ab-angle->ABCF A

Percentage Accurate: 80.0% → 79.7%

Time: 17.0s

Alternatives: 10

Speedup: N/A×

• 36.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left({\pi}^{0.16666666666666666} \cdot \left(\left(0.005555555555555556 \cdot \left(angle \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow
   (*
    a
    (sin
     (*
      (pow PI 0.16666666666666666)
      (*
       (* 0.005555555555555556 (* angle (pow PI 0.6666666666666666)))
       (pow PI 0.16666666666666666)))))
   2.0)
  (pow (* b 1.0) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin((pow(((double) M_PI), 0.16666666666666666) * ((0.005555555555555556 * (angle * pow(((double) M_PI), 0.6666666666666666))) * pow(((double) M_PI), 0.16666666666666666))))), 2.0) + pow((b * 1.0), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin((Math.pow(Math.PI, 0.16666666666666666) * ((0.005555555555555556 * (angle * Math.pow(Math.PI, 0.6666666666666666))) * Math.pow(Math.PI, 0.16666666666666666))))), 2.0) + Math.pow((b * 1.0), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin((math.pow(math.pi, 0.16666666666666666) * ((0.005555555555555556 * (angle * math.pow(math.pi, 0.6666666666666666))) * math.pow(math.pi, 0.16666666666666666))))), 2.0) + math.pow((b * 1.0), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64((pi ^ 0.16666666666666666) * Float64(Float64(0.005555555555555556 * Float64(angle * (pi ^ 0.6666666666666666))) * (pi ^ 0.16666666666666666))))) ^ 2.0) + (Float64(b * 1.0) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((pi ^ 0.16666666666666666) * ((0.005555555555555556 * (angle * (pi ^ 0.6666666666666666))) * (pi ^ 0.16666666666666666))))) ^ 2.0) + ((b * 1.0) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[Power[Pi, 0.16666666666666666], $MachinePrecision] * N[(N[(0.005555555555555556 * N[(angle * N[Power[Pi, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.5%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*r* N/A

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

   5. add-sqr-sqrt N/A

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

   6. cbrt-prod N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 77.6%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

5. Taylor expanded in angle around 0

   \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
6. Step-by-step derivation

7. Applied rewrites 77.6%

   \[\leadsto {\left(a \cdot \sin \left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

8. Final simplification 77.6%

   \[\leadsto {\left(a \cdot \sin \left({\pi}^{0.16666666666666666} \cdot \left(\left(0.005555555555555556 \cdot \left(angle \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
9. Add Preprocessing

Alternative 2: 79.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*r* N/A

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

   5. add-sqr-sqrt N/A

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

   6. cbrt-prod N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 77.6%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

5. Taylor expanded in angle around 0

   \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
6. Step-by-step derivation

7. Applied rewrites 77.6%

   \[\leadsto {\left(a \cdot \sin \left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   3. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   5. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. associate-*l* N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{2}{3}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   8. lift-pow.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{2}{3}}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. lift-pow.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{2}{3}} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   10. pow-prod-up N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{2}{3} + \frac{1}{6}\right)}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   11. metadata-eval N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{\frac{5}{6}}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   12. lift-pow.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\frac{5}{6}}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   13. associate-*l* N/A

       \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{5}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   14. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{5}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   15. lift-pow.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{5}{6}}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   16. lift-pow.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{5}{6}} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   17. pow-prod-up N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{5}{6} + \frac{1}{6}\right)}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   18. metadata-eval N/A

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

9. Applied rewrites 77.6%

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

10. Taylor expanded in angle around 0

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

    1. unpow2 N/A

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

    2. lower-*.f64 77.6

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

12. Applied rewrites 77.6%

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

13. Final simplification 77.6%

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

Alternative 3: 67.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot \left(1 \cdot \left(b \cdot 1\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (* a (fma (cos (* angle (* PI 0.011111111111111112))) -0.5 0.5))
  a
  (* b (* 1.0 (* b 1.0)))))

C

double code(double a, double b, double angle) {
	return fma((a * fma(cos((angle * (((double) M_PI) * 0.011111111111111112))), -0.5, 0.5)), a, (b * (1.0 * (b * 1.0))));
}

Julia

function code(a, b, angle)
	return fma(Float64(a * fma(cos(Float64(angle * Float64(pi * 0.011111111111111112))), -0.5, 0.5)), a, Float64(b * Float64(1.0 * Float64(b * 1.0))))
end

Wolfram

code[a_, b_, angle_] := N[(N[(a * N[(N[Cos[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * a + N[(b * N[(1.0 * N[(b * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.5%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*r* N/A

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

   5. add-sqr-sqrt N/A

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

   6. cbrt-prod N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 77.6%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

5. Taylor expanded in angle around 0

   \[\leadsto {\left(a \cdot \sin \left(\left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
6. Step-by-step derivation

7. Applied rewrites 77.6%

   \[\leadsto {\left(a \cdot \sin \left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

9. Applied rewrites 68.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot \left(1 \cdot \left(b \cdot 1\right)\right)\right)} \]
10. Add Preprocessing

Alternative 4: 57.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{fma}\left(a \cdot a, 3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle, b \cdot b\right)\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+131}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(angle, angle \cdot \left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right), \mathsf{fma}\left(a \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot \left(angle \cdot angle\right)\right)\right)\right), \frac{3.08641975308642 \cdot 10^{-5}}{b \cdot b}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 7.2e-109)
   (fma
    (*
     (* angle PI)
     (* PI (fma (* a a) 3.08641975308642e-5 (* (* b b) -3.08641975308642e-5))))
    angle
    (* b b))
   (if (<= b 6.5e+131)
     (*
      (* b b)
      (fma
       angle
       (* angle (* -3.08641975308642e-5 (* PI PI)))
       (fma
        (* a (* a (* PI (* PI (* angle angle)))))
        (/ 3.08641975308642e-5 (* b b))
        1.0)))
     (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 7.2e-109) {
		tmp = fma(((angle * ((double) M_PI)) * (((double) M_PI) * fma((a * a), 3.08641975308642e-5, ((b * b) * -3.08641975308642e-5)))), angle, (b * b));
	} else if (b <= 6.5e+131) {
		tmp = (b * b) * fma(angle, (angle * (-3.08641975308642e-5 * (((double) M_PI) * ((double) M_PI)))), fma((a * (a * (((double) M_PI) * (((double) M_PI) * (angle * angle))))), (3.08641975308642e-5 / (b * b)), 1.0));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 7.2e-109)
		tmp = fma(Float64(Float64(angle * pi) * Float64(pi * fma(Float64(a * a), 3.08641975308642e-5, Float64(Float64(b * b) * -3.08641975308642e-5)))), angle, Float64(b * b));
	elseif (b <= 6.5e+131)
		tmp = Float64(Float64(b * b) * fma(angle, Float64(angle * Float64(-3.08641975308642e-5 * Float64(pi * pi))), fma(Float64(a * Float64(a * Float64(pi * Float64(pi * Float64(angle * angle))))), Float64(3.08641975308642e-5 / Float64(b * b)), 1.0)));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 7.2e-109], N[(N[(N[(angle * Pi), $MachinePrecision] * N[(Pi * N[(N[(a * a), $MachinePrecision] * 3.08641975308642e-5 + N[(N[(b * b), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle + N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e+131], N[(N[(b * b), $MachinePrecision] * N[(angle * N[(angle * N[(-3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(a * N[(Pi * N[(Pi * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(3.08641975308642e-5 / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 7.2000000000000001e-109

   1. Initial program 77.0%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 42.5%

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

   7. Applied rewrites 44.9%

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

   if 7.2000000000000001e-109 < b < 6.5e131

   1. Initial program 64.3%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 57.6%

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

   if 6.5e131 < b

   1. Initial program 95.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 95.6

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

   5. Applied rewrites 95.6%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{fma}\left(a \cdot a, 3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle, b \cdot b\right)\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+131}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(angle, angle \cdot \left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right), \mathsf{fma}\left(a \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot \left(angle \cdot angle\right)\right)\right)\right), \frac{3.08641975308642 \cdot 10^{-5}}{b \cdot b}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-171}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(angle \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e-171)
   (* b b)
   (if (<= (/ angle 180.0) 5e+143)
     (fma
      (* angle angle)
      (* PI (* PI (* a (* a 3.08641975308642e-5))))
      (* b b))
     (* a (* a (* 3.08641975308642e-5 (* PI (* angle (* angle PI)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 5e-171) {
		tmp = b * b;
	} else if ((angle / 180.0) <= 5e+143) {
		tmp = fma((angle * angle), (((double) M_PI) * (((double) M_PI) * (a * (a * 3.08641975308642e-5)))), (b * b));
	} else {
		tmp = a * (a * (3.08641975308642e-5 * (((double) M_PI) * (angle * (angle * ((double) M_PI))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5e-171)
		tmp = Float64(b * b);
	elseif (Float64(angle / 180.0) <= 5e+143)
		tmp = fma(Float64(angle * angle), Float64(pi * Float64(pi * Float64(a * Float64(a * 3.08641975308642e-5)))), Float64(b * b));
	else
		tmp = Float64(a * Float64(a * Float64(3.08641975308642e-5 * Float64(pi * Float64(angle * Float64(angle * pi))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e-171], N[(b * b), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+143], N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * N[(a * N[(a * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(3.08641975308642e-5 * N[(Pi * N[(angle * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999992e-171

   1. Initial program 81.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.9

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

   5. Applied rewrites 57.9%

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

   if 4.99999999999999992e-171 < (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000012e143

   1. Initial program 79.8%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 70.0%

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

   if 5.00000000000000012e143 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 59.4%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 19.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 41.0%

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

   10. Applied rewrites 49.5%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-171}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(angle \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-171}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(angle \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e-171)
   (* b b)
   (if (<= (/ angle 180.0) 5e+132)
     (fma
      (* angle angle)
      (* a (* a (* 3.08641975308642e-5 (* PI PI))))
      (* b b))
     (* a (* a (* 3.08641975308642e-5 (* PI (* angle (* angle PI)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 5e-171) {
		tmp = b * b;
	} else if ((angle / 180.0) <= 5e+132) {
		tmp = fma((angle * angle), (a * (a * (3.08641975308642e-5 * (((double) M_PI) * ((double) M_PI))))), (b * b));
	} else {
		tmp = a * (a * (3.08641975308642e-5 * (((double) M_PI) * (angle * (angle * ((double) M_PI))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5e-171)
		tmp = Float64(b * b);
	elseif (Float64(angle / 180.0) <= 5e+132)
		tmp = fma(Float64(angle * angle), Float64(a * Float64(a * Float64(3.08641975308642e-5 * Float64(pi * pi)))), Float64(b * b));
	else
		tmp = Float64(a * Float64(a * Float64(3.08641975308642e-5 * Float64(pi * Float64(angle * Float64(angle * pi))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e-171], N[(b * b), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+132], N[(N[(angle * angle), $MachinePrecision] * N[(a * N[(a * N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(3.08641975308642e-5 * N[(Pi * N[(angle * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999992e-171

   1. Initial program 81.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.9

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

   5. Applied rewrites 57.9%

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

   if 4.99999999999999992e-171 < (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000001e132

   1. Initial program 79.4%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 69.5%

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

   if 5.0000000000000001e132 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 60.4%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 19.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 42.6%

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

   10. Applied rewrites 50.8%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-171}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(angle \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{fma}\left(a \cdot a, 3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.05e+117)
   (fma
    (*
     (* angle PI)
     (* PI (fma (* a a) 3.08641975308642e-5 (* (* b b) -3.08641975308642e-5))))
    angle
    (* b b))
   (* b b)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.05e+117) {
		tmp = fma(((angle * ((double) M_PI)) * (((double) M_PI) * fma((a * a), 3.08641975308642e-5, ((b * b) * -3.08641975308642e-5)))), angle, (b * b));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.05e+117)
		tmp = fma(Float64(Float64(angle * pi) * Float64(pi * fma(Float64(a * a), 3.08641975308642e-5, Float64(Float64(b * b) * -3.08641975308642e-5)))), angle, Float64(b * b));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.05e+117], N[(N[(N[(angle * Pi), $MachinePrecision] * N[(Pi * N[(N[(a * a), $MachinePrecision] * 3.08641975308642e-5 + N[(N[(b * b), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0500000000000001e117

   1. Initial program 73.9%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 43.2%

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

   7. Applied rewrites 45.3%

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

   if 1.0500000000000001e117 < b

   1. Initial program 96.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 91.8

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

   5. Applied rewrites 91.8%

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

Alternative 8: 61.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.9 \cdot 10^{+152}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(angle \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.9e+152)
   (* b b)
   (* a (* a (* 3.08641975308642e-5 (* PI (* angle (* angle PI))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.9e+152) {
		tmp = b * b;
	} else {
		tmp = a * (a * (3.08641975308642e-5 * (((double) M_PI) * (angle * (angle * ((double) M_PI))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.9e+152) {
		tmp = b * b;
	} else {
		tmp = a * (a * (3.08641975308642e-5 * (Math.PI * (angle * (angle * Math.PI)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 3.9e+152:
		tmp = b * b
	else:
		tmp = a * (a * (3.08641975308642e-5 * (math.pi * (angle * (angle * math.pi)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.9e+152)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * Float64(a * Float64(3.08641975308642e-5 * Float64(pi * Float64(angle * Float64(angle * pi))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 3.9e+152)
		tmp = b * b;
	else
		tmp = a * (a * (3.08641975308642e-5 * (pi * (angle * (angle * pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.9e+152], N[(b * b), $MachinePrecision], N[(a * N[(a * N[(3.08641975308642e-5 * N[(Pi * N[(angle * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.90000000000000011e152

   1. Initial program 74.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

   if 3.90000000000000011e152 < a

   1. Initial program 96.5%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 39.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 46.0%

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

   10. Applied rewrites 55.9%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.9 \cdot 10^{+152}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(angle \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.9 \cdot 10^{+152}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(angle \cdot \pi\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.9e+152)
   (* b b)
   (* (* angle (* angle PI)) (* 3.08641975308642e-5 (* PI (* a a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.9e+152) {
		tmp = b * b;
	} else {
		tmp = (angle * (angle * ((double) M_PI))) * (3.08641975308642e-5 * (((double) M_PI) * (a * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.9e+152) {
		tmp = b * b;
	} else {
		tmp = (angle * (angle * Math.PI)) * (3.08641975308642e-5 * (Math.PI * (a * a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 3.9e+152:
		tmp = b * b
	else:
		tmp = (angle * (angle * math.pi)) * (3.08641975308642e-5 * (math.pi * (a * a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.9e+152)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(angle * Float64(angle * pi)) * Float64(3.08641975308642e-5 * Float64(pi * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 3.9e+152)
		tmp = b * b;
	else
		tmp = (angle * (angle * pi)) * (3.08641975308642e-5 * (pi * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.9e+152], N[(b * b), $MachinePrecision], N[(N[(angle * N[(angle * Pi), $MachinePrecision]), $MachinePrecision] * N[(3.08641975308642e-5 * N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.90000000000000011e152

   1. Initial program 74.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

   if 3.90000000000000011e152 < a

   1. Initial program 96.5%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 39.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 46.0%

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

   10. Applied rewrites 46.3%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.9 \cdot 10^{+152}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(angle \cdot \pi\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.9% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 55.2

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

5. Applied rewrites 55.2%

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

Reproduce

herbie shell --seed 2024234 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))