ab-angle->ABCF C

Percentage Accurate: 79.4% → 79.4%

Time: 16.0s

Alternatives: 12

Speedup: N/A×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.4

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

5. Applied rewrites 81.4%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-PI.f64 N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. lift-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lift-PI.f64 N/A

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

   16. lower-sqrt.f64 81.4

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

7. Applied rewrites 81.4%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. sqrt-unprod N/A

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

   10. rem-square-sqrt N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. associate-*r* N/A

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

   14. lift-*.f64 N/A

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

   15. sqrt-unprod N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lift-sqrt.f64 N/A

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

9. Applied rewrites 81.4%

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

Alternative 2: 79.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.4

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

5. Applied rewrites 81.4%

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

Alternative 3: 79.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.4

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

5. Applied rewrites 81.4%

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

   1. lift-PI.f64 N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 81.0

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

7. Applied rewrites 81.0%

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

8. Final simplification 81.0%

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

Alternative 4: 76.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+24}:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right), b, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e+24)
   (+
    (* a a)
    (pow
     (*
      b
      (*
       angle
       (*
        PI
        (fma
         (* (* angle angle) -2.8577960676726107e-8)
         (* PI PI)
         0.005555555555555556))))
     2.0))
   (fma
    (* b (fma (cos (* (* angle PI) 0.011111111111111112)) -0.5 0.5))
    b
    (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+24) {
		tmp = (a * a) + pow((b * (angle * (((double) M_PI) * fma(((angle * angle) * -2.8577960676726107e-8), (((double) M_PI) * ((double) M_PI)), 0.005555555555555556)))), 2.0);
	} else {
		tmp = fma((b * fma(cos(((angle * ((double) M_PI)) * 0.011111111111111112)), -0.5, 0.5)), b, (a * a));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e+24)
		tmp = Float64(Float64(a * a) + (Float64(b * Float64(angle * Float64(pi * fma(Float64(Float64(angle * angle) * -2.8577960676726107e-8), Float64(pi * pi), 0.005555555555555556)))) ^ 2.0));
	else
		tmp = fma(Float64(b * fma(cos(Float64(Float64(angle * pi) * 0.011111111111111112)), -0.5, 0.5)), b, Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+24], N[(N[(a * a), $MachinePrecision] + N[Power[N[(b * N[(angle * 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[(b * N[(N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * b + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e24

   1. Initial program 88.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow3 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

   8. Applied rewrites 83.7%

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

   if 2e24 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 59.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+24}:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right), b, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;a \cdot a + {\left(0.005555555555555556 \cdot \left(b \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right), b, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e-5)
   (+ (* a a) (pow (* 0.005555555555555556 (* b (* angle PI))) 2.0))
   (fma
    (* b (fma (cos (* (* angle PI) 0.011111111111111112)) -0.5 0.5))
    b
    (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e-5) {
		tmp = (a * a) + pow((0.005555555555555556 * (b * (angle * ((double) M_PI)))), 2.0);
	} else {
		tmp = fma((b * fma(cos(((angle * ((double) M_PI)) * 0.011111111111111112)), -0.5, 0.5)), b, (a * a));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e-5)
		tmp = Float64(Float64(a * a) + (Float64(0.005555555555555556 * Float64(b * Float64(angle * pi))) ^ 2.0));
	else
		tmp = fma(Float64(b * fma(cos(Float64(Float64(angle * pi) * 0.011111111111111112)), -0.5, 0.5)), b, Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e-5], N[(N[(a * a), $MachinePrecision] + N[Power[N[(0.005555555555555556 * N[(b * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * b + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000016e-5

   1. Initial program 88.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 84.9

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

   8. Applied rewrites 84.9%

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

   if 2.00000000000000016e-5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 61.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.0

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

   5. Applied rewrites 62.0%

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

   7. Applied rewrites 61.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;a \cdot a + {\left(0.005555555555555556 \cdot \left(b \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right), b, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-110}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(0.005555555555555556 \cdot \left(b \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.2e-110)
   (* a a)
   (+ (* a a) (pow (* 0.005555555555555556 (* b (* angle PI))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.2e-110) {
		tmp = a * a;
	} else {
		tmp = (a * a) + pow((0.005555555555555556 * (b * (angle * ((double) M_PI)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.2e-110) {
		tmp = a * a;
	} else {
		tmp = (a * a) + Math.pow((0.005555555555555556 * (b * (angle * Math.PI))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 3.2e-110:
		tmp = a * a
	else:
		tmp = (a * a) + math.pow((0.005555555555555556 * (b * (angle * math.pi))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.2e-110)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + (Float64(0.005555555555555556 * Float64(b * Float64(angle * pi))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 3.2e-110)
		tmp = a * a;
	else
		tmp = (a * a) + ((0.005555555555555556 * (b * (angle * pi))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.2e-110], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(0.005555555555555556 * N[(b * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.20000000000000028e-110

   1. Initial program 78.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 3.20000000000000028e-110 < b

   1. Initial program 85.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 82.7

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

   8. Applied rewrites 82.7%

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

Alternative 7: 62.5% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-110}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot \left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b \cdot \pi\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.2e-110)
   (* a a)
   (if (<= b 1.1e+140)
     (fma
      3.08641975308642e-5
      (* (* b b) (* angle (* angle (* PI PI))))
      (* a a))
     (* b (* (* b PI) (* (* angle angle) (* PI 3.08641975308642e-5)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.2e-110) {
		tmp = a * a;
	} else if (b <= 1.1e+140) {
		tmp = fma(3.08641975308642e-5, ((b * b) * (angle * (angle * (((double) M_PI) * ((double) M_PI))))), (a * a));
	} else {
		tmp = b * ((b * ((double) M_PI)) * ((angle * angle) * (((double) M_PI) * 3.08641975308642e-5)));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.2e-110)
		tmp = Float64(a * a);
	elseif (b <= 1.1e+140)
		tmp = fma(3.08641975308642e-5, Float64(Float64(b * b) * Float64(angle * Float64(angle * Float64(pi * pi)))), Float64(a * a));
	else
		tmp = Float64(b * Float64(Float64(b * pi) * Float64(Float64(angle * angle) * Float64(pi * 3.08641975308642e-5))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.2e-110], N[(a * a), $MachinePrecision], If[LessEqual[b, 1.1e+140], N[(3.08641975308642e-5 * N[(N[(b * b), $MachinePrecision] * N[(angle * N[(angle * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(b * Pi), $MachinePrecision] * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.20000000000000028e-110

   1. Initial program 78.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 3.20000000000000028e-110 < b < 1.0999999999999999e140

   1. Initial program 72.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 72.7

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 65.9

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

   8. Applied rewrites 65.9%

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

   if 1.0999999999999999e140 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 56.0

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

   8. Applied rewrites 56.0%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. metadata-eval N/A

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

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

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

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

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

       13. metadata-eval N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

   11. Applied rewrites 56.0%

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

       1. lift-PI.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

   13. Applied rewrites 66.2%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-110}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot \left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b \cdot \pi\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;b \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(b \cdot b\right), t\_0 \cdot 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, b \cdot \left(b \cdot \left(angle \cdot t\_0\right)\right), a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle PI))))
   (if (<= b 5e+110)
     (fma (* angle (* b b)) (* t_0 3.08641975308642e-5) (* a a))
     (fma 3.08641975308642e-5 (* b (* b (* angle t_0))) (* a a)))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle * ((double) M_PI));
	double tmp;
	if (b <= 5e+110) {
		tmp = fma((angle * (b * b)), (t_0 * 3.08641975308642e-5), (a * a));
	} else {
		tmp = fma(3.08641975308642e-5, (b * (b * (angle * t_0))), (a * a));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * pi))
	tmp = 0.0
	if (b <= 5e+110)
		tmp = fma(Float64(angle * Float64(b * b)), Float64(t_0 * 3.08641975308642e-5), Float64(a * a));
	else
		tmp = fma(3.08641975308642e-5, Float64(b * Float64(b * Float64(angle * t_0))), Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 5e+110], N[(N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * 3.08641975308642e-5), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(3.08641975308642e-5 * N[(b * N[(b * N[(angle * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.99999999999999978e110

   1. Initial program 78.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 63.2

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

   8. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

   10. Applied rewrites 70.1%

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

   if 4.99999999999999978e110 < b

   1. Initial program 94.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 54.7

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

   8. Applied rewrites 54.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 85.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 85.5

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

   10. Applied rewrites 85.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(b \cdot b\right), \left(\pi \cdot \left(angle \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, b \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(angle \cdot \pi\right)\right)\right)\right), a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.0% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-110}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, b \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(angle \cdot \pi\right)\right)\right)\right), a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.2e-110)
   (* a a)
   (fma
    3.08641975308642e-5
    (* b (* b (* angle (* PI (* angle PI)))))
    (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.2e-110) {
		tmp = a * a;
	} else {
		tmp = fma(3.08641975308642e-5, (b * (b * (angle * (((double) M_PI) * (angle * ((double) M_PI)))))), (a * a));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.2e-110)
		tmp = Float64(a * a);
	else
		tmp = fma(3.08641975308642e-5, Float64(b * Float64(b * Float64(angle * Float64(pi * Float64(angle * pi))))), Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.2e-110], N[(a * a), $MachinePrecision], N[(3.08641975308642e-5 * N[(b * N[(b * N[(angle * N[(Pi * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.20000000000000028e-110

   1. Initial program 78.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 3.20000000000000028e-110 < b

   1. Initial program 85.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.3

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

   8. Applied rewrites 61.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 77.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 77.4

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

   10. Applied rewrites 77.4%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-110}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, b \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(angle \cdot \pi\right)\right)\right)\right), a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{+98}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b \cdot \pi\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.55e+98)
   (* a a)
   (* b (* (* b PI) (* (* angle angle) (* PI 3.08641975308642e-5))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.55e+98) {
		tmp = a * a;
	} else {
		tmp = b * ((b * ((double) M_PI)) * ((angle * angle) * (((double) M_PI) * 3.08641975308642e-5)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.55e+98) {
		tmp = a * a;
	} else {
		tmp = b * ((b * Math.PI) * ((angle * angle) * (Math.PI * 3.08641975308642e-5)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.55e+98:
		tmp = a * a
	else:
		tmp = b * ((b * math.pi) * ((angle * angle) * (math.pi * 3.08641975308642e-5)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.55e+98)
		tmp = Float64(a * a);
	else
		tmp = Float64(b * Float64(Float64(b * pi) * Float64(Float64(angle * angle) * Float64(pi * 3.08641975308642e-5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.55e+98)
		tmp = a * a;
	else
		tmp = b * ((b * pi) * ((angle * angle) * (pi * 3.08641975308642e-5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.55e+98], N[(a * a), $MachinePrecision], N[(b * N[(N[(b * Pi), $MachinePrecision] * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5500000000000001e98

   1. Initial program 78.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.3

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

   5. Applied rewrites 63.3%

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

   if 1.5500000000000001e98 < b

   1. Initial program 91.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 54.5

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

   8. Applied rewrites 54.5%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. metadata-eval N/A

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

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

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

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

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

       13. metadata-eval N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

   11. Applied rewrites 54.5%

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

       1. lift-PI.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

   13. Applied rewrites 62.4%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{+98}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b \cdot \pi\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.6% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.12 \cdot 10^{+160}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.12e+160)
   (* a a)
   (* (* b b) (* (* PI PI) (* (* angle angle) 3.08641975308642e-5)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.12e+160) {
		tmp = a * a;
	} else {
		tmp = (b * b) * ((((double) M_PI) * ((double) M_PI)) * ((angle * angle) * 3.08641975308642e-5));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.12e+160) {
		tmp = a * a;
	} else {
		tmp = (b * b) * ((Math.PI * Math.PI) * ((angle * angle) * 3.08641975308642e-5));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.12e+160:
		tmp = a * a
	else:
		tmp = (b * b) * ((math.pi * math.pi) * ((angle * angle) * 3.08641975308642e-5))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.12e+160)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(pi * pi) * Float64(Float64(angle * angle) * 3.08641975308642e-5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.12e+160)
		tmp = a * a;
	else
		tmp = (b * b) * ((pi * pi) * ((angle * angle) * 3.08641975308642e-5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.12e+160], N[(a * a), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(angle * angle), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.12e160

   1. Initial program 77.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.2

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

   5. Applied rewrites 61.2%

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

   if 1.12e160 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 57.7

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. metadata-eval N/A

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

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

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

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

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

       13. metadata-eval N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

   11. Applied rewrites 57.7%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.12 \cdot 10^{+160}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.5% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 57.1

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

5. Applied rewrites 57.1%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))