ab-angle->ABCF A

Percentage Accurate: 79.9% → 79.7%

Time: 17.1s

Alternatives: 14

Speedup: 1.2×

• Could not converge on a ground truth

  1. a = 2.6362394664149683e+222
  2. b = 1.3446754729452727e-61
  3. angle = 2.2537262012846156e+286

• 36.2% 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: 79.9% 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, 0.8× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow
   (* a (sin (pow (* (sqrt (* angle_m PI)) (sqrt 0.005555555555555556)) 2.0)))
   2.0)
  (pow (* b (cos (* PI (/ angle_m 180.0)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[Power[N[(N[Sqrt[N[(angle$95$m * Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.005555555555555556], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 80.9

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

4. Applied rewrites 80.9%

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

   1. lift-/.f64 N/A

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

   2. inv-pow N/A

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

   3. sqr-pow N/A

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

   4. pow2 N/A

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

   5. lower-pow.f64 N/A

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

6. Applied rewrites 41.6%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.5}\right)}^{2}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. pow-prod-down N/A

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

   6. pow1/2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. pow1/2 N/A

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

   11. lower-sqrt.f64 41.6

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

8. Applied rewrites 41.6%

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

9. Final simplification 41.6%

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

Alternative 2: 79.7% accurate, 0.8× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (cos (* PI (/ angle_m 180.0)))) 2.0)
  (pow
   (* a (sin (pow (* (sqrt (* angle_m 0.005555555555555556)) (sqrt PI)) 2.0)))
   2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Sin[N[Power[N[(N[Sqrt[N[(angle$95$m * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 80.9

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

4. Applied rewrites 80.9%

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

   1. lift-/.f64 N/A

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

   2. inv-pow N/A

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

   3. sqr-pow N/A

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

   4. pow2 N/A

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

   5. lower-pow.f64 N/A

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

6. Applied rewrites 41.6%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.5}\right)}^{2}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. unpow-prod-down N/A

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

   8. pow1/2 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. pow1/2 N/A

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

   12. lower-sqrt.f64 41.6

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

8. Applied rewrites 41.6%

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

9. Final simplification 41.6%

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

Alternative 3: 79.8% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (cos (* PI (/ angle_m 180.0)))) 2.0)
  (pow (* a (sin (/ 1.0 (/ 180.0 (* angle_m PI))))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Sin[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 80.9

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

4. Applied rewrites 80.9%

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

5. Final simplification 80.9%

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

Alternative 4: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := a \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\\ \mathsf{fma}\left(t\_0, t\_0, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{0.005555555555555556}{\frac{1}{angle\_m \cdot \pi}}\right)\right)\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* a (sin (* PI (* angle_m 0.005555555555555556))))))
   (fma
    t_0
    t_0
    (*
     (* b b)
     (+
      0.5
      (*
       0.5
       (cos (* 2.0 (/ 0.005555555555555556 (/ 1.0 (* angle_m PI)))))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = a * sin((((double) M_PI) * (angle_m * 0.005555555555555556)));
	return fma(t_0, t_0, ((b * b) * (0.5 + (0.5 * cos((2.0 * (0.005555555555555556 / (1.0 / (angle_m * ((double) M_PI))))))))));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(a * sin(Float64(pi * Float64(angle_m * 0.005555555555555556))))
	return fma(t_0, t_0, Float64(Float64(b * b) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(0.005555555555555556 / Float64(1.0 / Float64(angle_m * pi)))))))))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(a * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0 + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.005555555555555556 / N[(1.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.9%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 80.9

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 80.4

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 80.9

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

4. Applied rewrites 80.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. clear-num N/A

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

   8. div-inv N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 80.9

       \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{0.005555555555555556}{\color{blue}{\frac{1}{angle \cdot \pi}}}\right)\right)\right) \]

6. Applied rewrites 80.9%

   \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}}\right)\right)\right) \]

7. Final simplification 80.9%

   \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right)\right)\right) \]
8. Add Preprocessing

Alternative 5: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := a \cdot \sin t\_0\\ \mathsf{fma}\left(t\_1, t\_1, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right)\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))) (t_1 (* a (sin t_0))))
   (fma t_1 t_1 (* (* b b) (+ 0.5 (* 0.5 (cos (* 2.0 t_0))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = a * sin(t_0);
	return fma(t_1, t_1, ((b * b) * (0.5 + (0.5 * cos((2.0 * t_0))))));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = Float64(a * sin(t_0))
	return fma(t_1, t_1, Float64(Float64(b * b) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_0))))))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$1 * t$95$1 + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 80.9%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 80.9

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 80.4

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 80.9

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

4. Applied rewrites 80.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)} \]

5. Final simplification 80.9%

   \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \]
6. Add Preprocessing

Alternative 6: 79.7% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (sin (/ 1.0 (/ 180.0 (* angle_m PI))))) 2.0)
  (pow (* b 1.0) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 80.9

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

4. Applied rewrites 80.9%

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

5. Taylor expanded in angle around 0

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

7. Applied rewrites 80.8%

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

Alternative 7: 79.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := a \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\\ \mathsf{fma}\left(t\_0, t\_0, \left(b \cdot b\right) \cdot 1\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* a (sin (* PI (* angle_m 0.005555555555555556))))))
   (fma t_0 t_0 (* (* b b) 1.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = a * sin((((double) M_PI) * (angle_m * 0.005555555555555556)));
	return fma(t_0, t_0, ((b * b) * 1.0));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(a * sin(Float64(pi * Float64(angle_m * 0.005555555555555556))))
	return fma(t_0, t_0, Float64(Float64(b * b) * 1.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(a * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0 + N[(N[(b * b), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.9%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 80.9

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 80.4

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 80.9

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

4. Applied rewrites 80.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)} \]

5. Taylor expanded in angle around 0

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

7. Applied rewrites 80.7%

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

8. Final simplification 80.7%

   \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \left(b \cdot b\right) \cdot 1\right) \]
9. Add Preprocessing

Alternative 8: 56.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(a, a \cdot 3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 2.5e+147)
   (fma
    (*
     (* PI PI)
     (*
      angle_m
      (fma a (* a 3.08641975308642e-5) (* (* b b) -3.08641975308642e-5))))
    angle_m
    (* b b))
   (* (* b b) (fma 0.5 (cos (* angle_m (* PI 0.011111111111111112))) 0.5))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.5e+147) {
		tmp = fma(((((double) M_PI) * ((double) M_PI)) * (angle_m * fma(a, (a * 3.08641975308642e-5), ((b * b) * -3.08641975308642e-5)))), angle_m, (b * b));
	} else {
		tmp = (b * b) * fma(0.5, cos((angle_m * (((double) M_PI) * 0.011111111111111112))), 0.5);
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 2.5e+147)
		tmp = fma(Float64(Float64(pi * pi) * Float64(angle_m * fma(a, Float64(a * 3.08641975308642e-5), Float64(Float64(b * b) * -3.08641975308642e-5)))), angle_m, Float64(b * b));
	else
		tmp = Float64(Float64(b * b) * fma(0.5, cos(Float64(angle_m * Float64(pi * 0.011111111111111112))), 0.5));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 2.5e+147], N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(angle$95$m * N[(a * N[(a * 3.08641975308642e-5), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(0.5 * N[Cos[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5000000000000001e147

   1. Initial program 76.8%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 51.4%

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

   7. Applied rewrites 55.1%

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

   if 2.5000000000000001e147 < b

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 100.0

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{2}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)}, \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)}, \frac{1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)}, \frac{1}{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right), \frac{1}{2}\right) \]

      11. lower-PI.f64 100.0

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\color{blue}{\pi} \cdot 0.011111111111111112\right)\right), 0.5\right) \]

   7. Applied rewrites 100.0%

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

Alternative 9: 56.0% accurate, 8.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(a, a \cdot 3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 2.5e+147)
   (fma
    (*
     (* PI PI)
     (*
      angle_m
      (fma a (* a 3.08641975308642e-5) (* (* b b) -3.08641975308642e-5))))
    angle_m
    (* b b))
   (* b b)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.5e+147) {
		tmp = fma(((((double) M_PI) * ((double) M_PI)) * (angle_m * fma(a, (a * 3.08641975308642e-5), ((b * b) * -3.08641975308642e-5)))), angle_m, (b * b));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 2.5e+147)
		tmp = fma(Float64(Float64(pi * pi) * Float64(angle_m * fma(a, Float64(a * 3.08641975308642e-5), Float64(Float64(b * b) * -3.08641975308642e-5)))), angle_m, Float64(b * b));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 2.5e+147], N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(angle$95$m * N[(a * N[(a * 3.08641975308642e-5), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5000000000000001e147

   1. Initial program 76.8%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 51.4%

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

   7. Applied rewrites 55.1%

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

   if 2.5000000000000001e147 < b

   1. Initial program 100.0%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 10: 64.0% accurate, 9.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\pi \cdot \pi\right)\\ \mathbf{if}\;a \leq 2.7 \cdot 10^{-43}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot t\_0, a \cdot \left(a \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(angle\_m \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* PI PI))))
   (if (<= a 2.7e-43)
     (* b b)
     (if (<= a 8.8e+147)
       (fma (* angle_m t_0) (* a (* a 3.08641975308642e-5)) (* b b))
       (* t_0 (* angle_m (* 3.08641975308642e-5 (* a a))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = angle_m * (((double) M_PI) * ((double) M_PI));
	double tmp;
	if (a <= 2.7e-43) {
		tmp = b * b;
	} else if (a <= 8.8e+147) {
		tmp = fma((angle_m * t_0), (a * (a * 3.08641975308642e-5)), (b * b));
	} else {
		tmp = t_0 * (angle_m * (3.08641975308642e-5 * (a * a)));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(angle_m * Float64(pi * pi))
	tmp = 0.0
	if (a <= 2.7e-43)
		tmp = Float64(b * b);
	elseif (a <= 8.8e+147)
		tmp = fma(Float64(angle_m * t_0), Float64(a * Float64(a * 3.08641975308642e-5)), Float64(b * b));
	else
		tmp = Float64(t_0 * Float64(angle_m * Float64(3.08641975308642e-5 * Float64(a * a))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 2.7e-43], N[(b * b), $MachinePrecision], If[LessEqual[a, 8.8e+147], N[(N[(angle$95$m * t$95$0), $MachinePrecision] * N[(a * N[(a * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(angle$95$m * N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 2.69999999999999991e-43

   1. Initial program 80.0%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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 63.3

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

   5. Applied rewrites 63.3%

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

   if 2.69999999999999991e-43 < a < 8.8000000000000007e147

   1. Initial program 73.4%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 29.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 69.0%

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

   if 8.8000000000000007e147 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 59.0%

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

   10. Applied rewrites 74.4%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.7 \cdot 10^{-43}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right), a \cdot \left(a \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{+128}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(angle\_m \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.25e+128)
   (* b b)
   (* (* angle_m (* PI PI)) (* angle_m (* 3.08641975308642e-5 (* a a))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.25e+128) {
		tmp = b * b;
	} else {
		tmp = (angle_m * (((double) M_PI) * ((double) M_PI))) * (angle_m * (3.08641975308642e-5 * (a * a)));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.25e+128) {
		tmp = b * b;
	} else {
		tmp = (angle_m * (Math.PI * Math.PI)) * (angle_m * (3.08641975308642e-5 * (a * a)));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 1.25e+128:
		tmp = b * b
	else:
		tmp = (angle_m * (math.pi * math.pi)) * (angle_m * (3.08641975308642e-5 * (a * a)))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.25e+128)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(angle_m * Float64(pi * pi)) * Float64(angle_m * Float64(3.08641975308642e-5 * Float64(a * a))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.25e+128)
		tmp = b * b;
	else
		tmp = (angle_m * (pi * pi)) * (angle_m * (3.08641975308642e-5 * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.25e+128], N[(b * b), $MachinePrecision], N[(N[(angle$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.25e128

   1. Initial program 78.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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 63.3

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

   5. Applied rewrites 63.3%

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

   if 1.25e128 < a

   1. Initial program 96.9%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 58.6%

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

   10. Applied rewrites 72.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{+128}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{+128}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(angle\_m \cdot \left(a \cdot \left(angle\_m \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.25e+128)
   (* b b)
   (* 3.08641975308642e-5 (* a (* angle_m (* a (* angle_m (* PI PI))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.25e+128) {
		tmp = b * b;
	} else {
		tmp = 3.08641975308642e-5 * (a * (angle_m * (a * (angle_m * (((double) M_PI) * ((double) M_PI))))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.25e+128) {
		tmp = b * b;
	} else {
		tmp = 3.08641975308642e-5 * (a * (angle_m * (a * (angle_m * (Math.PI * Math.PI)))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 1.25e+128:
		tmp = b * b
	else:
		tmp = 3.08641975308642e-5 * (a * (angle_m * (a * (angle_m * (math.pi * math.pi)))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.25e+128)
		tmp = Float64(b * b);
	else
		tmp = Float64(3.08641975308642e-5 * Float64(a * Float64(angle_m * Float64(a * Float64(angle_m * Float64(pi * pi))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.25e+128)
		tmp = b * b;
	else
		tmp = 3.08641975308642e-5 * (a * (angle_m * (a * (angle_m * (pi * pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.25e+128], N[(b * b), $MachinePrecision], N[(3.08641975308642e-5 * N[(a * N[(angle$95$m * N[(a * N[(angle$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.25e128

   1. Initial program 78.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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 63.3

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

   5. Applied rewrites 63.3%

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

   if 1.25e128 < a

   1. Initial program 96.9%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 58.6%

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

   10. Applied rewrites 66.3%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{+128}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{+128}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot \left(angle\_m \cdot angle\_m\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.3e+128)
   (* b b)
   (* 3.08641975308642e-5 (* a (* a (* PI (* PI (* angle_m angle_m))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.3e+128) {
		tmp = b * b;
	} else {
		tmp = 3.08641975308642e-5 * (a * (a * (((double) M_PI) * (((double) M_PI) * (angle_m * angle_m)))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.3e+128) {
		tmp = b * b;
	} else {
		tmp = 3.08641975308642e-5 * (a * (a * (Math.PI * (Math.PI * (angle_m * angle_m)))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 1.3e+128:
		tmp = b * b
	else:
		tmp = 3.08641975308642e-5 * (a * (a * (math.pi * (math.pi * (angle_m * angle_m)))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.3e+128)
		tmp = Float64(b * b);
	else
		tmp = Float64(3.08641975308642e-5 * Float64(a * Float64(a * Float64(pi * Float64(pi * Float64(angle_m * angle_m))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.3e+128)
		tmp = b * b;
	else
		tmp = 3.08641975308642e-5 * (a * (a * (pi * (pi * (angle_m * angle_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.3e+128], N[(b * b), $MachinePrecision], N[(3.08641975308642e-5 * N[(a * N[(a * N[(Pi * N[(Pi * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.3e128

   1. Initial program 78.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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 63.3

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

   5. Applied rewrites 63.3%

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

   if 1.3e128 < a

   1. Initial program 96.9%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 58.6%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{+128}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot \left(angle \cdot angle\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.0% accurate, 74.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ b \cdot b \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m) :precision binary64 (* b b))

C

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

Fortran

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

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return b * b;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return b * b

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 80.9%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
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 60.2

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

5. Applied rewrites 60.2%

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

Reproduce

herbie shell --seed 2024227 
(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)))