b * b;
	} else {
		tmp = Math.pow(b, 2.0) + ((a * angle) * (Math.PI * ((Math.PI * (a * angle)) * 3.08641975308642e-5)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.25e-154:
		tmp = b * b
	else:
		tmp = math.pow(b, 2.0) + ((a * angle) * (math.pi * ((math.pi * (a * angle)) * 3.08641975308642e-5)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.25e-154)
		tmp = Float64(b * b);
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(a * angle) * Float64(pi * Float64(Float64(pi * Float64(a * angle)) * 3.08641975308642e-5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.25e-154)
		tmp = b * b;
	else
		tmp = (b ^ 2.0) + ((a * angle) * (pi * ((pi * (a * angle)) * 3.08641975308642e-5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.25e-154], N[(b * b), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * angle), $MachinePrecision] * N[(Pi * N[(N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.25000000000000005e-154

   1. Initial program 80.2%

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

      1. associate-*l/ 80.8%

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

      2. associate-*r/ 80.8%

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

      3. associate-*l/ 80.9%

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

      4. associate-*r/ 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in angle around 0 80.2%

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

   5. Taylor expanded in angle around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-*l* 73.2%

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

   7. Simplified 73.2%

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

      1. unpow-prod-down 73.3%

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

      2. *-commutative 73.3%

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

      3. associate-*r* 73.2%

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

      4. *-commutative 73.2%

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

      5. associate-*l* 73.3%

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

      6. metadata-eval 73.3%

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

   9. Applied egg-rr 73.3%

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

   10. Taylor expanded in angle around 0 61.0%

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

       1. unpow2 61.0%

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

   12. Simplified 61.0%

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

   if 1.25000000000000005e-154 < a

   1. Initial program 80.2%

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

      1. associate-*l/ 80.2%

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

      2. associate-*r/ 80.2%

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

      3. associate-*l/ 80.2%

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

      4. associate-*r/ 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in angle around 0 79.5%

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

   5. Taylor expanded in angle around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*l* 76.7%

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

   7. Simplified 76.7%

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

      1. unpow2 76.7%

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

      2. associate-*r* 76.7%

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

      3. *-commutative 76.7%

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

      4. associate-*r* 76.8%

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

      5. *-commutative 76.8%

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

      6. associate-*l* 76.8%

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

      7. associate-*r* 76.9%

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

      8. *-commutative 76.9%

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

      9. *-commutative 76.9%

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

      10. associate-*l* 76.8%

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

      11. associate-*r* 76.9%

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

      12. *-commutative 76.9%

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

      13. associate-*l* 76.8%

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

      14. metadata-eval 76.8%

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

   9. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{\left(\left(\left(\pi \cdot \left(angle \cdot a\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \pi\right) \cdot \left(angle \cdot a\right)} + {\left(b \cdot 1\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{-154}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(a \cdot angle\right) \cdot \left(\pi \cdot \left(\left(\pi \cdot \left(a \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \]

Alternative 9: 67.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(a \cdot angle\right)\\ \mathbf{if}\;a \leq 1.25 \cdot 10^{-154}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_0 \cdot 3.08641975308642 \cdot 10^{-5}, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* a angle))))
   (if (<= a 1.25e-154)
     (* b b)
     (fma t_0 (* t_0 3.08641975308642e-5) (* b b)))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (a * angle);
	double tmp;
	if (a <= 1.25e-154) {
		tmp = b * b;
	} else {
		tmp = fma(t_0, (t_0 * 3.08641975308642e-5), (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(a * angle))
	tmp = 0.0
	if (a <= 1.25e-154)
		tmp = Float64(b * b);
	else
		tmp = fma(t_0, Float64(t_0 * 3.08641975308642e-5), Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.25e-154], N[(b * b), $MachinePrecision], N[(t$95$0 * N[(t$95$0 * 3.08641975308642e-5), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.25000000000000005e-154

   1. Initial program 80.2%

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

      1. associate-*l/ 80.8%

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

      2. associate-*r/ 80.8%

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

      3. associate-*l/ 80.9%

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

      4. associate-*r/ 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in angle around 0 80.2%

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

   5. Taylor expanded in angle around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-*l* 73.2%

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

   7. Simplified 73.2%

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

      1. unpow-prod-down 73.3%

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

      2. *-commutative 73.3%

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

      3. associate-*r* 73.2%

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

      4. *-commutative 73.2%

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

      5. associate-*l* 73.3%

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

      6. metadata-eval 73.3%

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

   9. Applied egg-rr 73.3%

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

   10. Taylor expanded in angle around 0 61.0%

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

       1. unpow2 61.0%

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

   12. Simplified 61.0%

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

   if 1.25000000000000005e-154 < a

   1. Initial program 80.2%

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

      1. associate-*l/ 80.2%

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

      2. associate-*r/ 80.2%

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

      3. associate-*l/ 80.2%

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

      4. associate-*r/ 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in angle around 0 79.5%

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

   5. Taylor expanded in angle around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*l* 76.7%

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

   7. Simplified 76.7%

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

      1. unpow-prod-down 76.7%

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

      2. *-commutative 76.7%

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

      3. associate-*r* 76.7%

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

      4. *-commutative 76.7%

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

      5. associate-*l* 76.7%

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

      6. metadata-eval 76.7%

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

   9. Applied egg-rr 76.7%

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

       1. unpow2 76.7%

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

       2. associate-*l* 76.8%

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

       3. fma-def 76.8%

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

       4. associate-*r* 76.8%

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

       5. *-commutative 76.8%

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

       6. associate-*l* 76.8%

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

       7. *-commutative 76.8%

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

       8. associate-*r* 76.8%

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

       9. *-rgt-identity 76.8%

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

       10. unpow2 76.8%

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

   11. Applied egg-rr 76.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot \left(\pi \cdot a\right), \left(3.08641975308642 \cdot 10^{-5} \cdot \pi\right) \cdot \left(angle \cdot a\right), b \cdot b\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 76.8%

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

       2. *-commutative 76.8%

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

       3. associate-*r* 76.7%

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

       4. associate-*l* 76.8%

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

   13. Simplified 76.8%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{-154}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot \left(a \cdot angle\right), \left(\pi \cdot \left(a \cdot angle\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}, b \cdot b\right)\\ \end{array} \]

Alternative 10: 67.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{-154}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot angle\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right), angle \cdot \left(a \cdot \pi\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.25e-154)
   (* b b)
   (fma
    (* (* a angle) (* PI 3.08641975308642e-5))
    (* angle (* a PI))
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.25e-154) {
		tmp = b * b;
	} else {
		tmp = fma(((a * angle) * (((double) M_PI) * 3.08641975308642e-5)), (angle * (a * ((double) M_PI))), (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.25e-154)
		tmp = Float64(b * b);
	else
		tmp = fma(Float64(Float64(a * angle) * Float64(pi * 3.08641975308642e-5)), Float64(angle * Float64(a * pi)), Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.25e-154], N[(b * b), $MachinePrecision], N[(N[(N[(a * angle), $MachinePrecision] * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.25000000000000005e-154

   1. Initial program 80.2%

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

      1. associate-*l/ 80.8%

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

      2. associate-*r/ 80.8%

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

      3. associate-*l/ 80.9%

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

      4. associate-*r/ 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in angle around 0 80.2%

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

   5. Taylor expanded in angle around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-*l* 73.2%

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

   7. Simplified 73.2%

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

      1. unpow-prod-down 73.3%

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

      2. *-commutative 73.3%

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

      3. associate-*r* 73.2%

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

      4. *-commutative 73.2%

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

      5. associate-*l* 73.3%

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

      6. metadata-eval 73.3%

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

   9. Applied egg-rr 73.3%

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

   10. Taylor expanded in angle around 0 61.0%

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

       1. unpow2 61.0%

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

   12. Simplified 61.0%

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

   if 1.25000000000000005e-154 < a

   1. Initial program 80.2%

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

      1. associate-*l/ 80.2%

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

      2. associate-*r/ 80.2%

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

      3. associate-*l/ 80.2%

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

      4. associate-*r/ 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in angle around 0 79.5%

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

   5. Taylor expanded in angle around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*l* 76.7%

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

   7. Simplified 76.7%

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

      1. unpow-prod-down 76.7%

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

      2. *-commutative 76.7%

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

      3. associate-*r* 76.7%

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

      4. *-commutative 76.7%

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

      5. associate-*l* 76.7%

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

      6. metadata-eval 76.7%

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

   9. Applied egg-rr 76.7%

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

       1. *-commutative 76.7%

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

       2. unpow2 76.7%

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

       3. associate-*r* 76.8%

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

       4. fma-def 76.8%

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

       5. associate-*r* 76.7%

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

       6. associate-*r* 76.8%

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

       7. *-commutative 76.8%

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

       8. associate-*l* 76.8%

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

       9. *-rgt-identity 76.8%

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

       10. unpow2 76.8%

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

   11. Applied egg-rr 76.8%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{-154}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot angle\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right), angle \cdot \left(a \cdot \pi\right), b \cdot b\right)\\ \end{array} \]

Alternative 11: 67.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{-154}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2}, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.25e-154)
   (* b b)
   (fma 3.08641975308642e-5 (pow (* PI (* a angle)) 2.0) (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.25e-154) {
		tmp = b * b;
	} else {
		tmp = fma(3.08641975308642e-5, pow((((double) M_PI) * (a * angle)), 2.0), (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.25e-154)
		tmp = Float64(b * b);
	else
		tmp = fma(3.08641975308642e-5, (Float64(pi * Float64(a * angle)) ^ 2.0), Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.25e-154], N[(b * b), $MachinePrecision], N[(3.08641975308642e-5 * N[Power[N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.25000000000000005e-154

   1. Initial program 80.2%

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

      1. associate-*l/ 80.8%

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

      2. associate-*r/ 80.8%

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

      3. associate-*l/ 80.9%

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

      4. associate-*r/ 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in angle around 0 80.2%

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

   5. Taylor expanded in angle around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-*l* 73.2%

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

   7. Simplified 73.2%

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

      1. unpow-prod-down 73.3%

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

      2. *-commutative 73.3%

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

      3. associate-*r* 73.2%

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

      4. *-commutative 73.2%

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

      5. associate-*l* 73.3%

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

      6. metadata-eval 73.3%

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

   9. Applied egg-rr 73.3%

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

   10. Taylor expanded in angle around 0 61.0%

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

       1. unpow2 61.0%

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

   12. Simplified 61.0%

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

   if 1.25000000000000005e-154 < a

   1. Initial program 80.2%

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

      1. associate-*l/ 80.2%

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

      2. associate-*r/ 80.2%

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

      3. associate-*l/ 80.2%

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

      4. associate-*r/ 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in angle around 0 79.5%

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

   5. Taylor expanded in angle around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*l* 76.7%

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

   7. Simplified 76.7%

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

      1. unpow-prod-down 76.7%

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

      2. *-commutative 76.7%

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

      3. associate-*r* 76.7%

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

      4. *-commutative 76.7%

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

      5. associate-*l* 76.7%

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

      6. metadata-eval 76.7%

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

   9. Applied egg-rr 76.7%

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

   10. Taylor expanded in angle around 0 57.0%

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

       1. fma-def 57.0%

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

       2. unpow2 57.0%

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

       3. *-commutative 57.0%

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

       4. unpow2 57.0%

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

       5. unpow2 57.0%

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

       6. unpow2 57.0%

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

       7. swap-sqr 57.1%

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

       8. swap-sqr 76.7%

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

       9. associate-*r* 76.8%

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

       10. associate-*r* 76.7%

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

       11. unpow2 76.7%

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

       12. associate-*r* 76.7%

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

       13. *-commutative 76.7%

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

       14. associate-*r* 76.7%

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

   12. Simplified 76.7%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{-154}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2}, b \cdot b\right)\\ \end{array} \]

Alternative 12: 67.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{-154}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.25e-154)
   (* b b)
   (fma b b (* 3.08641975308642e-5 (pow (* angle (* a PI)) 2.0)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.25e-154) {
		tmp = b * b;
	} else {
		tmp = fma(b, b, (3.08641975308642e-5 * pow((angle * (a * ((double) M_PI))), 2.0)));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.25e-154)
		tmp = Float64(b * b);
	else
		tmp = fma(b, b, Float64(3.08641975308642e-5 * (Float64(angle * Float64(a * pi)) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.25e-154], N[(b * b), $MachinePrecision], N[(b * b + N[(3.08641975308642e-5 * N[Power[N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.25000000000000005e-154

   1. Initial program 80.2%

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

      1. associate-*l/ 80.8%

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

      2. associate-*r/ 80.8%

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

      3. associate-*l/ 80.9%

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

      4. associate-*r/ 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in angle around 0 80.2%

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

   5. Taylor expanded in angle around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-*l* 73.2%

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

   7. Simplified 73.2%

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

      1. unpow-prod-down 73.3%

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

      2. *-commutative 73.3%

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

      3. associate-*r* 73.2%

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

      4. *-commutative 73.2%

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

      5. associate-*l* 73.3%

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

      6. metadata-eval 73.3%

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

   9. Applied egg-rr 73.3%

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

   10. Taylor expanded in angle around 0 61.0%

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

       1. unpow2 61.0%

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

   12. Simplified 61.0%

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

   if 1.25000000000000005e-154 < a

   1. Initial program 80.2%

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

      1. associate-*l/ 80.2%

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

      2. associate-*r/ 80.2%

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

      3. associate-*l/ 80.2%

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

      4. associate-*r/ 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in angle around 0 79.5%

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

   5. Taylor expanded in angle around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*l* 76.7%

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

   7. Simplified 76.7%

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

      1. unpow-prod-down 76.7%

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

      2. *-commutative 76.7%

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

      3. associate-*r* 76.7%

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

      4. *-commutative 76.7%

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

      5. associate-*l* 76.7%

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

      6. metadata-eval 76.7%

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

   9. Applied egg-rr 76.7%

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

       1. +-commutative 76.7%

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

       2. *-rgt-identity 76.7%

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

       3. unpow2 76.7%

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

       4. fma-def 76.7%

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

       5. associate-*r* 76.7%

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

       6. *-commutative 76.7%

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

       7. associate-*l* 76.7%

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

   11. Applied egg-rr 76.7%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{-154}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2}\right)\\ \end{array} \]

Alternative 13: 57.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ b \cdot b \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (* b b))

C

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

Fortran

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

Java

public static double code(double a, double b, double angle) {
	return b * b;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. associate-*l/ 80.6%

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

   2. associate-*r/ 80.6%

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

   3. associate-*l/ 80.6%

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

   4. associate-*r/ 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in angle around 0 79.9%

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

5. Taylor expanded in angle around 0 74.6%

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

   1. *-commutative 74.6%

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

   2. associate-*l* 74.6%

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

7. Simplified 74.6%

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

   1. unpow-prod-down 74.6%

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

   2. *-commutative 74.6%

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

   3. associate-*r* 74.6%

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

   4. *-commutative 74.6%

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

   5. associate-*l* 74.6%

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

   6. metadata-eval 74.6%

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

9. Applied egg-rr 74.6%

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

10. Taylor expanded in angle around 0 56.1%

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

    1. unpow2 56.1%

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

12. Simplified 56.1%

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

13. Final simplification 56.1%

    \[\leadsto b \cdot b \]

Reproduce

herbie shell --seed 2023297 
(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)))