ab-angle->ABCF A

Percentage Accurate: 80.1% → 79.9%

Time: 14.7s

Alternatives: 14

Speedup: N/A×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 85.9

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

4. Applied rewrites 85.9%

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

5. Taylor expanded in angle around 0

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

7. Applied rewrites 86.0%

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

8. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 86.0

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

10. Applied rewrites 86.0%

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

Alternative 2: 79.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 85.9

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

4. Applied rewrites 85.9%

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

5. Taylor expanded in angle around 0

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

7. Applied rewrites 86.0%

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

8. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 86.0

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

10. Applied rewrites 86.0%

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

    1. lift-/.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-/r/ N/A

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

    4. metadata-eval N/A

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

    5. lift-*.f64 N/A

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

    6. *-commutative N/A

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

    7. associate-*r* N/A

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

    8. *-commutative N/A

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

    9. lift-*.f64 N/A

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

    10. lower-*.f64 85.9

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

12. Applied rewrites 85.9%

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

13. Final simplification 85.9%

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

Alternative 3: 79.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 85.9

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

4. Applied rewrites 85.9%

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

5. Taylor expanded in angle around 0

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

7. Applied rewrites 86.0%

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

8. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 86.0

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

10. Applied rewrites 86.0%

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

    1. lift-/.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. clear-num N/A

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

    4. lift-*.f64 N/A

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

    5. associate-*l/ N/A

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

    6. lower-*.f64 N/A

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

    7. div-inv N/A

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

    8. metadata-eval N/A

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

    9. lower-*.f64 85.8

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

12. Applied rewrites 85.8%

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

13. Final simplification 85.8%

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

Alternative 4: 76.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;b \cdot b + {\left(angle \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(a \cdot a, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right), \left(b \cdot b\right) \cdot \left(1 \cdot 1\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 4e-6)
   (+
    (* b b)
    (pow
     (*
      angle
      (*
       a
       (*
        PI
        (fma
         (* (* angle angle) -2.8577960676726107e-8)
         (* PI PI)
         0.005555555555555556))))
     2.0))
   (/
    1.0
    (/
     1.0
     (fma
      (* a a)
      (- 0.5 (* 0.5 (cos (* 2.0 (* angle (* PI 0.005555555555555556))))))
      (* (* b b) (* 1.0 1.0)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 4e-6) {
		tmp = (b * b) + pow((angle * (a * (((double) M_PI) * fma(((angle * angle) * -2.8577960676726107e-8), (((double) M_PI) * ((double) M_PI)), 0.005555555555555556)))), 2.0);
	} else {
		tmp = 1.0 / (1.0 / fma((a * a), (0.5 - (0.5 * cos((2.0 * (angle * (((double) M_PI) * 0.005555555555555556)))))), ((b * b) * (1.0 * 1.0))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 4e-6)
		tmp = Float64(Float64(b * b) + (Float64(angle * Float64(a * Float64(pi * fma(Float64(Float64(angle * angle) * -2.8577960676726107e-8), Float64(pi * pi), 0.005555555555555556)))) ^ 2.0));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(Float64(a * a), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(pi * 0.005555555555555556)))))), Float64(Float64(b * b) * Float64(1.0 * 1.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 4e-6], N[(N[(b * b), $MachinePrecision] + N[Power[N[(angle * N[(a * N[(Pi * N[(N[(N[(angle * angle), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(N[(a * a), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999982e-6

   1. Initial program 89.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 89.3

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

   4. Applied rewrites 89.3%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 89.4

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

   10. Applied rewrites 89.4%

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

   11. Taylor expanded in angle around 0

       \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]

   13. Applied rewrites 86.1%

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

   if 3.99999999999999982e-6 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 74.3%

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

   9. Applied rewrites 73.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(a \cdot a, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right), \left(b \cdot b\right) \cdot \left(1 \cdot 1\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;b \cdot b + {\left(angle \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(a \cdot a, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right), \left(b \cdot b\right) \cdot \left(1 \cdot 1\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;b \cdot b + {\left(angle \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot \left(b \cdot 1\right), b, a \cdot \left(a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 4e-6)
   (+
    (* b b)
    (pow
     (*
      angle
      (*
       a
       (*
        PI
        (fma
         (* (* angle angle) -2.8577960676726107e-8)
         (* PI PI)
         0.005555555555555556))))
     2.0))
   (fma
    (* 1.0 (* b 1.0))
    b
    (*
     a
     (*
      a
      (- 0.5 (* 0.5 (cos (* 2.0 (* angle (* PI 0.005555555555555556)))))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 4e-6) {
		tmp = (b * b) + pow((angle * (a * (((double) M_PI) * fma(((angle * angle) * -2.8577960676726107e-8), (((double) M_PI) * ((double) M_PI)), 0.005555555555555556)))), 2.0);
	} else {
		tmp = fma((1.0 * (b * 1.0)), b, (a * (a * (0.5 - (0.5 * cos((2.0 * (angle * (((double) M_PI) * 0.005555555555555556)))))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 4e-6)
		tmp = Float64(Float64(b * b) + (Float64(angle * Float64(a * Float64(pi * fma(Float64(Float64(angle * angle) * -2.8577960676726107e-8), Float64(pi * pi), 0.005555555555555556)))) ^ 2.0));
	else
		tmp = fma(Float64(1.0 * Float64(b * 1.0)), b, Float64(a * Float64(a * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(pi * 0.005555555555555556)))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 4e-6], N[(N[(b * b), $MachinePrecision] + N[Power[N[(angle * N[(a * N[(Pi * N[(N[(N[(angle * angle), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[(b * 1.0), $MachinePrecision]), $MachinePrecision] * b + N[(a * N[(a * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999982e-6

   1. Initial program 89.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 89.3

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

   4. Applied rewrites 89.3%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 89.4

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

   10. Applied rewrites 89.4%

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

   11. Taylor expanded in angle around 0

       \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]

   13. Applied rewrites 86.1%

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

   if 3.99999999999999982e-6 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 74.3%

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

   9. Applied rewrites 73.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(b \cdot 1\right) \cdot 1, b, a \cdot \left(a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;b \cdot b + {\left(angle \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot \left(b \cdot 1\right), b, a \cdot \left(a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;b \cdot b + {\left(angle \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 4e-6)
   (+
    (* b b)
    (pow
     (*
      angle
      (*
       a
       (*
        PI
        (fma
         (* (* angle angle) -2.8577960676726107e-8)
         (* PI PI)
         0.005555555555555556))))
     2.0))
   (fma
    a
    (* a (- 0.5 (* 0.5 (cos (* 2.0 (* (* angle PI) 0.005555555555555556))))))
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 4e-6) {
		tmp = (b * b) + pow((angle * (a * (((double) M_PI) * fma(((angle * angle) * -2.8577960676726107e-8), (((double) M_PI) * ((double) M_PI)), 0.005555555555555556)))), 2.0);
	} else {
		tmp = fma(a, (a * (0.5 - (0.5 * cos((2.0 * ((angle * ((double) M_PI)) * 0.005555555555555556)))))), (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 4e-6)
		tmp = Float64(Float64(b * b) + (Float64(angle * Float64(a * Float64(pi * fma(Float64(Float64(angle * angle) * -2.8577960676726107e-8), Float64(pi * pi), 0.005555555555555556)))) ^ 2.0));
	else
		tmp = fma(a, Float64(a * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(angle * pi) * 0.005555555555555556)))))), Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 4e-6], N[(N[(b * b), $MachinePrecision] + N[Power[N[(angle * N[(a * N[(Pi * N[(N[(N[(angle * angle), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999982e-6

   1. Initial program 89.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 89.3

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

   4. Applied rewrites 89.3%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 89.4

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

   10. Applied rewrites 89.4%

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

   11. Taylor expanded in angle around 0

       \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]

   13. Applied rewrites 86.1%

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

   if 3.99999999999999982e-6 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 74.3

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

   10. Applied rewrites 74.3%

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

       1. lift-+.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. unpow-prod-down N/A

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

       5. pow2 N/A

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

       6. lift-sin.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. associate-/r/ N/A

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

       10. metadata-eval N/A

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

       11. lift-*.f64 N/A

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

       12. lift-PI.f64 N/A

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

       13. associate-*l* N/A

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

   12. Applied rewrites 74.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right), b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;b \cdot b + {\left(angle \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;b \cdot b + {\left(\left(angle \cdot \pi\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 4e-6)
   (+ (* b b) (pow (* (* angle PI) (* a 0.005555555555555556)) 2.0))
   (fma
    a
    (* a (- 0.5 (* 0.5 (cos (* 2.0 (* (* angle PI) 0.005555555555555556))))))
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 4e-6) {
		tmp = (b * b) + pow(((angle * ((double) M_PI)) * (a * 0.005555555555555556)), 2.0);
	} else {
		tmp = fma(a, (a * (0.5 - (0.5 * cos((2.0 * ((angle * ((double) M_PI)) * 0.005555555555555556)))))), (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 4e-6)
		tmp = Float64(Float64(b * b) + (Float64(Float64(angle * pi) * Float64(a * 0.005555555555555556)) ^ 2.0));
	else
		tmp = fma(a, Float64(a * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(angle * pi) * 0.005555555555555556)))))), Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 4e-6], N[(N[(b * b), $MachinePrecision] + N[Power[N[(N[(angle * Pi), $MachinePrecision] * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999982e-6

   1. Initial program 89.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 89.3

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

   4. Applied rewrites 89.3%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 89.4

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

   10. Applied rewrites 89.4%

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

   11. Taylor expanded in angle around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 N/A

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

       6. lower-*.f64 86.8

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

   13. Applied rewrites 86.8%

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

   if 3.99999999999999982e-6 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 74.3

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

   10. Applied rewrites 74.3%

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

       1. lift-+.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. unpow-prod-down N/A

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

       5. pow2 N/A

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

       6. lift-sin.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. associate-/r/ N/A

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

       10. metadata-eval N/A

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

       11. lift-*.f64 N/A

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

       12. lift-PI.f64 N/A

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

       13. associate-*l* N/A

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

   12. Applied rewrites 74.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right), b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;b \cdot b + {\left(\left(angle \cdot \pi\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.4 \cdot 10^{-148}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(\left(angle \cdot \pi\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.4e-148)
   (* b b)
   (+ (* b b) (pow (* (* angle PI) (* a 0.005555555555555556)) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.4e-148) {
		tmp = b * b;
	} else {
		tmp = (b * b) + pow(((angle * ((double) M_PI)) * (a * 0.005555555555555556)), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.4e-148) {
		tmp = b * b;
	} else {
		tmp = (b * b) + Math.pow(((angle * Math.PI) * (a * 0.005555555555555556)), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 4.4e-148:
		tmp = b * b
	else:
		tmp = (b * b) + math.pow(((angle * math.pi) * (a * 0.005555555555555556)), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 4.4e-148)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(b * b) + (Float64(Float64(angle * pi) * Float64(a * 0.005555555555555556)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 4.4e-148)
		tmp = b * b;
	else
		tmp = (b * b) + (((angle * pi) * (a * 0.005555555555555556)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 4.4e-148], N[(b * b), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[Power[N[(N[(angle * Pi), $MachinePrecision] * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.40000000000000034e-148

   1. Initial program 85.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if 4.40000000000000034e-148 < a

   1. Initial program 86.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 86.3%

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

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.3

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

   10. Applied rewrites 86.3%

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

   11. Taylor expanded in angle around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 N/A

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

       6. lower-*.f64 84.1

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

   13. Applied rewrites 84.1%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.4 \cdot 10^{-148}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(\left(angle \cdot \pi\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(a, a \cdot 3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle, b \cdot b\right)\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+141}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(angle, angle \cdot \left(\left(\pi \cdot \pi\right) \cdot -3.08641975308642 \cdot 10^{-5}\right), \mathsf{fma}\left(\pi \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)\right)\right), \frac{3.08641975308642 \cdot 10^{-5}}{b \cdot b}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 7e-92)
   (fma
    (*
     angle
     (*
      (* PI PI)
      (fma a (* a 3.08641975308642e-5) (* (* b b) -3.08641975308642e-5))))
    angle
    (* b b))
   (if (<= b 4e+141)
     (*
      (* b b)
      (fma
       angle
       (* angle (* (* PI PI) -3.08641975308642e-5))
       (fma
        (* PI (* PI (* a (* a (* angle angle)))))
        (/ 3.08641975308642e-5 (* b b))
        1.0)))
     (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 7e-92) {
		tmp = fma((angle * ((((double) M_PI) * ((double) M_PI)) * fma(a, (a * 3.08641975308642e-5), ((b * b) * -3.08641975308642e-5)))), angle, (b * b));
	} else if (b <= 4e+141) {
		tmp = (b * b) * fma(angle, (angle * ((((double) M_PI) * ((double) M_PI)) * -3.08641975308642e-5)), fma((((double) M_PI) * (((double) M_PI) * (a * (a * (angle * angle))))), (3.08641975308642e-5 / (b * b)), 1.0));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 7e-92)
		tmp = fma(Float64(angle * Float64(Float64(pi * pi) * fma(a, Float64(a * 3.08641975308642e-5), Float64(Float64(b * b) * -3.08641975308642e-5)))), angle, Float64(b * b));
	elseif (b <= 4e+141)
		tmp = Float64(Float64(b * b) * fma(angle, Float64(angle * Float64(Float64(pi * pi) * -3.08641975308642e-5)), fma(Float64(pi * Float64(pi * Float64(a * Float64(a * Float64(angle * angle))))), Float64(3.08641975308642e-5 / Float64(b * b)), 1.0)));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 7e-92], N[(N[(angle * N[(N[(Pi * Pi), $MachinePrecision] * N[(a * N[(a * 3.08641975308642e-5), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle + N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e+141], N[(N[(b * b), $MachinePrecision] * N[(angle * N[(angle * N[(N[(Pi * Pi), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] + N[(N[(Pi * N[(Pi * N[(a * N[(a * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(3.08641975308642e-5 / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 7e-92

   1. Initial program 84.6%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 38.7%

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

   7. Applied rewrites 43.7%

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

   if 7e-92 < b < 4.00000000000000007e141

   1. Initial program 79.3%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 16.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 71.4%

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

   if 4.00000000000000007e141 < b

   1. Initial program 98.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{b \cdot b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(a, a \cdot 3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle, b \cdot b\right)\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+141}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(angle, angle \cdot \left(\left(\pi \cdot \pi\right) \cdot -3.08641975308642 \cdot 10^{-5}\right), \mathsf{fma}\left(\pi \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)\right)\right), \frac{3.08641975308642 \cdot 10^{-5}}{b \cdot b}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5}\\ \mathbf{if}\;b \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(a, a \cdot 3.08641975308642 \cdot 10^{-5}, t\_0\right)\right), angle, b \cdot b\right)\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right), angle, \mathsf{fma}\left(a, \left(a \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right), b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* b b) -3.08641975308642e-5)))
   (if (<= b 4e-94)
     (fma
      (* angle (* (* PI PI) (fma a (* a 3.08641975308642e-5) t_0)))
      angle
      (* b b))
     (if (<= b 7.5e+91)
       (fma
        (* t_0 (* angle (* PI PI)))
        angle
        (fma
         a
         (* (* a 3.08641975308642e-5) (* (* angle PI) (* angle PI)))
         (* b b)))
       (* b b)))))

C

double code(double a, double b, double angle) {
	double t_0 = (b * b) * -3.08641975308642e-5;
	double tmp;
	if (b <= 4e-94) {
		tmp = fma((angle * ((((double) M_PI) * ((double) M_PI)) * fma(a, (a * 3.08641975308642e-5), t_0))), angle, (b * b));
	} else if (b <= 7.5e+91) {
		tmp = fma((t_0 * (angle * (((double) M_PI) * ((double) M_PI)))), angle, fma(a, ((a * 3.08641975308642e-5) * ((angle * ((double) M_PI)) * (angle * ((double) M_PI)))), (b * b)));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(b * b) * -3.08641975308642e-5)
	tmp = 0.0
	if (b <= 4e-94)
		tmp = fma(Float64(angle * Float64(Float64(pi * pi) * fma(a, Float64(a * 3.08641975308642e-5), t_0))), angle, Float64(b * b));
	elseif (b <= 7.5e+91)
		tmp = fma(Float64(t_0 * Float64(angle * Float64(pi * pi))), angle, fma(a, Float64(Float64(a * 3.08641975308642e-5) * Float64(Float64(angle * pi) * Float64(angle * pi))), Float64(b * b)));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]}, If[LessEqual[b, 4e-94], N[(N[(angle * N[(N[(Pi * Pi), $MachinePrecision] * N[(a * N[(a * 3.08641975308642e-5), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle + N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.5e+91], N[(N[(t$95$0 * N[(angle * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle + N[(a * N[(N[(a * 3.08641975308642e-5), $MachinePrecision] * N[(N[(angle * Pi), $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.9999999999999998e-94

   1. Initial program 84.6%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 38.7%

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

   7. Applied rewrites 43.7%

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

   if 3.9999999999999998e-94 < b < 7.50000000000000033e91

   1. Initial program 85.9%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 74.9%

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

   7. Applied rewrites 80.4%

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

   if 7.50000000000000033e91 < b

   1. Initial program 89.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{b \cdot b} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 55.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1250000000000:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(a, a \cdot 3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1250000000000.0)
   (fma
    (*
     angle
     (*
      (* PI PI)
      (fma a (* a 3.08641975308642e-5) (* (* b b) -3.08641975308642e-5))))
    angle
    (* b b))
   (* b b)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1250000000000.0) {
		tmp = fma((angle * ((((double) M_PI) * ((double) M_PI)) * fma(a, (a * 3.08641975308642e-5), ((b * b) * -3.08641975308642e-5)))), angle, (b * b));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1250000000000.0)
		tmp = fma(Float64(angle * Float64(Float64(pi * pi) * fma(a, Float64(a * 3.08641975308642e-5), Float64(Float64(b * b) * -3.08641975308642e-5)))), angle, Float64(b * b));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1250000000000.0], N[(N[(angle * N[(N[(Pi * Pi), $MachinePrecision] * N[(a * N[(a * 3.08641975308642e-5), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.25e12

   1. Initial program 85.3%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 42.4%

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

   7. Applied rewrites 47.0%

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

   if 1.25e12 < b

   1. Initial program 87.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.9

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

   5. Applied rewrites 80.9%

      \[\leadsto \color{blue}{b \cdot b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 52.4% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-145}:\\ \;\;\;\;angle \cdot \left(\left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right), \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.1e-145)
   (* angle (* (* angle 3.08641975308642e-5) (* PI (* PI (* a a)))))
   (if (<= b 4e+87)
     (fma
      (* angle (* angle (* PI PI)))
      (* (* a a) 3.08641975308642e-5)
      (* b b))
     (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.1e-145) {
		tmp = angle * ((angle * 3.08641975308642e-5) * (((double) M_PI) * (((double) M_PI) * (a * a))));
	} else if (b <= 4e+87) {
		tmp = fma((angle * (angle * (((double) M_PI) * ((double) M_PI)))), ((a * a) * 3.08641975308642e-5), (b * b));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.1e-145)
		tmp = Float64(angle * Float64(Float64(angle * 3.08641975308642e-5) * Float64(pi * Float64(pi * Float64(a * a)))));
	elseif (b <= 4e+87)
		tmp = fma(Float64(angle * Float64(angle * Float64(pi * pi))), Float64(Float64(a * a) * 3.08641975308642e-5), Float64(b * b));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.1e-145], N[(angle * N[(N[(angle * 3.08641975308642e-5), $MachinePrecision] * N[(Pi * N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e+87], N[(N[(angle * N[(angle * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.1e-145

   1. Initial program 86.1%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 36.5%

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

   10. Applied rewrites 43.7%

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

   if 3.1e-145 < b < 3.9999999999999998e87

   1. Initial program 79.3%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 69.3%

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

   if 3.9999999999999998e87 < b

   1. Initial program 89.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

      \[\leadsto \color{blue}{b \cdot b} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 49.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-137}:\\ \;\;\;\;angle \cdot \left(\left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.35e-137)
   (* angle (* (* angle 3.08641975308642e-5) (* PI (* PI (* a a)))))
   (* b b)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.35e-137) {
		tmp = angle * ((angle * 3.08641975308642e-5) * (((double) M_PI) * (((double) M_PI) * (a * a))));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.35e-137) {
		tmp = angle * ((angle * 3.08641975308642e-5) * (Math.PI * (Math.PI * (a * a))));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.35e-137:
		tmp = angle * ((angle * 3.08641975308642e-5) * (math.pi * (math.pi * (a * a))))
	else:
		tmp = b * b
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.35e-137)
		tmp = Float64(angle * Float64(Float64(angle * 3.08641975308642e-5) * Float64(pi * Float64(pi * Float64(a * a)))));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.35e-137)
		tmp = angle * ((angle * 3.08641975308642e-5) * (pi * (pi * (a * a))));
	else
		tmp = b * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.35e-137], N[(angle * N[(N[(angle * 3.08641975308642e-5), $MachinePrecision] * N[(Pi * N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.34999999999999996e-137

   1. Initial program 86.1%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 36.5%

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

   10. Applied rewrites 43.7%

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

   if 1.34999999999999996e-137 < b

   1. Initial program 85.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

      \[\leadsto \color{blue}{b \cdot b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 57.4% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 64.2

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

5. Applied rewrites 64.2%

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

Reproduce

herbie shell --seed 2024233 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))