ab-angle->ABCF A

Percentage Accurate: 79.6% → 79.5%

Time: 15.2s

Alternatives: 10

Speedup: N/A×

• 35.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% 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: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\frac{{\pi}^{0.6666666666666666}}{180} \cdot \frac{\sqrt[3]{\pi}}{\frac{1}{angle}}\right)\right)}^{2} + b \cdot b \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow
   (*
    a
    (sin
     (* (/ (pow PI 0.6666666666666666) 180.0) (/ (cbrt PI) (/ 1.0 angle)))))
   2.0)
  (* b b)))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((pow(((double) M_PI), 0.6666666666666666) / 180.0) * (cbrt(((double) M_PI)) / (1.0 / angle))))), 2.0) + (b * b);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((Math.pow(Math.PI, 0.6666666666666666) / 180.0) * (Math.cbrt(Math.PI) / (1.0 / angle))))), 2.0) + (b * b);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64((pi ^ 0.6666666666666666) / 180.0) * Float64(cbrt(pi) / Float64(1.0 / angle))))) ^ 2.0) + Float64(b * b))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(N[Power[Pi, 0.6666666666666666], $MachinePrecision] / 180.0), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 77.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lift-PI.f64 N/A

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

   7. add-cube-cbrt N/A

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

   8. div-inv N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow2 N/A

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

   13. lift-PI.f64 N/A

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

   14. pow1/3 N/A

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

   15. pow-pow N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-pow.f64 N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f64 N/A

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

   21. lift-PI.f64 N/A

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

   22. lower-cbrt.f64 N/A

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

   23. lower-/.f64 77.3

       \[\leadsto {\left(a \cdot \sin \left(\frac{{\pi}^{0.6666666666666666}}{180} \cdot \frac{\sqrt[3]{\pi}}{\color{blue}{\frac{1}{angle}}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

7. Applied rewrites 77.3%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{{\pi}^{0.6666666666666666}}{180} \cdot \frac{\sqrt[3]{\pi}}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

8. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 77.3

      \[\leadsto {\left(a \cdot \sin \left(\frac{{\pi}^{0.6666666666666666}}{180} \cdot \frac{\sqrt[3]{\pi}}{\frac{1}{angle}}\right)\right)}^{2} + \color{blue}{b \cdot b} \]

10. Applied rewrites 77.3%

    \[\leadsto {\left(a \cdot \sin \left(\frac{{\pi}^{0.6666666666666666}}{180} \cdot \frac{\sqrt[3]{\pi}}{\frac{1}{angle}}\right)\right)}^{2} + \color{blue}{b \cdot b} \]
11. Add Preprocessing

Alternative 2: 77.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;{\left(0.005555555555555556 \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e-10)
   (+
    (pow (* 0.005555555555555556 (* a (* PI angle))) 2.0)
    (pow (* b 1.0) 2.0))
   (fma
    a
    (* a (- 0.5 (* 0.5 (cos (* (* PI angle) 0.011111111111111112)))))
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 5e-10) {
		tmp = pow((0.005555555555555556 * (a * (((double) M_PI) * angle))), 2.0) + pow((b * 1.0), 2.0);
	} else {
		tmp = fma(a, (a * (0.5 - (0.5 * cos(((((double) M_PI) * angle) * 0.011111111111111112))))), (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5e-10)
		tmp = Float64((Float64(0.005555555555555556 * Float64(a * Float64(pi * angle))) ^ 2.0) + (Float64(b * 1.0) ^ 2.0));
	else
		tmp = fma(a, Float64(a * Float64(0.5 - Float64(0.5 * cos(Float64(Float64(pi * angle) * 0.011111111111111112))))), Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e-10], N[(N[Power[N[(0.005555555555555556 * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(0.5 - N[(0.5 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000031e-10

   1. Initial program 86.6%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 83.4

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

   8. Applied rewrites 83.4%

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

   if 5.00000000000000031e-10 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 58.9%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 58.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-PI.f64 N/A

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

      7. add-cube-cbrt N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 N/A

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

      22. lower-cbrt.f64 N/A

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

      23. lower-/.f64 58.6

          \[\leadsto {\left(a \cdot \sin \left(\frac{{\pi}^{0.6666666666666666}}{180} \cdot \frac{\sqrt[3]{\pi}}{\color{blue}{\frac{1}{angle}}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Applied rewrites 58.6%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{{\pi}^{0.6666666666666666}}{180} \cdot \frac{\sqrt[3]{\pi}}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.6

         \[\leadsto {\left(a \cdot \sin \left(\frac{{\pi}^{0.6666666666666666}}{180} \cdot \frac{\sqrt[3]{\pi}}{\frac{1}{angle}}\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   10. Applied rewrites 58.6%

       \[\leadsto {\left(a \cdot \sin \left(\frac{{\pi}^{0.6666666666666666}}{180} \cdot \frac{\sqrt[3]{\pi}}{\frac{1}{angle}}\right)\right)}^{2} + \color{blue}{b \cdot b} \]

   12. Applied rewrites 58.4%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;{\left(0.005555555555555556 \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024229 
(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)))