ab-angle->ABCF A

Percentage Accurate: 79.9% → 79.8%

Time: 7.6s

Alternatives: 5

Speedup: N/A×

• 35.7% 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: 79.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \left(b \cdot b\right) \cdot {\sin \left(angle \cdot \mathsf{fma}\left(0.5, \frac{\pi}{angle}, -0.005555555555555556 \cdot \pi\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
  (*
   (* b b)
   (pow
    (sin (* angle (fma 0.5 (/ PI angle) (* -0.005555555555555556 PI))))
    2.0))))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + ((b * b) * pow(sin((angle * fma(0.5, (((double) M_PI) / angle), (-0.005555555555555556 * ((double) M_PI))))), 2.0));
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + Float64(Float64(b * b) * (sin(Float64(angle * fma(0.5, Float64(pi / angle), Float64(-0.005555555555555556 * pi)))) ^ 2.0)))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[Power[N[Sin[N[(angle * N[(0.5 * N[(Pi / angle), $MachinePrecision] + N[(-0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-PI.f64 76.4

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in angle around inf

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

   1. lower-*.f64 N/A

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 76.5

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \mathsf{fma}\left(0.5, \frac{\pi}{angle}, -0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} \]

7. Applied rewrites 76.5%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. unpow-prod-down N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-pow.f64 76.5

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \left(b \cdot b\right) \cdot \color{blue}{{\sin \left(angle \cdot \mathsf{fma}\left(0.5, \frac{\pi}{angle}, -0.005555555555555556 \cdot \pi\right)\right)}^{2}} \]

9. Applied rewrites 76.5%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{\left(b \cdot b\right) \cdot {\sin \left(angle \cdot \mathsf{fma}\left(0.5, \frac{\pi}{angle}, -0.005555555555555556 \cdot \pi\right)\right)}^{2}} \]
10. Add Preprocessing

Alternative 2: 79.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* 0.005555555555555556 (* angle PI)))) 2.0)
  (pow (* b (sin (fma 0.5 PI (* -0.005555555555555556 (* angle PI))))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin((0.005555555555555556 * (angle * ((double) M_PI))))), 2.0) + pow((b * sin(fma(0.5, ((double) M_PI), (-0.005555555555555556 * (angle * ((double) M_PI)))))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0) + (Float64(b * sin(fma(0.5, pi, Float64(-0.005555555555555556 * Float64(angle * pi))))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(0.5 * Pi + N[(-0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-PI.f64 76.4

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

   2. pow-prod-down N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

7. Applied rewrites 76.1%

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

8. Final simplification 76.1%

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

Alternative 3: 79.8% accurate, N/A× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (+ (pow (* (sin t_0) a) 2.0) (pow (* (sin (fma 0.5 PI t_0)) b) 2.0))))

C

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

Julia

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

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(N[Power[N[(N[Sin[t$95$0], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[N[(0.5 * Pi + t$95$0), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.5%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. sin-sum N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 76.5%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 76.1%

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

Alternative 4: 43.5% accurate, N/A× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (*
    (+ (/ (pow (* (sin (fma 0.5 PI t_0)) b) 2.0) (* a a)) (pow (sin t_0) 2.0))
    (* a a))))

C

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

Julia

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

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(N[(N[(N[Power[N[(N[Sin[N[(0.5 * Pi + t$95$0), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.5%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. sin-sum N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 76.5%

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

5. Taylor expanded in a around inf

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

7. Applied rewrites 44.7%

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

Alternative 5: 21.1% accurate, N/A× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (*
    (+
     (/ (pow (* (sin (fma 0.5 PI t_0)) b) 2.0) (exp (* (log a) 2.0)))
     (pow (sin t_0) 2.0))
    (* a a))))

C

double code(double a, double b, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	return ((pow((sin(fma(0.5, ((double) M_PI), t_0)) * b), 2.0) / exp((log(a) * 2.0))) + pow(sin(t_0), 2.0)) * (a * a);
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	return Float64(Float64(Float64((Float64(sin(fma(0.5, pi, t_0)) * b) ^ 2.0) / exp(Float64(log(a) * 2.0))) + (sin(t_0) ^ 2.0)) * Float64(a * a))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(N[(N[(N[Power[N[(N[Sin[N[(0.5 * Pi + t$95$0), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision] / N[Exp[N[(N[Log[a], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.5%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. sin-sum N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 76.5%

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

5. Taylor expanded in a around inf

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

7. Applied rewrites 44.7%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. pow-to-exp N/A

      \[\leadsto \left(\frac{{\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot b\right)}^{2}}{e^{\log a \cdot 2}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right) \cdot \left(a \cdot a\right) \]

   4. lower-exp.f64 N/A

      \[\leadsto \left(\frac{{\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot b\right)}^{2}}{e^{\log a \cdot 2}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right) \cdot \left(a \cdot a\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{{\left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot b\right)}^{2}}{e^{\log a \cdot 2}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right) \cdot \left(a \cdot a\right) \]

   6. lower-log.f64 24.3

      \[\leadsto \left(\frac{{\left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot b\right)}^{2}}{e^{\log a \cdot 2}} + {\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right) \cdot \left(a \cdot a\right) \]

9. Applied rewrites 24.3%

   \[\leadsto \left(\frac{{\left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot b\right)}^{2}}{e^{\log a \cdot 2}} + {\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right) \cdot \left(a \cdot a\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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)))