ab-angle->ABCF B

Percentage Accurate: 54.1% → 61.2%

Time: 6.4s

Alternatives: 13

Speedup: 1.4×

• 32.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 61.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ t_1 := \sqrt{0.005555555555555556} \cdot \sqrt{\left|angle\right|}\\ \mathbf{if}\;t\_0 \leq 10^{-311}:\\ \;\;\;\;\left(\left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right) \cdot a\right) \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left|angle\right|\right)\right) \cdot -2\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\mathsf{fma}\left(t\_1 \cdot t\_1, \pi, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\frac{1}{a} + \frac{-a}{b \cdot b}\right) \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot 2\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a 2.0))))
        (t_1 (* (sqrt 0.005555555555555556) (sqrt (fabs angle)))))
   (if (<= t_0 1e-311)
     (*
      (*
       (* (* (sin (* (* PI angle) 0.005555555555555556)) a) a)
       (cos (* (* PI 0.005555555555555556) (fabs angle))))
      -2.0)
     (if (<= t_0 INFINITY)
       (*
        (* (* 2.0 (* b b)) (sin (* PI (/ angle 180.0))))
        (sin (fma (* t_1 t_1) PI (/ PI 2.0))))
       (*
        (*
         (* (* (* (+ (/ 1.0 a) (/ (- a) (* b b))) (* b b)) a) 2.0)
         (sin (* (* 0.005555555555555556 angle) PI)))
        1.0)))))

C

double code(double a, double b, double angle) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double t_1 = sqrt(0.005555555555555556) * sqrt(fabs(angle));
	double tmp;
	if (t_0 <= 1e-311) {
		tmp = (((sin(((((double) M_PI) * angle) * 0.005555555555555556)) * a) * a) * cos(((((double) M_PI) * 0.005555555555555556) * fabs(angle)))) * -2.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((2.0 * (b * b)) * sin((((double) M_PI) * (angle / 180.0)))) * sin(fma((t_1 * t_1), ((double) M_PI), (((double) M_PI) / 2.0)));
	} else {
		tmp = ((((((1.0 / a) + (-a / (b * b))) * (b * b)) * a) * 2.0) * sin(((0.005555555555555556 * angle) * ((double) M_PI)))) * 1.0;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	t_1 = Float64(sqrt(0.005555555555555556) * sqrt(abs(angle)))
	tmp = 0.0
	if (t_0 <= 1e-311)
		tmp = Float64(Float64(Float64(Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * a) * a) * cos(Float64(Float64(pi * 0.005555555555555556) * abs(angle)))) * -2.0);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(2.0 * Float64(b * b)) * sin(Float64(pi * Float64(angle / 180.0)))) * sin(fma(Float64(t_1 * t_1), pi, Float64(pi / 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 / a) + Float64(Float64(-a) / Float64(b * b))) * Float64(b * b)) * a) * 2.0) * sin(Float64(Float64(0.005555555555555556 * angle) * pi))) * 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[0.005555555555555556], $MachinePrecision] * N[Sqrt[N[Abs[angle], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-311], N[(N[(N[(N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * N[Cos[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(t$95$1 * t$95$1), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(1.0 / a), $MachinePrecision] + N[((-a) / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 9.99999999999948e-312

   1. Initial program 54.1%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. fabs-mul N/A

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

      9. add-exp-log N/A

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

      10. exp-fabs N/A

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

      11. add-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-PI.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 54.3

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

   3. Applied rewrites 54.3%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 26.7

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

   5. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. fabs-mul N/A

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

      7. metadata-eval N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-fabs.f64 26.7

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

   7. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. fabs-mul N/A

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

      7. metadata-eval N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-fabs.f64 54.1

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

   9. Applied rewrites 54.1%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 40.7%

       \[\leadsto \color{blue}{\left(\left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right) \cdot a\right) \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left|angle\right|\right)\right) \cdot -2} \]

   if 9.99999999999948e-312 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < +inf.0

   1. Initial program 54.1%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. fabs-mul N/A

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

      9. add-exp-log N/A

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

      10. exp-fabs N/A

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

      11. add-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-PI.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 54.3

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

   3. Applied rewrites 54.3%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 26.7

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

   5. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. fabs-mul N/A

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

      7. metadata-eval N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-fabs.f64 26.7

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

   7. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. fabs-mul N/A

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

      7. metadata-eval N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-fabs.f64 54.1

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

   9. Applied rewrites 54.1%

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

   10. Taylor expanded in a around 0

       \[\leadsto \left(\left(2 \cdot \color{blue}{{b}^{2}}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\left(\sqrt{\frac{1}{180}} \cdot \sqrt{\left|angle\right|}\right) \cdot \left(\sqrt{\frac{1}{180}} \cdot \sqrt{\left|angle\right|}\right), \pi, \frac{\pi}{2}\right)\right) \]
   11. Step-by-step derivation

       1. pow2 N/A

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

       2. lower-*.f64 36.7

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

   12. Applied rewrites 36.7%

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

   if +inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-flip N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 47.2

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 46.3%

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

   9. Applied rewrites 49.9%

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

   10. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. associate-*r/ N/A

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

       7. lower-/.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. pow2 N/A

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

       11. lower-*.f64 N/A

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

       12. pow2 N/A

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

       13. lower-*.f64 36.1

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

   12. Applied rewrites 36.1%

       \[\leadsto \left(\left(\left(\left(\left(\frac{1}{a} + \frac{-a}{b \cdot b}\right) \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot 2\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 61.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\\ t_1 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ t_2 := \sqrt{0.005555555555555556 \cdot angle}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-168}:\\ \;\;\;\;\left(\left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right) \cdot a\right) \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left|angle\right|\right)\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\left(t\_0 \cdot \left(b \cdot b\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(t\_2 \cdot t\_2, \pi, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\frac{1}{a} + \frac{-a}{b \cdot b}\right) \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot 2\right) \cdot t\_0\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sin (* (* 0.005555555555555556 angle) PI)))
        (t_1 (* 2.0 (- (pow b 2.0) (pow a 2.0))))
        (t_2 (sqrt (* 0.005555555555555556 angle))))
   (if (<= t_1 2e-168)
     (*
      (*
       (* (* (sin (* (* PI angle) 0.005555555555555556)) a) a)
       (cos (* (* PI 0.005555555555555556) (fabs angle))))
      -2.0)
     (if (<= t_1 INFINITY)
       (* (* (* t_0 (* b b)) 2.0) (sin (fma (* t_2 t_2) PI (/ PI 2.0))))
       (*
        (* (* (* (* (+ (/ 1.0 a) (/ (- a) (* b b))) (* b b)) a) 2.0) t_0)
        1.0)))))

C

double code(double a, double b, double angle) {
	double t_0 = sin(((0.005555555555555556 * angle) * ((double) M_PI)));
	double t_1 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double t_2 = sqrt((0.005555555555555556 * angle));
	double tmp;
	if (t_1 <= 2e-168) {
		tmp = (((sin(((((double) M_PI) * angle) * 0.005555555555555556)) * a) * a) * cos(((((double) M_PI) * 0.005555555555555556) * fabs(angle)))) * -2.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * (b * b)) * 2.0) * sin(fma((t_2 * t_2), ((double) M_PI), (((double) M_PI) / 2.0)));
	} else {
		tmp = ((((((1.0 / a) + (-a / (b * b))) * (b * b)) * a) * 2.0) * t_0) * 1.0;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = sin(Float64(Float64(0.005555555555555556 * angle) * pi))
	t_1 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	t_2 = sqrt(Float64(0.005555555555555556 * angle))
	tmp = 0.0
	if (t_1 <= 2e-168)
		tmp = Float64(Float64(Float64(Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * a) * a) * cos(Float64(Float64(pi * 0.005555555555555556) * abs(angle)))) * -2.0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * Float64(b * b)) * 2.0) * sin(fma(Float64(t_2 * t_2), pi, Float64(pi / 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 / a) + Float64(Float64(-a) / Float64(b * b))) * Float64(b * b)) * a) * 2.0) * t_0) * 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(0.005555555555555556 * angle), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 2e-168], N[(N[(N[(N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * N[Cos[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 * N[(b * b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[(N[(t$95$2 * t$95$2), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(1.0 / a), $MachinePrecision] + N[((-a) / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 2.0000000000000001e-168

   1. Initial program 54.1%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. fabs-mul N/A

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

      9. add-exp-log N/A

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

      10. exp-fabs N/A

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

      11. add-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-PI.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 54.3

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

   3. Applied rewrites 54.3%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 26.7

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

   5. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. fabs-mul N/A

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

      7. metadata-eval N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-fabs.f64 26.7

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

   7. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. fabs-mul N/A

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

      7. metadata-eval N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-fabs.f64 54.1

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

   9. Applied rewrites 54.1%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 40.7%

       \[\leadsto \color{blue}{\left(\left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right) \cdot a\right) \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left|angle\right|\right)\right) \cdot -2} \]

   if 2.0000000000000001e-168 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < +inf.0

   1. Initial program 54.1%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. fabs-mul N/A

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

      9. add-exp-log N/A

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

      10. exp-fabs N/A

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

      11. add-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-PI.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 54.3

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

   3. Applied rewrites 54.3%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 26.7

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

   5. Applied rewrites 26.7%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. rem-square-sqrt N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 17.9

          \[\leadsto \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(b \cdot b\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\sqrt{0.005555555555555556 \cdot angle} \cdot \sqrt{0.005555555555555556 \cdot angle}, \pi, \frac{\pi}{2}\right)\right) \]

   8. Applied rewrites 17.9%

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

   if +inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-flip N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 47.2

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 46.3%

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

   9. Applied rewrites 49.9%

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

   10. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. associate-*r/ N/A

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

       7. lower-/.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. pow2 N/A

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

       11. lower-*.f64 N/A

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

       12. pow2 N/A

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

       13. lower-*.f64 36.1

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

   12. Applied rewrites 36.1%

       \[\leadsto \left(\left(\left(\left(\left(\frac{1}{a} + \frac{-a}{b \cdot b}\right) \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot 2\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 61.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.005555555555555556 \cdot angle}\\ \mathbf{if}\;a \leq 3.3 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(2 \cdot \left(\frac{1}{\frac{1}{b \cdot b}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\mathsf{fma}\left(t\_0 \cdot t\_0, \pi, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sqrt (* 0.005555555555555556 angle))))
   (if (<= a 3.3e+153)
     (*
      (*
       (* 2.0 (- (/ 1.0 (/ 1.0 (* b b))) (pow a 2.0)))
       (sin (* PI (/ angle 180.0))))
      (sin (fma (* t_0 t_0) PI (/ PI 2.0))))
     (* (* -0.011111111111111112 a) (* a (* PI angle))))))

C

double code(double a, double b, double angle) {
	double t_0 = sqrt((0.005555555555555556 * angle));
	double tmp;
	if (a <= 3.3e+153) {
		tmp = ((2.0 * ((1.0 / (1.0 / (b * b))) - pow(a, 2.0))) * sin((((double) M_PI) * (angle / 180.0)))) * sin(fma((t_0 * t_0), ((double) M_PI), (((double) M_PI) / 2.0)));
	} else {
		tmp = (-0.011111111111111112 * a) * (a * (((double) M_PI) * angle));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = sqrt(Float64(0.005555555555555556 * angle))
	tmp = 0.0
	if (a <= 3.3e+153)
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(1.0 / Float64(1.0 / Float64(b * b))) - (a ^ 2.0))) * sin(Float64(pi * Float64(angle / 180.0)))) * sin(fma(Float64(t_0 * t_0), pi, Float64(pi / 2.0))));
	else
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(a * Float64(pi * angle)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Sqrt[N[(0.005555555555555556 * angle), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, 3.3e+153], N[(N[(N[(2.0 * N[(N[(1.0 / N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(t$95$0 * t$95$0), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.29999999999999994e153

   1. Initial program 54.1%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. fabs-mul N/A

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

      9. add-exp-log N/A

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

      10. exp-fabs N/A

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

      11. add-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-PI.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 54.3

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

   3. Applied rewrites 54.3%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 26.7

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

   5. Applied rewrites 26.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-neg N/A

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

      4. metadata-eval N/A

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

      5. pow-flip N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 26.7

         \[\leadsto \left(\left(2 \cdot \left(\frac{1}{\frac{1}{\color{blue}{b \cdot b}}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\sqrt{0.005555555555555556 \cdot angle} \cdot \sqrt{0.005555555555555556 \cdot angle}, \pi, \frac{\pi}{2}\right)\right) \]

   7. Applied rewrites 26.7%

      \[\leadsto \left(\left(2 \cdot \left(\color{blue}{\frac{1}{\frac{1}{b \cdot b}}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\sqrt{0.005555555555555556 \cdot angle} \cdot \sqrt{0.005555555555555556 \cdot angle}, \pi, \frac{\pi}{2}\right)\right) \]

   if 3.29999999999999994e153 < a

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 50.9

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 34.6

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 34.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 34.6

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 34.6%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       13. lift-PI.f64 37.8

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right) \]

   11. Applied rewrites 37.8%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 60.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 10^{-291}:\\ \;\;\;\;\left(\left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right) \cdot a\right) \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left|angle\right|\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 1e-291)
   (*
    (*
     (* (* (sin (* (* PI angle) 0.005555555555555556)) a) a)
     (cos (* (* PI 0.005555555555555556) (fabs angle))))
    -2.0)
   (*
    (* (* 2.0 (* b b)) (sin (* PI (/ angle 180.0))))
    (cos (* (* 0.005555555555555556 angle) PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 1e-291) {
		tmp = (((sin(((((double) M_PI) * angle) * 0.005555555555555556)) * a) * a) * cos(((((double) M_PI) * 0.005555555555555556) * fabs(angle)))) * -2.0;
	} else {
		tmp = ((2.0 * (b * b)) * sin((((double) M_PI) * (angle / 180.0)))) * cos(((0.005555555555555556 * angle) * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= 1e-291) {
		tmp = (((Math.sin(((Math.PI * angle) * 0.005555555555555556)) * a) * a) * Math.cos(((Math.PI * 0.005555555555555556) * Math.abs(angle)))) * -2.0;
	} else {
		tmp = ((2.0 * (b * b)) * Math.sin((Math.PI * (angle / 180.0)))) * Math.cos(((0.005555555555555556 * angle) * Math.PI));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= 1e-291:
		tmp = (((math.sin(((math.pi * angle) * 0.005555555555555556)) * a) * a) * math.cos(((math.pi * 0.005555555555555556) * math.fabs(angle)))) * -2.0
	else:
		tmp = ((2.0 * (b * b)) * math.sin((math.pi * (angle / 180.0)))) * math.cos(((0.005555555555555556 * angle) * math.pi))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 1e-291)
		tmp = Float64(Float64(Float64(Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * a) * a) * cos(Float64(Float64(pi * 0.005555555555555556) * abs(angle)))) * -2.0);
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(b * b)) * sin(Float64(pi * Float64(angle / 180.0)))) * cos(Float64(Float64(0.005555555555555556 * angle) * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= 1e-291)
		tmp = (((sin(((pi * angle) * 0.005555555555555556)) * a) * a) * cos(((pi * 0.005555555555555556) * abs(angle)))) * -2.0;
	else
		tmp = ((2.0 * (b * b)) * sin((pi * (angle / 180.0)))) * cos(((0.005555555555555556 * angle) * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-291], N[(N[(N[(N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * N[Cos[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 9.99999999999999962e-292

   1. Initial program 54.1%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. fabs-mul N/A

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

      9. add-exp-log N/A

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

      10. exp-fabs N/A

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

      11. add-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-PI.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 54.3

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

   3. Applied rewrites 54.3%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 26.7

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

   5. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. fabs-mul N/A

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

      7. metadata-eval N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-fabs.f64 26.7

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

   7. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. fabs-mul N/A

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

      7. metadata-eval N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-fabs.f64 54.1

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

   9. Applied rewrites 54.1%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 40.7%

       \[\leadsto \color{blue}{\left(\left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right) \cdot a\right) \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left|angle\right|\right)\right) \cdot -2} \]

   if 9.99999999999999962e-292 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. fabs-mul N/A

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

      9. add-exp-log N/A

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

      10. exp-fabs N/A

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

      11. add-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-PI.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 54.3

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

   3. Applied rewrites 54.3%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 26.7

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

   5. Applied rewrites 26.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. sin-+PI/2 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 54.0%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lower-*.f64 36.5

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

   10. Applied rewrites 36.5%

       \[\leadsto \left(\left(2 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 60.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 10^{-291}:\\ \;\;\;\;\cos t\_0 \cdot \left(\left(\left(\sin t\_0 \cdot a\right) \cdot a\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 1e-291)
     (* (cos t_0) (* (* (* (sin t_0) a) a) -2.0))
     (*
      (* (* 2.0 (* b b)) (sin (* PI (/ angle 180.0))))
      (cos (* (* 0.005555555555555556 angle) PI))))))

C

double code(double a, double b, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 1e-291) {
		tmp = cos(t_0) * (((sin(t_0) * a) * a) * -2.0);
	} else {
		tmp = ((2.0 * (b * b)) * sin((((double) M_PI) * (angle / 180.0)))) * cos(((0.005555555555555556 * angle) * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= 1e-291) {
		tmp = Math.cos(t_0) * (((Math.sin(t_0) * a) * a) * -2.0);
	} else {
		tmp = ((2.0 * (b * b)) * Math.sin((Math.PI * (angle / 180.0)))) * Math.cos(((0.005555555555555556 * angle) * Math.PI));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (math.pi * angle) * 0.005555555555555556
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= 1e-291:
		tmp = math.cos(t_0) * (((math.sin(t_0) * a) * a) * -2.0)
	else:
		tmp = ((2.0 * (b * b)) * math.sin((math.pi * (angle / 180.0)))) * math.cos(((0.005555555555555556 * angle) * math.pi))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 1e-291)
		tmp = Float64(cos(t_0) * Float64(Float64(Float64(sin(t_0) * a) * a) * -2.0));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(b * b)) * sin(Float64(pi * Float64(angle / 180.0)))) * cos(Float64(Float64(0.005555555555555556 * angle) * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (pi * angle) * 0.005555555555555556;
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= 1e-291)
		tmp = cos(t_0) * (((sin(t_0) * a) * a) * -2.0);
	else
		tmp = ((2.0 * (b * b)) * sin((pi * (angle / 180.0)))) * cos(((0.005555555555555556 * angle) * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-291], N[(N[Cos[t$95$0], $MachinePrecision] * N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 9.99999999999999962e-292

   1. Initial program 54.1%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 40.9%

      \[\leadsto \color{blue}{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right) \cdot a\right) \cdot -2\right)} \]

   if 9.99999999999999962e-292 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. fabs-mul N/A

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

      9. add-exp-log N/A

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

      10. exp-fabs N/A

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

      11. add-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-PI.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 54.3

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

   3. Applied rewrites 54.3%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 26.7

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

   5. Applied rewrites 26.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. sin-+PI/2 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 54.0%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lower-*.f64 36.5

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

   10. Applied rewrites 36.5%

       \[\leadsto \left(\left(2 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 60.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -4 \cdot 10^{-146}:\\ \;\;\;\;\cos t\_0 \cdot \left(\left(\left(\sin t\_0 \cdot a\right) \cdot a\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(b \cdot b\right)\right) \cdot 2\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -4e-146)
     (* (cos t_0) (* (* (* (sin t_0) a) a) -2.0))
     (*
      (* (* (sin (* PI (* 0.005555555555555556 angle))) (* b b)) 2.0)
      (cos (* PI (/ angle 180.0)))))))

C

double code(double a, double b, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= -4e-146) {
		tmp = cos(t_0) * (((sin(t_0) * a) * a) * -2.0);
	} else {
		tmp = ((sin((((double) M_PI) * (0.005555555555555556 * angle))) * (b * b)) * 2.0) * cos((((double) M_PI) * (angle / 180.0)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= -4e-146) {
		tmp = Math.cos(t_0) * (((Math.sin(t_0) * a) * a) * -2.0);
	} else {
		tmp = ((Math.sin((Math.PI * (0.005555555555555556 * angle))) * (b * b)) * 2.0) * Math.cos((Math.PI * (angle / 180.0)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (math.pi * angle) * 0.005555555555555556
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= -4e-146:
		tmp = math.cos(t_0) * (((math.sin(t_0) * a) * a) * -2.0)
	else:
		tmp = ((math.sin((math.pi * (0.005555555555555556 * angle))) * (b * b)) * 2.0) * math.cos((math.pi * (angle / 180.0)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= -4e-146)
		tmp = Float64(cos(t_0) * Float64(Float64(Float64(sin(t_0) * a) * a) * -2.0));
	else
		tmp = Float64(Float64(Float64(sin(Float64(pi * Float64(0.005555555555555556 * angle))) * Float64(b * b)) * 2.0) * cos(Float64(pi * Float64(angle / 180.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (pi * angle) * 0.005555555555555556;
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= -4e-146)
		tmp = cos(t_0) * (((sin(t_0) * a) * a) * -2.0);
	else
		tmp = ((sin((pi * (0.005555555555555556 * angle))) * (b * b)) * 2.0) * cos((pi * (angle / 180.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-146], N[(N[Cos[t$95$0], $MachinePrecision] * N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.0000000000000001e-146

   1. Initial program 54.1%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 40.9%

      \[\leadsto \color{blue}{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right) \cdot a\right) \cdot -2\right)} \]

   if -4.0000000000000001e-146 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(b \cdot b\right)\right) \cdot 2\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := \sin t\_0\\ \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -4 \cdot 10^{-146}:\\ \;\;\;\;\cos t\_0 \cdot \left(\left(\left(t\_1 \cdot a\right) \cdot a\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 \cdot \left(b \cdot b\right)\right) \cdot 2\right) \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)) (t_1 (sin t_0)))
   (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -4e-146)
     (* (cos t_0) (* (* (* t_1 a) a) -2.0))
     (* (* (* t_1 (* b b)) 2.0) (cos (* (* 0.005555555555555556 angle) PI))))))

C

double code(double a, double b, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = sin(t_0);
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= -4e-146) {
		tmp = cos(t_0) * (((t_1 * a) * a) * -2.0);
	} else {
		tmp = ((t_1 * (b * b)) * 2.0) * cos(((0.005555555555555556 * angle) * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (Math.PI * angle) * 0.005555555555555556;
	double t_1 = Math.sin(t_0);
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= -4e-146) {
		tmp = Math.cos(t_0) * (((t_1 * a) * a) * -2.0);
	} else {
		tmp = ((t_1 * (b * b)) * 2.0) * Math.cos(((0.005555555555555556 * angle) * Math.PI));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (math.pi * angle) * 0.005555555555555556
	t_1 = math.sin(t_0)
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= -4e-146:
		tmp = math.cos(t_0) * (((t_1 * a) * a) * -2.0)
	else:
		tmp = ((t_1 * (b * b)) * 2.0) * math.cos(((0.005555555555555556 * angle) * math.pi))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = sin(t_0)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= -4e-146)
		tmp = Float64(cos(t_0) * Float64(Float64(Float64(t_1 * a) * a) * -2.0));
	else
		tmp = Float64(Float64(Float64(t_1 * Float64(b * b)) * 2.0) * cos(Float64(Float64(0.005555555555555556 * angle) * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (pi * angle) * 0.005555555555555556;
	t_1 = sin(t_0);
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= -4e-146)
		tmp = cos(t_0) * (((t_1 * a) * a) * -2.0);
	else
		tmp = ((t_1 * (b * b)) * 2.0) * cos(((0.005555555555555556 * angle) * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-146], N[(N[Cos[t$95$0], $MachinePrecision] * N[(N[(N[(t$95$1 * a), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * N[(b * b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.0000000000000001e-146

   1. Initial program 54.1%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 40.9%

      \[\leadsto \color{blue}{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right) \cdot a\right) \cdot -2\right)} \]

   if -4.0000000000000001e-146 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. fabs-mul N/A

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

      9. add-exp-log N/A

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

      10. exp-fabs N/A

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

      11. add-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-PI.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 54.3

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

   3. Applied rewrites 54.3%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 26.7

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

   5. Applied rewrites 26.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. sin-+PI/2 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 54.0%

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

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 36.6

          \[\leadsto \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(b \cdot b\right)\right) \cdot 2\right) \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \]

   10. Applied rewrites 36.6%

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

Alternative 8: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_1 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 10^{-291}:\\ \;\;\;\;\cos t\_1 \cdot \left(\left(\left(\sin t\_1 \cdot a\right) \cdot a\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\sin t\_0 \cdot \cos t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle)))
        (t_1 (* (* PI angle) 0.005555555555555556)))
   (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 1e-291)
     (* (cos t_1) (* (* (* (sin t_1) a) a) -2.0))
     (* (* (* b b) 2.0) (* (sin t_0) (cos t_0))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_1 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 1e-291) {
		tmp = cos(t_1) * (((sin(t_1) * a) * a) * -2.0);
	} else {
		tmp = ((b * b) * 2.0) * (sin(t_0) * cos(t_0));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (0.005555555555555556 * angle);
	double t_1 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= 1e-291) {
		tmp = Math.cos(t_1) * (((Math.sin(t_1) * a) * a) * -2.0);
	} else {
		tmp = ((b * b) * 2.0) * (Math.sin(t_0) * Math.cos(t_0));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (0.005555555555555556 * angle)
	t_1 = (math.pi * angle) * 0.005555555555555556
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= 1e-291:
		tmp = math.cos(t_1) * (((math.sin(t_1) * a) * a) * -2.0)
	else:
		tmp = ((b * b) * 2.0) * (math.sin(t_0) * math.cos(t_0))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_1 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 1e-291)
		tmp = Float64(cos(t_1) * Float64(Float64(Float64(sin(t_1) * a) * a) * -2.0));
	else
		tmp = Float64(Float64(Float64(b * b) * 2.0) * Float64(sin(t_0) * cos(t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (0.005555555555555556 * angle);
	t_1 = (pi * angle) * 0.005555555555555556;
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= 1e-291)
		tmp = cos(t_1) * (((sin(t_1) * a) * a) * -2.0);
	else
		tmp = ((b * b) * 2.0) * (sin(t_0) * cos(t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-291], N[(N[Cos[t$95$1], $MachinePrecision] * N[(N[(N[(N[Sin[t$95$1], $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 9.99999999999999962e-292

   1. Initial program 54.1%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 40.9%

      \[\leadsto \color{blue}{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right) \cdot a\right) \cdot -2\right)} \]

   if 9.99999999999999962e-292 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. mult-flip N/A

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

      12. lift-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-/.f64 N/A

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

   4. Applied rewrites 36.7%

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

Alternative 9: 56.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.6 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.6e+153)
   (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) 1.0)
   (* (* -0.011111111111111112 a) (* a (* PI angle)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.6e+153) {
		tmp = ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin((((double) M_PI) * (angle / 180.0)))) * 1.0;
	} else {
		tmp = (-0.011111111111111112 * a) * (a * (((double) M_PI) * angle));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.6e+153) {
		tmp = ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin((Math.PI * (angle / 180.0)))) * 1.0;
	} else {
		tmp = (-0.011111111111111112 * a) * (a * (Math.PI * angle));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2.6e+153:
		tmp = ((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin((math.pi * (angle / 180.0)))) * 1.0
	else:
		tmp = (-0.011111111111111112 * a) * (a * (math.pi * angle))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.6e+153)
		tmp = Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(Float64(pi * Float64(angle / 180.0)))) * 1.0);
	else
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(a * Float64(pi * angle)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.6e+153)
		tmp = ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin((pi * (angle / 180.0)))) * 1.0;
	else
		tmp = (-0.011111111111111112 * a) * (a * (pi * angle));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.6e+153], N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.5999999999999999e153

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 53.0%

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

   if 2.5999999999999999e153 < a

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 50.9

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 34.6

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 34.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 34.6

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 34.6%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       13. lift-PI.f64 37.8

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right) \]

   11. Applied rewrites 37.8%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 56.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ t_1 := \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a 2.0))))
        (t_1 (* (* -0.011111111111111112 a) (* a (* PI angle)))))
   (if (<= t_0 -2e-139)
     t_1
     (if (<= t_0 INFINITY)
       (* (* 0.011111111111111112 angle) (* PI (* b b)))
       t_1))))

C

double code(double a, double b, double angle) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double t_1 = (-0.011111111111111112 * a) * (a * (((double) M_PI) * angle));
	double tmp;
	if (t_0 <= -2e-139) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (0.011111111111111112 * angle) * (((double) M_PI) * (b * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	double t_1 = (-0.011111111111111112 * a) * (a * (Math.PI * angle));
	double tmp;
	if (t_0 <= -2e-139) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (0.011111111111111112 * angle) * (Math.PI * (b * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = 2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))
	t_1 = (-0.011111111111111112 * a) * (a * (math.pi * angle))
	tmp = 0
	if t_0 <= -2e-139:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = (0.011111111111111112 * angle) * (math.pi * (b * b))
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	t_1 = Float64(Float64(-0.011111111111111112 * a) * Float64(a * Float64(pi * angle)))
	tmp = 0.0
	if (t_0 <= -2e-139)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(0.011111111111111112 * angle) * Float64(pi * Float64(b * b)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = 2.0 * ((b ^ 2.0) - (a ^ 2.0));
	t_1 = (-0.011111111111111112 * a) * (a * (pi * angle));
	tmp = 0.0;
	if (t_0 <= -2e-139)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = (0.011111111111111112 * angle) * (pi * (b * b));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-139], t$95$1, If[LessEqual[t$95$0, Infinity], N[(N[(0.011111111111111112 * angle), $MachinePrecision] * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -2.00000000000000006e-139 or +inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 50.9

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 34.6

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 34.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 34.6

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 34.6%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       13. lift-PI.f64 37.8

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right) \]

   11. Applied rewrites 37.8%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \]

   if -2.00000000000000006e-139 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < +inf.0

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 50.9

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot {b}^{2}\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right) \]

      5. lift-*.f64 36.2

         \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right) \]

   7. Applied rewrites 36.2%

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

Alternative 11: 49.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.4 \cdot 10^{+140}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 5.4e+140)
   (* (* (* 0.011111111111111112 angle) PI) (* (+ b a) (- b a)))
   (* (* -0.011111111111111112 a) (* a (* PI angle)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.4e+140) {
		tmp = ((0.011111111111111112 * angle) * ((double) M_PI)) * ((b + a) * (b - a));
	} else {
		tmp = (-0.011111111111111112 * a) * (a * (((double) M_PI) * angle));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.4e+140) {
		tmp = ((0.011111111111111112 * angle) * Math.PI) * ((b + a) * (b - a));
	} else {
		tmp = (-0.011111111111111112 * a) * (a * (Math.PI * angle));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 5.4e+140:
		tmp = ((0.011111111111111112 * angle) * math.pi) * ((b + a) * (b - a))
	else:
		tmp = (-0.011111111111111112 * a) * (a * (math.pi * angle))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 5.4e+140)
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle) * pi) * Float64(Float64(b + a) * Float64(b - a)));
	else
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(a * Float64(pi * angle)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 5.4e+140)
		tmp = ((0.011111111111111112 * angle) * pi) * ((b + a) * (b - a));
	else
		tmp = (-0.011111111111111112 * a) * (a * (pi * angle));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 5.4e+140], N[(N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.40000000000000036e140

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 50.9

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\pi} \cdot \left(b \cdot b - a \cdot a\right)\right) \]

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left({b}^{\color{blue}{2}} - {a}^{2}\right) \]

      17. pow2 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right) \]

      18. pow2 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right) \]

      19. difference-of-squares N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   6. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   if 5.40000000000000036e140 < a

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 50.9

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 34.6

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 34.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 34.6

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 34.6%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       13. lift-PI.f64 37.8

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right) \]

   11. Applied rewrites 37.8%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 37.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ t_1 := \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a 2.0))))
        (t_1 (* (* -0.011111111111111112 a) (* a (* PI angle)))))
   (if (<= t_0 -2e-139)
     t_1
     (if (<= t_0 INFINITY)
       (* (* (* PI (* b b)) angle) 0.011111111111111112)
       t_1))))

C

double code(double a, double b, double angle) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double t_1 = (-0.011111111111111112 * a) * (a * (((double) M_PI) * angle));
	double tmp;
	if (t_0 <= -2e-139) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((((double) M_PI) * (b * b)) * angle) * 0.011111111111111112;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	double t_1 = (-0.011111111111111112 * a) * (a * (Math.PI * angle));
	double tmp;
	if (t_0 <= -2e-139) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = ((Math.PI * (b * b)) * angle) * 0.011111111111111112;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = 2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))
	t_1 = (-0.011111111111111112 * a) * (a * (math.pi * angle))
	tmp = 0
	if t_0 <= -2e-139:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = ((math.pi * (b * b)) * angle) * 0.011111111111111112
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	t_1 = Float64(Float64(-0.011111111111111112 * a) * Float64(a * Float64(pi * angle)))
	tmp = 0.0
	if (t_0 <= -2e-139)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(pi * Float64(b * b)) * angle) * 0.011111111111111112);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = 2.0 * ((b ^ 2.0) - (a ^ 2.0));
	t_1 = (-0.011111111111111112 * a) * (a * (pi * angle));
	tmp = 0.0;
	if (t_0 <= -2e-139)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = ((pi * (b * b)) * angle) * 0.011111111111111112;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-139], t$95$1, If[LessEqual[t$95$0, Infinity], N[(N[(N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -2.00000000000000006e-139 or +inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 50.9

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 34.6

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 34.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 34.6

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 34.6%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       13. lift-PI.f64 37.8

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right) \]

   11. Applied rewrites 37.8%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \]

   if -2.00000000000000006e-139 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < +inf.0

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 50.9

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      8. pow2 N/A

         \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      9. lift-*.f64 36.2

         \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112 \]

   7. Applied rewrites 36.2%

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

Alternative 13: 32.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* (* -0.011111111111111112 a) (* a (* PI angle))))

C

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

Java

public static double code(double a, double b, double angle) {
	return (-0.011111111111111112 * a) * (a * (Math.PI * angle));
}

Python

def code(a, b, angle):
	return (-0.011111111111111112 * a) * (a * (math.pi * angle))

Julia

function code(a, b, angle)
	return Float64(Float64(-0.011111111111111112 * a) * Float64(a * Float64(pi * angle)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (-0.011111111111111112 * a) * (a * (pi * angle));
end

Wolfram

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

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in angle around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

   6. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   10. lower-*.f64 50.9

       \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

5. Taylor expanded in a around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   4. pow2 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

   8. lift-PI.f64 34.6

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

7. Applied rewrites 34.6%

   \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   5. lower-*.f64 34.6

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

9. Applied rewrites 34.6%

   \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

    2. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

    4. lift-PI.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

    5. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

    6. associate-*l* N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

    7. lower-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

    8. lift-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

    10. lower-*.f64 N/A

        \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

    11. *-commutative N/A

        \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

    12. lift-*.f64 N/A

        \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

    13. lift-PI.f64 37.8

        \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right) \]

11. Applied rewrites 37.8%

    \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))