ab-angle->ABCF C

Percentage Accurate: 79.5% → 79.6%

Time: 27.1s

Alternatives: 11

Speedup: 1.0×

• 36.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* PI (/ angle 180.0)))) 2.0)
  (pow (* b (sin (* PI (* angle -0.005555555555555556)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * cos((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((b * sin((((double) M_PI) * (angle * -0.005555555555555556)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.cos((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle * -0.005555555555555556)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.cos((math.pi * (angle / 180.0)))), 2.0) + math.pow((b * math.sin((math.pi * (angle * -0.005555555555555556)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * cos(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle * -0.005555555555555556)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * cos((pi * (angle / 180.0)))) ^ 2.0) + ((b * sin((pi * (angle * -0.005555555555555556)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.1%

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

   1. unpow2 81.1%

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

   2. swap-sqr 72.4%

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

   3. sqr-neg 72.4%

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

   4. swap-sqr 81.1%

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

   5. unpow2 81.1%

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

   6. distribute-lft-neg-out 81.1%

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

   7. distribute-rgt-neg-in 81.1%

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

   8. sin-neg 81.1%

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

   9. distribute-rgt-neg-out 81.1%

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

   10. distribute-frac-neg 81.1%

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

   11. unpow2 81.1%

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

   12. associate-*l* 80.0%

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

3. Simplified 81.2%

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

4. Final simplification 81.2%

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

Alternative 2: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
  (pow (* a (cos (* 0.005555555555555556 (* PI angle)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((a * cos((0.005555555555555556 * (((double) M_PI) * angle)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((a * Math.cos((0.005555555555555556 * (Math.PI * angle)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((b * math.sin((math.pi * (angle / 180.0)))), 2.0) + math.pow((a * math.cos((0.005555555555555556 * (math.pi * angle)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * cos(Float64(0.005555555555555556 * Float64(pi * angle)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((b * sin((pi * (angle / 180.0)))) ^ 2.0) + ((a * cos((0.005555555555555556 * (pi * angle)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.1%

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

2. Taylor expanded in angle around inf 81.0%

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

3. Final simplification 81.0%

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

Alternative 3: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

2. Taylor expanded in angle around inf 81.0%

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

   1. *-commutative 81.0%

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

   2. associate-*r* 81.0%

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

   3. metadata-eval 81.0%

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

   4. associate-/r/ 81.0%

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

   5. associate-*l/ 81.0%

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

   6. *-lft-identity 81.0%

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

   7. associate-/l* 81.0%

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

   8. associate-*r/ 81.0%

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

4. Simplified 81.0%

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

5. Final simplification 81.0%

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

Alternative 4: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. unpow2 81.1%

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

   2. swap-sqr 72.4%

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

   3. sqr-neg 72.4%

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

   4. swap-sqr 81.1%

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

   5. unpow2 81.1%

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

   6. distribute-lft-neg-out 81.1%

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

   7. distribute-rgt-neg-in 81.1%

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

   8. sin-neg 81.1%

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

   9. distribute-rgt-neg-out 81.1%

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

   10. distribute-frac-neg 81.1%

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

   11. unpow2 81.1%

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

   12. associate-*l* 80.0%

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

3. Simplified 81.2%

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

4. Taylor expanded in angle around inf 81.1%

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

   1. associate-*r* 80.6%

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

   2. *-commutative 80.6%

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

   3. *-commutative 80.6%

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

   4. unpow1 80.6%

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

   5. sqr-pow 38.7%

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

   6. fabs-sqr 38.7%

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

   7. sqr-pow 80.6%

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

   8. unpow1 80.6%

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

   9. associate-*r* 80.5%

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

   10. *-commutative 80.5%

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

   11. associate-*r* 80.5%

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

   12. fabs-mul 80.5%

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

   13. fabs-mul 80.5%

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

   14. metadata-eval 80.5%

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

   15. unpow1 80.5%

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

   16. sqr-pow 80.3%

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

   17. fabs-sqr 80.3%

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

   18. sqr-pow 80.5%

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

   19. unpow1 80.5%

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

   20. unpow1 80.5%

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

   21. sqr-pow 41.7%

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

   22. fabs-sqr 41.7%

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

   23. sqr-pow 80.5%

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

   24. unpow1 80.5%

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

6. Simplified 81.1%

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

7. Final simplification 81.1%

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

Alternative 5: 79.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. unpow2 81.1%

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

   2. swap-sqr 72.4%

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

   3. sqr-neg 72.4%

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

   4. swap-sqr 81.1%

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

   5. unpow2 81.1%

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

   6. distribute-lft-neg-out 81.1%

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

   7. distribute-rgt-neg-in 81.1%

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

   8. sin-neg 81.1%

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

   9. distribute-rgt-neg-out 81.1%

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

   10. distribute-frac-neg 81.1%

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

   11. unpow2 81.1%

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

   12. associate-*l* 80.0%

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

3. Simplified 81.2%

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

4. Taylor expanded in angle around 0 80.6%

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

5. Taylor expanded in angle around inf 80.5%

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

   1. associate-*r* 80.6%

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

   2. *-commutative 80.6%

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

   3. *-commutative 80.6%

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

   4. unpow1 80.6%

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

   5. sqr-pow 38.7%

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

   6. fabs-sqr 38.7%

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

   7. sqr-pow 80.6%

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

   8. unpow1 80.6%

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

   9. associate-*r* 80.5%

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

   10. *-commutative 80.5%

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

   11. associate-*r* 80.5%

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

   12. fabs-mul 80.5%

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

   13. fabs-mul 80.5%

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

   14. metadata-eval 80.5%

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

   15. unpow1 80.5%

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

   16. sqr-pow 80.3%

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

   17. fabs-sqr 80.3%

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

   18. sqr-pow 80.5%

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

   19. unpow1 80.5%

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

   20. unpow1 80.5%

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

   21. sqr-pow 41.7%

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

   22. fabs-sqr 41.7%

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

   23. sqr-pow 80.5%

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

   24. unpow1 80.5%

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

7. Simplified 80.5%

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

8. Final simplification 80.5%

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

Alternative 6: 79.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. unpow2 81.1%

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

   2. swap-sqr 72.4%

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

   3. sqr-neg 72.4%

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

   4. swap-sqr 81.1%

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

   5. unpow2 81.1%

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

   6. distribute-lft-neg-out 81.1%

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

   7. distribute-rgt-neg-in 81.1%

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

   8. sin-neg 81.1%

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

   9. distribute-rgt-neg-out 81.1%

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

   10. distribute-frac-neg 81.1%

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

   11. unpow2 81.1%

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

   12. associate-*l* 80.0%

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

3. Simplified 81.2%

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

4. Taylor expanded in angle around 0 80.6%

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

5. Final simplification 80.6%

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

Alternative 7: 67.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-112}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.1e-112)
   (* a a)
   (+ (pow a 2.0) (pow (* b (* angle (/ PI 180.0))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.1e-112) {
		tmp = a * a;
	} else {
		tmp = pow(a, 2.0) + pow((b * (angle * (((double) M_PI) / 180.0))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.1e-112) {
		tmp = a * a;
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow((b * (angle * (Math.PI / 180.0))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.1e-112:
		tmp = a * a
	else:
		tmp = math.pow(a, 2.0) + math.pow((b * (angle * (math.pi / 180.0))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.1e-112)
		tmp = Float64(a * a);
	else
		tmp = Float64((a ^ 2.0) + (Float64(b * Float64(angle * Float64(pi / 180.0))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.1e-112)
		tmp = a * a;
	else
		tmp = (a ^ 2.0) + ((b * (angle * (pi / 180.0))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.1e-112], N[(a * a), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.10000000000000011e-112

   1. Initial program 81.2%

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

      1. unpow2 81.2%

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

      2. swap-sqr 76.6%

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

      3. sqr-neg 76.6%

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

      4. swap-sqr 81.2%

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

      5. unpow2 81.2%

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

      6. distribute-lft-neg-out 81.2%

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

      7. distribute-rgt-neg-in 81.2%

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

      8. sin-neg 81.2%

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

      9. distribute-rgt-neg-out 81.2%

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

      10. distribute-frac-neg 81.2%

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

      11. unpow2 81.2%

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

      12. associate-*l* 80.8%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in angle around 0 80.8%

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

   5. Taylor expanded in angle around 0 75.6%

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

      1. associate-*r* 75.6%

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

      2. *-commutative 75.6%

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

      3. *-commutative 75.6%

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

      4. unpow1 75.6%

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

      5. sqr-pow 36.2%

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

      6. fabs-sqr 36.2%

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

      7. sqr-pow 75.6%

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

      8. unpow1 75.6%

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

      9. associate-*r* 75.6%

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

      10. *-commutative 75.6%

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

      11. associate-*r* 75.6%

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

      12. fabs-mul 75.6%

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

      13. fabs-mul 75.6%

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

      14. metadata-eval 75.6%

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

      15. unpow1 75.6%

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

      16. sqr-pow 75.5%

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

      17. fabs-sqr 75.5%

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

      18. sqr-pow 75.6%

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

      19. unpow1 75.6%

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

      20. unpow1 75.6%

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

      21. sqr-pow 39.4%

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

      22. fabs-sqr 39.4%

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

      23. sqr-pow 75.6%

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

      24. unpow1 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in a around inf 65.2%

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

      1. unpow2 65.2%

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

   10. Simplified 65.2%

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

   if 1.10000000000000011e-112 < b

   1. Initial program 80.7%

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

      1. unpow2 80.7%

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

      2. swap-sqr 61.1%

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

      3. sqr-neg 61.1%

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

      4. swap-sqr 80.7%

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

      5. unpow2 80.7%

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

      6. distribute-lft-neg-out 80.7%

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

      7. distribute-rgt-neg-in 80.7%

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

      8. sin-neg 80.7%

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

      9. distribute-rgt-neg-out 80.7%

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

      10. distribute-frac-neg 80.7%

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

      11. unpow2 80.7%

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

      12. associate-*l* 78.1%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in angle around 0 80.0%

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

   5. Taylor expanded in angle around 0 75.1%

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

      1. associate-*r* 75.2%

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

      2. *-commutative 75.2%

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

      3. *-commutative 75.2%

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

      4. unpow1 75.2%

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

      5. sqr-pow 35.3%

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

      6. fabs-sqr 35.3%

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

      7. sqr-pow 75.2%

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

      8. unpow1 75.2%

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

      9. associate-*r* 75.1%

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

      10. *-commutative 75.1%

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

      11. associate-*r* 75.2%

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

      12. fabs-mul 75.2%

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

      13. fabs-mul 75.2%

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

      14. metadata-eval 75.2%

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

      15. unpow1 75.2%

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

      16. sqr-pow 74.9%

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

      17. fabs-sqr 74.9%

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

      18. sqr-pow 75.2%

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

      19. unpow1 75.2%

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

      20. unpow1 75.2%

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

      21. sqr-pow 39.8%

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

      22. fabs-sqr 39.8%

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

      23. sqr-pow 75.2%

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

      24. unpow1 75.2%

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

   7. Simplified 75.2%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-112}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \end{array} \]

Alternative 8: 67.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-112}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.8e-112)
   (* a a)
   (+ (pow a 2.0) (pow (* b (* PI (* angle -0.005555555555555556))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.8e-112) {
		tmp = a * a;
	} else {
		tmp = pow(a, 2.0) + pow((b * (((double) M_PI) * (angle * -0.005555555555555556))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.8e-112) {
		tmp = a * a;
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow((b * (Math.PI * (angle * -0.005555555555555556))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 4.8e-112:
		tmp = a * a
	else:
		tmp = math.pow(a, 2.0) + math.pow((b * (math.pi * (angle * -0.005555555555555556))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 4.8e-112)
		tmp = Float64(a * a);
	else
		tmp = Float64((a ^ 2.0) + (Float64(b * Float64(pi * Float64(angle * -0.005555555555555556))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 4.8e-112)
		tmp = a * a;
	else
		tmp = (a ^ 2.0) + ((b * (pi * (angle * -0.005555555555555556))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 4.8e-112], N[(a * a), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[(Pi * N[(angle * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.8000000000000001e-112

   1. Initial program 81.2%

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

      1. unpow2 81.2%

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

      2. swap-sqr 76.6%

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

      3. sqr-neg 76.6%

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

      4. swap-sqr 81.2%

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

      5. unpow2 81.2%

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

      6. distribute-lft-neg-out 81.2%

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

      7. distribute-rgt-neg-in 81.2%

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

      8. sin-neg 81.2%

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

      9. distribute-rgt-neg-out 81.2%

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

      10. distribute-frac-neg 81.2%

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

      11. unpow2 81.2%

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

      12. associate-*l* 80.8%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in angle around 0 80.8%

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

   5. Taylor expanded in angle around 0 75.6%

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

      1. associate-*r* 75.6%

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

      2. *-commutative 75.6%

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

      3. *-commutative 75.6%

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

      4. unpow1 75.6%

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

      5. sqr-pow 36.2%

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

      6. fabs-sqr 36.2%

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

      7. sqr-pow 75.6%

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

      8. unpow1 75.6%

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

      9. associate-*r* 75.6%

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

      10. *-commutative 75.6%

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

      11. associate-*r* 75.6%

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

      12. fabs-mul 75.6%

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

      13. fabs-mul 75.6%

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

      14. metadata-eval 75.6%

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

      15. unpow1 75.6%

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

      16. sqr-pow 75.5%

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

      17. fabs-sqr 75.5%

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

      18. sqr-pow 75.6%

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

      19. unpow1 75.6%

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

      20. unpow1 75.6%

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

      21. sqr-pow 39.4%

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

      22. fabs-sqr 39.4%

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

      23. sqr-pow 75.6%

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

      24. unpow1 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in a around inf 65.2%

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

      1. unpow2 65.2%

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

   10. Simplified 65.2%

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

   if 4.8000000000000001e-112 < b

   1. Initial program 80.7%

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

      1. unpow2 80.7%

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

      2. swap-sqr 61.1%

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

      3. sqr-neg 61.1%

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

      4. swap-sqr 80.7%

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

      5. unpow2 80.7%

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

      6. distribute-lft-neg-out 80.7%

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

      7. distribute-rgt-neg-in 80.7%

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

      8. sin-neg 80.7%

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

      9. distribute-rgt-neg-out 80.7%

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

      10. distribute-frac-neg 80.7%

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

      11. unpow2 80.7%

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

      12. associate-*l* 78.1%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in angle around 0 80.0%

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

   5. Taylor expanded in angle around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. *-commutative 75.1%

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

      3. associate-*r* 75.2%

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

      4. *-commutative 75.2%

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

   7. Simplified 75.2%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-112}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 9: 67.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-112}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.1e-112)
   (* a a)
   (pow (hypot (* angle (* b (* PI 0.005555555555555556))) a) 2.0)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.1e-112) {
		tmp = a * a;
	} else {
		tmp = pow(hypot((angle * (b * (((double) M_PI) * 0.005555555555555556))), a), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.1e-112) {
		tmp = a * a;
	} else {
		tmp = Math.pow(Math.hypot((angle * (b * (Math.PI * 0.005555555555555556))), a), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 3.1e-112:
		tmp = a * a
	else:
		tmp = math.pow(math.hypot((angle * (b * (math.pi * 0.005555555555555556))), a), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.1e-112)
		tmp = Float64(a * a);
	else
		tmp = hypot(Float64(angle * Float64(b * Float64(pi * 0.005555555555555556))), a) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 3.1e-112)
		tmp = a * a;
	else
		tmp = hypot((angle * (b * (pi * 0.005555555555555556))), a) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.1e-112], N[(a * a), $MachinePrecision], N[Power[N[Sqrt[N[(angle * N[(b * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + a ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.0999999999999998e-112

   1. Initial program 81.2%

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

      1. unpow2 81.2%

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

      2. swap-sqr 76.6%

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

      3. sqr-neg 76.6%

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

      4. swap-sqr 81.2%

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

      5. unpow2 81.2%

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

      6. distribute-lft-neg-out 81.2%

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

      7. distribute-rgt-neg-in 81.2%

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

      8. sin-neg 81.2%

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

      9. distribute-rgt-neg-out 81.2%

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

      10. distribute-frac-neg 81.2%

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

      11. unpow2 81.2%

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

      12. associate-*l* 80.8%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in angle around 0 80.8%

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

   5. Taylor expanded in angle around 0 75.6%

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

      1. associate-*r* 75.6%

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

      2. *-commutative 75.6%

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

      3. *-commutative 75.6%

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

      4. unpow1 75.6%

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

      5. sqr-pow 36.2%

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

      6. fabs-sqr 36.2%

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

      7. sqr-pow 75.6%

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

      8. unpow1 75.6%

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

      9. associate-*r* 75.6%

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

      10. *-commutative 75.6%

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

      11. associate-*r* 75.6%

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

      12. fabs-mul 75.6%

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

      13. fabs-mul 75.6%

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

      14. metadata-eval 75.6%

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

      15. unpow1 75.6%

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

      16. sqr-pow 75.5%

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

      17. fabs-sqr 75.5%

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

      18. sqr-pow 75.6%

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

      19. unpow1 75.6%

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

      20. unpow1 75.6%

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

      21. sqr-pow 39.4%

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

      22. fabs-sqr 39.4%

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

      23. sqr-pow 75.6%

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

      24. unpow1 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in a around inf 65.2%

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

      1. unpow2 65.2%

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

   10. Simplified 65.2%

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

   if 3.0999999999999998e-112 < b

   1. Initial program 80.7%

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

      1. unpow2 80.7%

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

      2. swap-sqr 61.1%

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

      3. sqr-neg 61.1%

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

      4. swap-sqr 80.7%

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

      5. unpow2 80.7%

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

      6. distribute-lft-neg-out 80.7%

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

      7. distribute-rgt-neg-in 80.7%

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

      8. sin-neg 80.7%

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

      9. distribute-rgt-neg-out 80.7%

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

      10. distribute-frac-neg 80.7%

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

      11. unpow2 80.7%

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

      12. associate-*l* 78.1%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in angle around 0 80.0%

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

   5. Taylor expanded in angle around 0 75.1%

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

      1. associate-*r* 75.2%

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

      2. *-commutative 75.2%

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

      3. *-commutative 75.2%

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

      4. unpow1 75.2%

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

      5. sqr-pow 35.3%

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

      6. fabs-sqr 35.3%

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

      7. sqr-pow 75.2%

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

      8. unpow1 75.2%

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

      9. associate-*r* 75.1%

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

      10. *-commutative 75.1%

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

      11. associate-*r* 75.2%

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

      12. fabs-mul 75.2%

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

      13. fabs-mul 75.2%

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

      14. metadata-eval 75.2%

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

      15. unpow1 75.2%

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

      16. sqr-pow 74.9%

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

      17. fabs-sqr 74.9%

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

      18. sqr-pow 75.2%

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

      19. unpow1 75.2%

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

      20. unpow1 75.2%

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

      21. sqr-pow 39.8%

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

      22. fabs-sqr 39.8%

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

      23. sqr-pow 75.2%

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

      24. unpow1 75.2%

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

   7. Simplified 75.2%

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

      1. add-sqr-sqrt 75.2%

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

      2. pow2 75.2%

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

      3. +-commutative 75.2%

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

      4. unpow2 75.2%

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

      5. *-rgt-identity 75.2%

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

      6. pow2 75.2%

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

      7. hypot-def 75.2%

         \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(b \cdot \left(angle \cdot \frac{\pi}{180}\right), a\right)\right)}}^{2} \]

      8. *-commutative 75.2%

         \[\leadsto {\left(\mathsf{hypot}\left(\color{blue}{\left(angle \cdot \frac{\pi}{180}\right) \cdot b}, a\right)\right)}^{2} \]

      9. associate-*l* 75.2%

         \[\leadsto {\left(\mathsf{hypot}\left(\color{blue}{angle \cdot \left(\frac{\pi}{180} \cdot b\right)}, a\right)\right)}^{2} \]

      10. div-inv 75.2%

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

      11. metadata-eval 75.2%

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

   9. Applied egg-rr 75.2%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-112}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a\right)\right)}^{2}\\ \end{array} \]

Alternative 10: 67.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-112}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(\pi \cdot \left(b \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.8e-112)
   (* a a)
   (+ (* a a) (pow (* angle (* PI (* b 0.005555555555555556))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.8e-112) {
		tmp = a * a;
	} else {
		tmp = (a * a) + pow((angle * (((double) M_PI) * (b * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.8e-112) {
		tmp = a * a;
	} else {
		tmp = (a * a) + Math.pow((angle * (Math.PI * (b * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 4.8e-112:
		tmp = a * a
	else:
		tmp = (a * a) + math.pow((angle * (math.pi * (b * 0.005555555555555556))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 4.8e-112)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + (Float64(angle * Float64(pi * Float64(b * 0.005555555555555556))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 4.8e-112)
		tmp = a * a;
	else
		tmp = (a * a) + ((angle * (pi * (b * 0.005555555555555556))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 4.8e-112], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(angle * N[(Pi * N[(b * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.8000000000000001e-112

   1. Initial program 81.2%

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

      1. unpow2 81.2%

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

      2. swap-sqr 76.6%

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

      3. sqr-neg 76.6%

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

      4. swap-sqr 81.2%

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

      5. unpow2 81.2%

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

      6. distribute-lft-neg-out 81.2%

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

      7. distribute-rgt-neg-in 81.2%

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

      8. sin-neg 81.2%

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

      9. distribute-rgt-neg-out 81.2%

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

      10. distribute-frac-neg 81.2%

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

      11. unpow2 81.2%

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

      12. associate-*l* 80.8%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in angle around 0 80.8%

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

   5. Taylor expanded in angle around 0 75.6%

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

      1. associate-*r* 75.6%

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

      2. *-commutative 75.6%

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

      3. *-commutative 75.6%

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

      4. unpow1 75.6%

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

      5. sqr-pow 36.2%

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

      6. fabs-sqr 36.2%

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

      7. sqr-pow 75.6%

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

      8. unpow1 75.6%

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

      9. associate-*r* 75.6%

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

      10. *-commutative 75.6%

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

      11. associate-*r* 75.6%

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

      12. fabs-mul 75.6%

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

      13. fabs-mul 75.6%

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

      14. metadata-eval 75.6%

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

      15. unpow1 75.6%

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

      16. sqr-pow 75.5%

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

      17. fabs-sqr 75.5%

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

      18. sqr-pow 75.6%

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

      19. unpow1 75.6%

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

      20. unpow1 75.6%

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

      21. sqr-pow 39.4%

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

      22. fabs-sqr 39.4%

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

      23. sqr-pow 75.6%

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

      24. unpow1 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in a around inf 65.2%

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

      1. unpow2 65.2%

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

   10. Simplified 65.2%

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

   if 4.8000000000000001e-112 < b

   1. Initial program 80.7%

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

      1. unpow2 80.7%

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

      2. swap-sqr 61.1%

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

      3. sqr-neg 61.1%

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

      4. swap-sqr 80.7%

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

      5. unpow2 80.7%

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

      6. distribute-lft-neg-out 80.7%

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

      7. distribute-rgt-neg-in 80.7%

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

      8. sin-neg 80.7%

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

      9. distribute-rgt-neg-out 80.7%

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

      10. distribute-frac-neg 80.7%

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

      11. unpow2 80.7%

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

      12. associate-*l* 78.1%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in angle around 0 80.0%

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

   5. Taylor expanded in angle around 0 75.1%

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

      1. associate-*r* 75.2%

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

      2. *-commutative 75.2%

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

      3. *-commutative 75.2%

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

      4. unpow1 75.2%

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

      5. sqr-pow 35.3%

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

      6. fabs-sqr 35.3%

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

      7. sqr-pow 75.2%

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

      8. unpow1 75.2%

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

      9. associate-*r* 75.1%

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

      10. *-commutative 75.1%

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

      11. associate-*r* 75.2%

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

      12. fabs-mul 75.2%

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

      13. fabs-mul 75.2%

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

      14. metadata-eval 75.2%

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

      15. unpow1 75.2%

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

      16. sqr-pow 74.9%

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

      17. fabs-sqr 74.9%

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

      18. sqr-pow 75.2%

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

      19. unpow1 75.2%

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

      20. unpow1 75.2%

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

      21. sqr-pow 39.8%

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

      22. fabs-sqr 39.8%

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

      23. sqr-pow 75.2%

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

      24. unpow1 75.2%

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

   7. Simplified 75.2%

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

      1. +-commutative 75.2%

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

      2. unpow2 75.2%

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

      3. fma-def 75.2%

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

      4. *-commutative 75.2%

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

      5. associate-*l* 75.2%

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

      6. div-inv 75.2%

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

      7. metadata-eval 75.2%

         \[\leadsto \mathsf{fma}\left(angle \cdot \left(\left(\pi \cdot \color{blue}{0.005555555555555556}\right) \cdot b\right), b \cdot \left(angle \cdot \frac{\pi}{180}\right), {\left(a \cdot 1\right)}^{2}\right) \]

      8. *-commutative 75.2%

         \[\leadsto \mathsf{fma}\left(angle \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot b\right), \color{blue}{\left(angle \cdot \frac{\pi}{180}\right) \cdot b}, {\left(a \cdot 1\right)}^{2}\right) \]

      9. associate-*l* 75.2%

         \[\leadsto \mathsf{fma}\left(angle \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot b\right), \color{blue}{angle \cdot \left(\frac{\pi}{180} \cdot b\right)}, {\left(a \cdot 1\right)}^{2}\right) \]

      10. div-inv 75.2%

          \[\leadsto \mathsf{fma}\left(angle \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot b\right), angle \cdot \left(\color{blue}{\left(\pi \cdot \frac{1}{180}\right)} \cdot b\right), {\left(a \cdot 1\right)}^{2}\right) \]

      11. metadata-eval 75.2%

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

      12. *-rgt-identity 75.2%

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

      13. pow2 75.2%

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

   9. Applied egg-rr 75.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot b\right), angle \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot b\right), a \cdot a\right)} \]
   10. Step-by-step derivation

       1. fma-udef 75.2%

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

       2. unpow2 75.2%

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

       3. associate-*l* 75.2%

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

   11. Simplified 75.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-112}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(\pi \cdot \left(b \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 11: 57.9% accurate, 205.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. unpow2 81.1%

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

   2. swap-sqr 72.4%

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

   3. sqr-neg 72.4%

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

   4. swap-sqr 81.1%

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

   5. unpow2 81.1%

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

   6. distribute-lft-neg-out 81.1%

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

   7. distribute-rgt-neg-in 81.1%

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

   8. sin-neg 81.1%

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

   9. distribute-rgt-neg-out 81.1%

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

   10. distribute-frac-neg 81.1%

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

   11. unpow2 81.1%

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

   12. associate-*l* 80.0%

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

3. Simplified 81.2%

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

4. Taylor expanded in angle around 0 80.6%

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

5. Taylor expanded in angle around 0 75.5%

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

   1. associate-*r* 75.5%

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

   2. *-commutative 75.5%

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

   3. *-commutative 75.5%

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

   4. unpow1 75.5%

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

   5. sqr-pow 35.9%

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

   6. fabs-sqr 35.9%

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

   7. sqr-pow 75.5%

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

   8. unpow1 75.5%

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

   9. associate-*r* 75.5%

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

   10. *-commutative 75.5%

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

   11. associate-*r* 75.5%

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

   12. fabs-mul 75.5%

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

   13. fabs-mul 75.5%

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

   14. metadata-eval 75.5%

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

   15. unpow1 75.5%

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

   16. sqr-pow 75.3%

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

   17. fabs-sqr 75.3%

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

   18. sqr-pow 75.5%

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

   19. unpow1 75.5%

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

   20. unpow1 75.5%

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

   21. sqr-pow 39.5%

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

   22. fabs-sqr 39.5%

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

   23. sqr-pow 75.5%

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

   24. unpow1 75.5%

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

7. Simplified 75.5%

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

8. Taylor expanded in a around inf 59.7%

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

   1. unpow2 59.7%

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

10. Simplified 59.7%

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

11. Final simplification 59.7%

    \[\leadsto a \cdot a \]

Reproduce

herbie shell --seed 2023271 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))