ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.1%

Time: 16.1s

Alternatives: 14

Speedup: N/A×

• 36.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{180}{angle} \cdot \left(\pi + -1\right)\\ a \cdot a + {\left(b \cdot \sin \left(\frac{\left(\pi \cdot \pi + -1\right) \cdot 180 - angle \cdot t\_0}{180 \cdot t\_0}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ 180.0 angle) (+ PI -1.0))))
   (+
    (* a a)
    (pow
     (*
      b
      (sin (/ (- (* (+ (* PI PI) -1.0) 180.0) (* angle t_0)) (* 180.0 t_0))))
     2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = (180.0 / angle) * (((double) M_PI) + -1.0);
	return (a * a) + pow((b * sin((((((((double) M_PI) * ((double) M_PI)) + -1.0) * 180.0) - (angle * t_0)) / (180.0 * t_0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (180.0 / angle) * (Math.PI + -1.0);
	return (a * a) + Math.pow((b * Math.sin((((((Math.PI * Math.PI) + -1.0) * 180.0) - (angle * t_0)) / (180.0 * t_0)))), 2.0);
}

Python

def code(a, b, angle):
	t_0 = (180.0 / angle) * (math.pi + -1.0)
	return (a * a) + math.pow((b * math.sin((((((math.pi * math.pi) + -1.0) * 180.0) - (angle * t_0)) / (180.0 * t_0)))), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(180.0 / angle) * Float64(pi + -1.0))
	return Float64(Float64(a * a) + (Float64(b * sin(Float64(Float64(Float64(Float64(Float64(pi * pi) + -1.0) * 180.0) - Float64(angle * t_0)) / Float64(180.0 * t_0)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = (180.0 / angle) * (pi + -1.0);
	tmp = (a * a) + ((b * sin((((((pi * pi) + -1.0) * 180.0) - (angle * t_0)) / (180.0 * t_0)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(180.0 / angle), $MachinePrecision] * N[(Pi + -1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(a * a), $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] + -1.0), $MachinePrecision] * 180.0), $MachinePrecision] - N[(angle * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(180.0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.7%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   2. *-lowering-*.f64 77.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

5. Simplified 77.1%

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

   1. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

   3. expm1-log1p-u N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

   4. expm1-undefine N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

   5. log1p-undefine N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{e^{\log \left(1 + \mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{e^{\log \left(\mathsf{PI}\left(\right) + 1\right)} - 1}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

   7. rem-exp-log N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) + 1\right) - 1}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

   8. sub-div N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) + 1}{\frac{180}{angle}} - \frac{1}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

   9. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) + 1}{\frac{180}{angle}} - \frac{angle}{180}\right)\right)\right), 2\right)\right) \]

   10. flip-+ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right) - 1 \cdot 1}{\mathsf{PI}\left(\right) - 1}}{\frac{180}{angle}} - \frac{angle}{180}\right)\right)\right), 2\right)\right) \]

   11. associate-/l/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right) - 1 \cdot 1}{\frac{180}{angle} \cdot \left(\mathsf{PI}\left(\right) - 1\right)} - \frac{angle}{180}\right)\right)\right), 2\right)\right) \]

   12. frac-sub N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right) - 1 \cdot 1\right) \cdot 180 - \left(\frac{180}{angle} \cdot \left(\mathsf{PI}\left(\right) - 1\right)\right) \cdot angle}{\left(\frac{180}{angle} \cdot \left(\mathsf{PI}\left(\right) - 1\right)\right) \cdot 180}\right)\right)\right), 2\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right) - 1 \cdot 1\right) \cdot 180 - \left(\frac{180}{angle} \cdot \left(\mathsf{PI}\left(\right) - 1\right)\right) \cdot angle\right), \left(\left(\frac{180}{angle} \cdot \left(\mathsf{PI}\left(\right) - 1\right)\right) \cdot 180\right)\right)\right)\right), 2\right)\right) \]

7. Applied egg-rr 77.3%

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

8. Final simplification 77.3%

   \[\leadsto a \cdot a + {\left(b \cdot \sin \left(\frac{\left(\pi \cdot \pi + -1\right) \cdot 180 - angle \cdot \left(\frac{180}{angle} \cdot \left(\pi + -1\right)\right)}{180 \cdot \left(\frac{180}{angle} \cdot \left(\pi + -1\right)\right)}\right)\right)}^{2} \]
9. Add Preprocessing

Alternative 2: 79.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ a \cdot a + {\left(b \cdot \sin \left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (* a a) (pow (* b (sin (/ (/ PI 180.0) (/ 1.0 angle)))) 2.0)))

C

double code(double a, double b, double angle) {
	return (a * a) + pow((b * sin(((((double) M_PI) / 180.0) / (1.0 / angle)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return (a * a) + Math.pow((b * Math.sin(((Math.PI / 180.0) / (1.0 / angle)))), 2.0);
}

Python

def code(a, b, angle):
	return (a * a) + math.pow((b * math.sin(((math.pi / 180.0) / (1.0 / angle)))), 2.0)

Julia

function code(a, b, angle)
	return Float64(Float64(a * a) + (Float64(b * sin(Float64(Float64(pi / 180.0) / Float64(1.0 / angle)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a * a) + ((b * sin(((pi / 180.0) / (1.0 / angle)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[(a * a), $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(Pi / 180.0), $MachinePrecision] / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   2. *-lowering-*.f64 77.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

5. Simplified 77.1%

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

   1. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

   3. div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180 \cdot \frac{1}{angle}}\right)\right)\right), 2\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{180}}{\frac{1}{angle}}\right)\right)\right), 2\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180}\right), \left(\frac{1}{angle}\right)\right)\right)\right), 2\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 180\right), \left(\frac{1}{angle}\right)\right)\right)\right), 2\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \left(\frac{1}{angle}\right)\right)\right)\right), 2\right)\right) \]

   8. /-lowering-/.f64 77.3%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), 2\right)\right) \]

7. Applied egg-rr 77.3%

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

Alternative 3: 79.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   2. *-lowering-*.f64 77.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

5. Simplified 77.1%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot b\right), 2\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), b\right), 2\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{1}{\frac{180}{angle}}\right), b\right), 2\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right), b\right), 2\right)\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), b\right), 2\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{180}{angle}\right)\right)\right), b\right), 2\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{180}{angle}\right)\right)\right), b\right), 2\right)\right) \]

   8. /-lowering-/.f64 77.2%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

7. Applied egg-rr 77.2%

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

8. Final simplification 77.2%

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

Alternative 4: 79.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[(a * a), $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 76.7%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   2. *-lowering-*.f64 77.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

5. Simplified 77.1%

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

Alternative 5: 76.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.195:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556 + angle \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right)\right)\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.195)
   (+
    (* a a)
    (pow
     (*
      b
      (*
       angle
       (+
        (* PI 0.005555555555555556)
        (* angle (* angle (* PI (* (* PI PI) -2.8577960676726107e-8)))))))
     2.0))
   (+
    (* a a)
    (* (- 0.5 (* 0.5 (cos (* 2.0 (/ PI (/ 180.0 angle)))))) (* b b)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.195) {
		tmp = (a * a) + pow((b * (angle * ((((double) M_PI) * 0.005555555555555556) + (angle * (angle * (((double) M_PI) * ((((double) M_PI) * ((double) M_PI)) * -2.8577960676726107e-8))))))), 2.0);
	} else {
		tmp = (a * a) + ((0.5 - (0.5 * cos((2.0 * (((double) M_PI) / (180.0 / angle)))))) * (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.195) {
		tmp = (a * a) + Math.pow((b * (angle * ((Math.PI * 0.005555555555555556) + (angle * (angle * (Math.PI * ((Math.PI * Math.PI) * -2.8577960676726107e-8))))))), 2.0);
	} else {
		tmp = (a * a) + ((0.5 - (0.5 * Math.cos((2.0 * (Math.PI / (180.0 / angle)))))) * (b * b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 0.195:
		tmp = (a * a) + math.pow((b * (angle * ((math.pi * 0.005555555555555556) + (angle * (angle * (math.pi * ((math.pi * math.pi) * -2.8577960676726107e-8))))))), 2.0)
	else:
		tmp = (a * a) + ((0.5 - (0.5 * math.cos((2.0 * (math.pi / (180.0 / angle)))))) * (b * b))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 0.195)
		tmp = Float64(Float64(a * a) + (Float64(b * Float64(angle * Float64(Float64(pi * 0.005555555555555556) + Float64(angle * Float64(angle * Float64(pi * Float64(Float64(pi * pi) * -2.8577960676726107e-8))))))) ^ 2.0));
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(pi / Float64(180.0 / angle)))))) * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 0.195)
		tmp = (a * a) + ((b * (angle * ((pi * 0.005555555555555556) + (angle * (angle * (pi * ((pi * pi) * -2.8577960676726107e-8))))))) ^ 2.0);
	else
		tmp = (a * a) + ((0.5 - (0.5 * cos((2.0 * (pi / (180.0 / angle)))))) * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 0.195], N[(N[(a * a), $MachinePrecision] + N[Power[N[(b * N[(angle * N[(N[(Pi * 0.005555555555555556), $MachinePrecision] + N[(angle * N[(angle * N[(Pi * N[(N[(Pi * Pi), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 0.19500000000000001

   1. Initial program 83.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 83.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 83.4%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right), 2\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right), 2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)\right)\right)\right), 2\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)\right)\right), 2\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)\right)\right), 2\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right), \left(\left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)\right)\right)\right), 2\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI}\left(\right)\right), \left(\left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)\right)\right)\right), 2\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \left(\left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)\right)\right)\right), 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \left({angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right), 2\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \left(\left(angle \cdot angle\right) \cdot \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right), 2\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \left(angle \cdot \left(angle \cdot \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(angle, \left(angle \cdot \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left({\mathsf{PI}\left(\right)}^{3} \cdot \frac{-1}{34992000}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      14. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{-1}{34992000}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{34992000}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{34992000}\right)\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{34992000}\right)\right)\right)\right)\right)\right)\right), 2\right)\right) \]

   8. Simplified 77.1%

      \[\leadsto a \cdot a + {\left(b \cdot \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot \pi + angle \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right)\right)\right)\right)\right)}\right)}^{2} \]

   if 0.19500000000000001 < angle

   1. Initial program 57.9%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      8. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 56.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot b\right)}^{2}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot b\right)}^{2}\right)\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}^{2} \cdot \color{blue}{{b}^{2}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{*.f64}\left(\left({\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}^{2}\right), \color{blue}{\left({b}^{2}\right)}\right)\right) \]

   6. Applied egg-rr 56.3%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right)}, \mathsf{*.f64}\left(b, b\right)\right)\right) \]

      2. *-lowering-*.f64 59.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right)}, \mathsf{*.f64}\left(b, b\right)\right)\right) \]

   9. Simplified 59.7%

      \[\leadsto \color{blue}{a \cdot a} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.195:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556 + angle \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right)\right)\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.0045:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.0045)
   (+ (* a a) (pow (* angle (* b (* PI 0.005555555555555556))) 2.0))
   (+
    (* a a)
    (* (- 0.5 (* 0.5 (cos (* 2.0 (/ PI (/ 180.0 angle)))))) (* b b)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.0045) {
		tmp = (a * a) + pow((angle * (b * (((double) M_PI) * 0.005555555555555556))), 2.0);
	} else {
		tmp = (a * a) + ((0.5 - (0.5 * cos((2.0 * (((double) M_PI) / (180.0 / angle)))))) * (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.0045) {
		tmp = (a * a) + Math.pow((angle * (b * (Math.PI * 0.005555555555555556))), 2.0);
	} else {
		tmp = (a * a) + ((0.5 - (0.5 * Math.cos((2.0 * (Math.PI / (180.0 / angle)))))) * (b * b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 0.0045:
		tmp = (a * a) + math.pow((angle * (b * (math.pi * 0.005555555555555556))), 2.0)
	else:
		tmp = (a * a) + ((0.5 - (0.5 * math.cos((2.0 * (math.pi / (180.0 / angle)))))) * (b * b))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 0.0045)
		tmp = Float64(Float64(a * a) + (Float64(angle * Float64(b * Float64(pi * 0.005555555555555556))) ^ 2.0));
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(pi / Float64(180.0 / angle)))))) * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 0.0045)
		tmp = (a * a) + ((angle * (b * (pi * 0.005555555555555556))) ^ 2.0);
	else
		tmp = (a * a) + ((0.5 - (0.5 * cos((2.0 * (pi / (180.0 / angle)))))) * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 0.0045], N[(N[(a * a), $MachinePrecision] + N[Power[N[(angle * N[(b * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 0.00449999999999999966

   1. Initial program 83.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 83.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 83.4%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, 2\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right), 2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\left(angle \cdot \frac{1}{180}\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right), 2\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(angle \cdot \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right), 2\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)\right), 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(b \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      10. PI-lowering-PI.f64 78.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right)\right) \]

   8. Simplified 78.8%

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

   if 0.00449999999999999966 < angle

   1. Initial program 57.9%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      8. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 56.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot b\right)}^{2}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot b\right)}^{2}\right)\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}^{2} \cdot \color{blue}{{b}^{2}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{*.f64}\left(\left({\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}^{2}\right), \color{blue}{\left({b}^{2}\right)}\right)\right) \]

   6. Applied egg-rr 56.3%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right)}, \mathsf{*.f64}\left(b, b\right)\right)\right) \]

      2. *-lowering-*.f64 59.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right)}, \mathsf{*.f64}\left(b, b\right)\right)\right) \]

   9. Simplified 59.7%

      \[\leadsto \color{blue}{a \cdot a} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.0045:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.0045:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + b \cdot \left(b \cdot \left(0.5 + \frac{\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}{-2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.0045)
   (+ (* a a) (pow (* angle (* b (* PI 0.005555555555555556))) 2.0))
   (+
    (* a a)
    (* b (* b (+ 0.5 (/ (cos (* PI (* angle 0.011111111111111112))) -2.0)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.0045) {
		tmp = (a * a) + pow((angle * (b * (((double) M_PI) * 0.005555555555555556))), 2.0);
	} else {
		tmp = (a * a) + (b * (b * (0.5 + (cos((((double) M_PI) * (angle * 0.011111111111111112))) / -2.0))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.0045) {
		tmp = (a * a) + Math.pow((angle * (b * (Math.PI * 0.005555555555555556))), 2.0);
	} else {
		tmp = (a * a) + (b * (b * (0.5 + (Math.cos((Math.PI * (angle * 0.011111111111111112))) / -2.0))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 0.0045:
		tmp = (a * a) + math.pow((angle * (b * (math.pi * 0.005555555555555556))), 2.0)
	else:
		tmp = (a * a) + (b * (b * (0.5 + (math.cos((math.pi * (angle * 0.011111111111111112))) / -2.0))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 0.0045)
		tmp = Float64(Float64(a * a) + (Float64(angle * Float64(b * Float64(pi * 0.005555555555555556))) ^ 2.0));
	else
		tmp = Float64(Float64(a * a) + Float64(b * Float64(b * Float64(0.5 + Float64(cos(Float64(pi * Float64(angle * 0.011111111111111112))) / -2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 0.0045)
		tmp = (a * a) + ((angle * (b * (pi * 0.005555555555555556))) ^ 2.0);
	else
		tmp = (a * a) + (b * (b * (0.5 + (cos((pi * (angle * 0.011111111111111112))) / -2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 0.0045], N[(N[(a * a), $MachinePrecision] + N[Power[N[(angle * N[(b * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(b * N[(b * N[(0.5 + N[(N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 0.00449999999999999966

   1. Initial program 83.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 83.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 83.4%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, 2\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right), 2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\left(angle \cdot \frac{1}{180}\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right), 2\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(angle \cdot \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right), 2\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)\right), 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(b \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      10. PI-lowering-PI.f64 78.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right)\right) \]

   8. Simplified 78.8%

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

   if 0.00449999999999999966 < angle

   1. Initial program 57.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 59.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 59.6%

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

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{b}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{b}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{b}\right)\right) \]

   7. Applied egg-rr 59.7%

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

   9. Applied egg-rr 59.6%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.0045:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + b \cdot \left(b \cdot \left(0.5 + \frac{\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}{-2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+14}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.8e+14)
   (* a a)
   (+ (* a a) (pow (* angle (* b (* PI 0.005555555555555556))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.8e+14) {
		tmp = a * a;
	} else {
		tmp = (a * a) + pow((angle * (b * (((double) M_PI) * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.8e+14) {
		tmp = a * a;
	} else {
		tmp = (a * a) + Math.pow((angle * (b * (Math.PI * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.8e+14:
		tmp = a * a
	else:
		tmp = (a * a) + math.pow((angle * (b * (math.pi * 0.005555555555555556))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.8e+14)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + (Float64(angle * Float64(b * Float64(pi * 0.005555555555555556))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.8e+14)
		tmp = a * a;
	else
		tmp = (a * a) + ((angle * (b * (pi * 0.005555555555555556))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.8e+14], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(angle * N[(b * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8e14

   1. Initial program 74.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 58.9%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 58.9%

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

   if 1.8e14 < b

   1. Initial program 86.3%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 86.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 86.2%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, 2\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right), 2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\left(angle \cdot \frac{1}{180}\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right), 2\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(angle \cdot \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right), 2\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)\right), 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(b \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      10. PI-lowering-PI.f64 84.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right)\right) \]

   8. Simplified 84.8%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+14}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{+14}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(0.005555555555555556 \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.32e+14)
   (* a a)
   (+ (* a a) (pow (* 0.005555555555555556 (* PI (* b angle))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.32e+14) {
		tmp = a * a;
	} else {
		tmp = (a * a) + pow((0.005555555555555556 * (((double) M_PI) * (b * angle))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.32e+14) {
		tmp = a * a;
	} else {
		tmp = (a * a) + Math.pow((0.005555555555555556 * (Math.PI * (b * angle))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.32e+14:
		tmp = a * a
	else:
		tmp = (a * a) + math.pow((0.005555555555555556 * (math.pi * (b * angle))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.32e+14)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + (Float64(0.005555555555555556 * Float64(pi * Float64(b * angle))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.32e+14)
		tmp = a * a;
	else
		tmp = (a * a) + ((0.005555555555555556 * (pi * (b * angle))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.32e+14], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(0.005555555555555556 * N[(Pi * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.32e14

   1. Initial program 74.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 58.9%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 58.9%

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

   if 1.32e14 < b

   1. Initial program 86.3%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 86.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 86.2%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180 \cdot \frac{1}{angle}}\right)\right)\right), 2\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{180}}{\frac{1}{angle}}\right)\right)\right), 2\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180}\right), \left(\frac{1}{angle}\right)\right)\right)\right), 2\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 180\right), \left(\frac{1}{angle}\right)\right)\right)\right), 2\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \left(\frac{1}{angle}\right)\right)\right)\right), 2\right)\right) \]

      8. /-lowering-/.f64 86.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), 2\right)\right) \]

   7. Applied egg-rr 86.6%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, 2\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(\left(angle \cdot b\right) \cdot \mathsf{PI}\left(\right)\right)\right), 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot b\right)\right)\right), 2\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(angle \cdot b\right)\right)\right), 2\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(angle \cdot b\right)\right)\right), 2\right)\right) \]

      6. *-lowering-*.f64 84.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, b\right)\right)\right), 2\right)\right) \]

   10. Simplified 84.8%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{+14}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(0.005555555555555556 \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.1% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.46 \cdot 10^{+14}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+148}:\\ \;\;\;\;a \cdot a + \left(b \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.46e+14)
   (* a a)
   (if (<= b 9.5e+148)
     (+
      (* a a)
      (* (* b (* b (* PI PI))) (* 3.08641975308642e-5 (* angle angle))))
     (* (* (* b angle) (* b angle)) (* (* PI PI) 3.08641975308642e-5)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.46e+14) {
		tmp = a * a;
	} else if (b <= 9.5e+148) {
		tmp = (a * a) + ((b * (b * (((double) M_PI) * ((double) M_PI)))) * (3.08641975308642e-5 * (angle * angle)));
	} else {
		tmp = ((b * angle) * (b * angle)) * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.46e+14) {
		tmp = a * a;
	} else if (b <= 9.5e+148) {
		tmp = (a * a) + ((b * (b * (Math.PI * Math.PI))) * (3.08641975308642e-5 * (angle * angle)));
	} else {
		tmp = ((b * angle) * (b * angle)) * ((Math.PI * Math.PI) * 3.08641975308642e-5);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.46e+14:
		tmp = a * a
	elif b <= 9.5e+148:
		tmp = (a * a) + ((b * (b * (math.pi * math.pi))) * (3.08641975308642e-5 * (angle * angle)))
	else:
		tmp = ((b * angle) * (b * angle)) * ((math.pi * math.pi) * 3.08641975308642e-5)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.46e+14)
		tmp = Float64(a * a);
	elseif (b <= 9.5e+148)
		tmp = Float64(Float64(a * a) + Float64(Float64(b * Float64(b * Float64(pi * pi))) * Float64(3.08641975308642e-5 * Float64(angle * angle))));
	else
		tmp = Float64(Float64(Float64(b * angle) * Float64(b * angle)) * Float64(Float64(pi * pi) * 3.08641975308642e-5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.46e+14)
		tmp = a * a;
	elseif (b <= 9.5e+148)
		tmp = (a * a) + ((b * (b * (pi * pi))) * (3.08641975308642e-5 * (angle * angle)));
	else
		tmp = ((b * angle) * (b * angle)) * ((pi * pi) * 3.08641975308642e-5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.46e+14], N[(a * a), $MachinePrecision], If[LessEqual[b, 9.5e+148], N[(N[(a * a), $MachinePrecision] + N[(N[(b * N[(b * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(3.08641975308642e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * angle), $MachinePrecision] * N[(b * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.46e14

   1. Initial program 74.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 58.9%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 58.9%

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

   if 1.46e14 < b < 9.5000000000000002e148

   1. Initial program 68.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 68.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 68.2%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \]

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{{angle}^{2}}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) \cdot {\color{blue}{angle}}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right)}\right)\right) \]

   8. Simplified 55.8%

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

   if 9.5000000000000002e148 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 99.7%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \]

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{{angle}^{2}}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) \cdot {\color{blue}{angle}}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right)}\right)\right) \]

   8. Simplified 72.2%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}} \]

       2. associate-*r* N/A

          \[\leadsto \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

       3. associate-*l* N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)} \]

       4. metadata-eval N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \left(\mathsf{neg}\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{32400}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({angle}^{2} \cdot \left(b \cdot b\right)\right), \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({angle}^{2} \cdot b\right) \cdot b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({angle}^{2} \cdot b\right), b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({angle}^{2}\right), b\right), b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400}} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot angle\right), b\right), b\right), \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       14. distribute-lft-neg-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \left(\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \left(\frac{1}{32400} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       19. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       20. PI-lowering-PI.f64 72.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   11. Simplified 72.4%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(angle \cdot angle\right) \cdot \left(b \cdot b\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{32400}}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{32400}}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot b\right), \left(angle \cdot b\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{32400}}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \left(angle \cdot b\right)\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

       5. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(angle, b\right)\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   13. Applied egg-rr 86.2%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.46 \cdot 10^{+14}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+148}:\\ \;\;\;\;a \cdot a + \left(b \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.9% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+165}:\\ \;\;\;\;a \cdot a + angle \cdot \left(angle \cdot \left(\left(b \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.8e+165)
   (+
    (* a a)
    (* angle (* angle (* (* b 3.08641975308642e-5) (* b (* PI PI))))))
   (* (* (* b angle) (* b angle)) (* (* PI PI) 3.08641975308642e-5))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.8e+165) {
		tmp = (a * a) + (angle * (angle * ((b * 3.08641975308642e-5) * (b * (((double) M_PI) * ((double) M_PI))))));
	} else {
		tmp = ((b * angle) * (b * angle)) * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.8e+165) {
		tmp = (a * a) + (angle * (angle * ((b * 3.08641975308642e-5) * (b * (Math.PI * Math.PI)))));
	} else {
		tmp = ((b * angle) * (b * angle)) * ((Math.PI * Math.PI) * 3.08641975308642e-5);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.8e+165:
		tmp = (a * a) + (angle * (angle * ((b * 3.08641975308642e-5) * (b * (math.pi * math.pi)))))
	else:
		tmp = ((b * angle) * (b * angle)) * ((math.pi * math.pi) * 3.08641975308642e-5)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.8e+165)
		tmp = Float64(Float64(a * a) + Float64(angle * Float64(angle * Float64(Float64(b * 3.08641975308642e-5) * Float64(b * Float64(pi * pi))))));
	else
		tmp = Float64(Float64(Float64(b * angle) * Float64(b * angle)) * Float64(Float64(pi * pi) * 3.08641975308642e-5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.8e+165)
		tmp = (a * a) + (angle * (angle * ((b * 3.08641975308642e-5) * (b * (pi * pi)))));
	else
		tmp = ((b * angle) * (b * angle)) * ((pi * pi) * 3.08641975308642e-5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.8e+165], N[(N[(a * a), $MachinePrecision] + N[(angle * N[(angle * N[(N[(b * 3.08641975308642e-5), $MachinePrecision] * N[(b * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * angle), $MachinePrecision] * N[(b * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.7999999999999999e165

   1. Initial program 74.2%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 74.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 74.7%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \]

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{{angle}^{2}}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) \cdot {\color{blue}{angle}}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right)}\right)\right) \]

   8. Simplified 60.6%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(b \cdot \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \frac{1}{32400}\right) \cdot \color{blue}{\left(angle \cdot angle\right)}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\left(b \cdot \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \frac{1}{32400}\right) \cdot angle\right) \cdot \color{blue}{angle}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\left(\left(b \cdot \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \frac{1}{32400}\right) \cdot angle\right), \color{blue}{angle}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(b \cdot \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \frac{1}{32400}\right), angle\right), angle\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\right) \cdot \frac{1}{32400}\right), angle\right), angle\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot \frac{1}{32400}\right)\right), angle\right), angle\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \left(b \cdot \frac{1}{32400}\right)\right), angle\right), angle\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right), \left(b \cdot \frac{1}{32400}\right)\right), angle\right), angle\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), b\right), \left(b \cdot \frac{1}{32400}\right)\right), angle\right), angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), b\right), \left(b \cdot \frac{1}{32400}\right)\right), angle\right), angle\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), b\right), \left(b \cdot \frac{1}{32400}\right)\right), angle\right), angle\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), b\right), \left(b \cdot \frac{1}{32400}\right)\right), angle\right), angle\right)\right) \]

      13. *-lowering-*.f64 67.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), b\right), \mathsf{*.f64}\left(b, \frac{1}{32400}\right)\right), angle\right), angle\right)\right) \]

   10. Applied egg-rr 67.6%

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

   if 1.7999999999999999e165 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 99.7%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \]

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{{angle}^{2}}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) \cdot {\color{blue}{angle}}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right)}\right)\right) \]

   8. Simplified 72.8%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}} \]

       2. associate-*r* N/A

          \[\leadsto \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

       3. associate-*l* N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)} \]

       4. metadata-eval N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \left(\mathsf{neg}\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{32400}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({angle}^{2} \cdot \left(b \cdot b\right)\right), \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({angle}^{2} \cdot b\right) \cdot b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({angle}^{2} \cdot b\right), b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({angle}^{2}\right), b\right), b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400}} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot angle\right), b\right), b\right), \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       14. distribute-lft-neg-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \left(\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \left(\frac{1}{32400} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       19. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       20. PI-lowering-PI.f64 76.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   11. Simplified 76.6%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(angle \cdot angle\right) \cdot \left(b \cdot b\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{32400}}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{32400}}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot b\right), \left(angle \cdot b\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{32400}}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \left(angle \cdot b\right)\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

       5. *-lowering-*.f64 92.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(angle, b\right)\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   13. Applied egg-rr 92.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+165}:\\ \;\;\;\;a \cdot a + angle \cdot \left(angle \cdot \left(\left(b \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.7% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.7 \cdot 10^{+135}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 5.7e+135)
   (* a a)
   (* (* (* b angle) (* b angle)) (* (* PI PI) 3.08641975308642e-5))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.7e+135) {
		tmp = a * a;
	} else {
		tmp = ((b * angle) * (b * angle)) * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.7e+135) {
		tmp = a * a;
	} else {
		tmp = ((b * angle) * (b * angle)) * ((Math.PI * Math.PI) * 3.08641975308642e-5);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 5.7e+135:
		tmp = a * a
	else:
		tmp = ((b * angle) * (b * angle)) * ((math.pi * math.pi) * 3.08641975308642e-5)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 5.7e+135)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(Float64(b * angle) * Float64(b * angle)) * Float64(Float64(pi * pi) * 3.08641975308642e-5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 5.7e+135)
		tmp = a * a;
	else
		tmp = ((b * angle) * (b * angle)) * ((pi * pi) * 3.08641975308642e-5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 5.7e+135], N[(a * a), $MachinePrecision], N[(N[(N[(b * angle), $MachinePrecision] * N[(b * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.7000000000000002e135

   1. Initial program 73.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 57.7%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 57.7%

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

   if 5.7000000000000002e135 < b

   1. Initial program 95.1%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 95.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 95.0%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \]

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{{angle}^{2}}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) \cdot {\color{blue}{angle}}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right)}\right)\right) \]

   8. Simplified 69.8%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}} \]

       2. associate-*r* N/A

          \[\leadsto \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

       3. associate-*l* N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)} \]

       4. metadata-eval N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \left(\mathsf{neg}\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{32400}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({angle}^{2} \cdot \left(b \cdot b\right)\right), \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({angle}^{2} \cdot b\right) \cdot b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({angle}^{2} \cdot b\right), b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({angle}^{2}\right), b\right), b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400}} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot angle\right), b\right), b\right), \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       14. distribute-lft-neg-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \left(\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \left(\frac{1}{32400} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       19. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       20. PI-lowering-PI.f64 64.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   11. Simplified 64.8%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(angle \cdot angle\right) \cdot \left(b \cdot b\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{32400}}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{32400}}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot b\right), \left(angle \cdot b\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{32400}}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \left(angle \cdot b\right)\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

       5. *-lowering-*.f64 78.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(angle, b\right)\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   13. Applied egg-rr 78.6%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.7 \cdot 10^{+135}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.2% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+136}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(b \cdot angle\right) \cdot \left(b \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.4e+136)
   (* a a)
   (* angle (* (* b angle) (* b (* PI (* PI 3.08641975308642e-5)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.4e+136) {
		tmp = a * a;
	} else {
		tmp = angle * ((b * angle) * (b * (((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.4e+136) {
		tmp = a * a;
	} else {
		tmp = angle * ((b * angle) * (b * (Math.PI * (Math.PI * 3.08641975308642e-5))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.4e+136:
		tmp = a * a
	else:
		tmp = angle * ((b * angle) * (b * (math.pi * (math.pi * 3.08641975308642e-5))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.4e+136)
		tmp = Float64(a * a);
	else
		tmp = Float64(angle * Float64(Float64(b * angle) * Float64(b * Float64(pi * Float64(pi * 3.08641975308642e-5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.4e+136)
		tmp = a * a;
	else
		tmp = angle * ((b * angle) * (b * (pi * (pi * 3.08641975308642e-5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.4e+136], N[(a * a), $MachinePrecision], N[(angle * N[(N[(b * angle), $MachinePrecision] * N[(b * N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4000000000000001e136

   1. Initial program 73.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 57.7%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 57.7%

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

   if 1.4000000000000001e136 < b

   1. Initial program 95.1%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 95.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 95.0%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \]

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{{angle}^{2}}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) \cdot {\color{blue}{angle}}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right)}\right)\right) \]

   8. Simplified 69.8%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}} \]

       2. associate-*r* N/A

          \[\leadsto \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

       3. associate-*l* N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)} \]

       4. metadata-eval N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \left(\mathsf{neg}\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{32400}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \left({angle}^{2} \cdot {b}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({angle}^{2} \cdot \left(b \cdot b\right)\right), \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({angle}^{2} \cdot b\right) \cdot b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({angle}^{2} \cdot b\right), b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({angle}^{2}\right), b\right), b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400}} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot angle\right), b\right), b\right), \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       14. distribute-lft-neg-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \left(\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \left(\frac{1}{32400} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       19. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       20. PI-lowering-PI.f64 64.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), b\right), b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   11. Simplified 64.8%

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

       1. associate-*l* N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot b\right) \cdot \color{blue}{\left(b \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

       2. associate-*l* N/A

          \[\leadsto \left(angle \cdot \left(angle \cdot b\right)\right) \cdot \left(\color{blue}{b} \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto angle \cdot \color{blue}{\left(\left(angle \cdot b\right) \cdot \left(b \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(angle, \color{blue}{\left(\left(angle \cdot b\right) \cdot \left(b \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\left(angle \cdot b\right), \color{blue}{\left(b \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \left(\color{blue}{b} \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(b, \left(\left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(b, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right) \]

       11. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\frac{1}{32400}} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\frac{1}{32400}}\right)\right)\right)\right)\right) \]

       14. PI-lowering-PI.f64 73.6%

           \[\leadsto \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right)\right)\right)\right) \]

   13. Applied egg-rr 73.6%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+136}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(b \cdot angle\right) \cdot \left(b \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.8% accurate, 139.0× 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 76.7%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 54.6%

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

5. Simplified 54.6%

   \[\leadsto \color{blue}{a \cdot a} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(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)))