ab-angle->ABCF B

Percentage Accurate: 54.4% → 67.7%

Time: 9.6s

Alternatives: 16

Speedup: 5.2×

• 33.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 54.4% accurate, 1.0× speedup

Math

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

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: 67.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot \left|angle\right|\right) \cdot \pi\\ t_1 := \sin t\_0 \cdot \left(\left|a\right| + b\right)\\ t_2 := b - \left|a\right|\\ t_3 := \pi \cdot \left|angle\right|\\ t_4 := t\_1 \cdot t\_2\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|a\right| \leq 2.5 \cdot 10^{+25}:\\ \;\;\;\;\left(\left(t\_2 \cdot \sin \left(\mathsf{fma}\left(t\_3, 0.005555555555555556, 0.5 \cdot \pi\right)\right)\right) \cdot 2\right) \cdot t\_1\\ \mathbf{elif}\;\left|a\right| \leq 2.02 \cdot 10^{+192}:\\ \;\;\;\;t\_4 \cdot \left(\left(-\sin \left(-0.5 \cdot \pi - \left|\left|angle\right|\right| \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(-0.005555555555555556, \left|angle\right| \cdot \pi, t\_0\right)}{2}\right) \cdot \sin \left(\frac{\mathsf{fma}\left(t\_3, -0.011111111111111112, \pi\right)}{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 (fabs angle)) PI))
        (t_1 (* (sin t_0) (+ (fabs a) b)))
        (t_2 (- b (fabs a)))
        (t_3 (* PI (fabs angle)))
        (t_4 (* t_1 t_2)))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs a) 2.5e+25)
      (* (* (* t_2 (sin (fma t_3 0.005555555555555556 (* 0.5 PI)))) 2.0) t_1)
      (if (<= (fabs a) 2.02e+192)
        (*
         t_4
         (*
          (-
           (sin
            (-
             (* -0.5 PI)
             (* (fabs (fabs angle)) (* PI -0.005555555555555556)))))
          2.0))
        (*
         t_4
         (*
          2.0
          (*
           (cos (/ (fma -0.005555555555555556 (* (fabs angle) PI) t_0) 2.0))
           (sin (/ (fma t_3 -0.011111111111111112 PI) 2.0))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 * fabs(angle)) * ((double) M_PI);
	double t_1 = sin(t_0) * (fabs(a) + b);
	double t_2 = b - fabs(a);
	double t_3 = ((double) M_PI) * fabs(angle);
	double t_4 = t_1 * t_2;
	double tmp;
	if (fabs(a) <= 2.5e+25) {
		tmp = ((t_2 * sin(fma(t_3, 0.005555555555555556, (0.5 * ((double) M_PI))))) * 2.0) * t_1;
	} else if (fabs(a) <= 2.02e+192) {
		tmp = t_4 * (-sin(((-0.5 * ((double) M_PI)) - (fabs(fabs(angle)) * (((double) M_PI) * -0.005555555555555556)))) * 2.0);
	} else {
		tmp = t_4 * (2.0 * (cos((fma(-0.005555555555555556, (fabs(angle) * ((double) M_PI)), t_0) / 2.0)) * sin((fma(t_3, -0.011111111111111112, ((double) M_PI)) / 2.0))));
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(0.005555555555555556 * abs(angle)) * pi)
	t_1 = Float64(sin(t_0) * Float64(abs(a) + b))
	t_2 = Float64(b - abs(a))
	t_3 = Float64(pi * abs(angle))
	t_4 = Float64(t_1 * t_2)
	tmp = 0.0
	if (abs(a) <= 2.5e+25)
		tmp = Float64(Float64(Float64(t_2 * sin(fma(t_3, 0.005555555555555556, Float64(0.5 * pi)))) * 2.0) * t_1);
	elseif (abs(a) <= 2.02e+192)
		tmp = Float64(t_4 * Float64(Float64(-sin(Float64(Float64(-0.5 * pi) - Float64(abs(abs(angle)) * Float64(pi * -0.005555555555555556))))) * 2.0));
	else
		tmp = Float64(t_4 * Float64(2.0 * Float64(cos(Float64(fma(-0.005555555555555556, Float64(abs(angle) * pi), t_0) / 2.0)) * sin(Float64(fma(t_3, -0.011111111111111112, pi) / 2.0)))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[t$95$0], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * t$95$2), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[a], $MachinePrecision], 2.5e+25], N[(N[(N[(t$95$2 * N[Sin[N[(t$95$3 * 0.005555555555555556 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Abs[a], $MachinePrecision], 2.02e+192], N[(t$95$4 * N[((-N[Sin[N[(N[(-0.5 * Pi), $MachinePrecision] - N[(N[Abs[N[Abs[angle], $MachinePrecision]], $MachinePrecision] * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * 2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$4 * N[(2.0 * N[(N[Cos[N[(N[(-0.005555555555555556 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$3 * -0.011111111111111112 + Pi), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 2.50000000000000012e25

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 68.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. cos-neg-rev N/A

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

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

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

      11. lower-sin.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-neg-in N/A

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

      18. lift-*.f64 N/A

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

   7. Applied rewrites 67.3%

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

   if 2.50000000000000012e25 < a < 2.02e192

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 68.2%

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

   5. Applied rewrites 67.2%

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

   if 2.02e192 < a

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 68.2%

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

   5. Applied rewrites 67.6%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 67.4%

      \[\leadsto \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(-0.005555555555555556, angle \cdot \pi, \left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{fma}\left(\pi \cdot angle, -0.011111111111111112, \pi\right)}{2}\right)}\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot \left|angle\right|\right) \cdot \pi\\ t_1 := \left|b\right| - \left|a\right|\\ t_2 := \left|a\right| + \left|b\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 7 \cdot 10^{+129}:\\ \;\;\;\;\left(\left(t\_1 \cdot \cos \left(-0.005555555555555556 \cdot \left(\left|angle\right| \cdot \pi\right)\right)\right) \cdot 2\right) \cdot \left(\sin \left(\frac{1}{\frac{180}{\left|angle\right|}} \cdot \pi\right) \cdot t\_2\right)\\ \mathbf{elif}\;\left|angle\right| \leq 2.2 \cdot 10^{+287}:\\ \;\;\;\;\left(\left(\sin t\_0 \cdot t\_2\right) \cdot t\_1\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, t\_0\right)\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\left(\left|angle\right| \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, {\left(\left|angle\right|\right)}^{2} \cdot {\pi}^{3}, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left|b\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 (fabs angle)) PI))
        (t_1 (- (fabs b) (fabs a)))
        (t_2 (+ (fabs a) (fabs b))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 7e+129)
      (*
       (* (* t_1 (cos (* -0.005555555555555556 (* (fabs angle) PI)))) 2.0)
       (* (sin (* (/ 1.0 (/ 180.0 (fabs angle))) PI)) t_2))
      (if (<= (fabs angle) 2.2e+287)
        (* (* (* (sin t_0) t_2) t_1) (* (sin (fma 0.5 PI t_0)) 2.0))
        (*
         t_1
         (*
          (*
           (fabs angle)
           (fma
            -2.2862368541380886e-7
            (* (pow (fabs angle) 2.0) (pow PI 3.0))
            (* 0.011111111111111112 PI)))
          (fabs b))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 * fabs(angle)) * ((double) M_PI);
	double t_1 = fabs(b) - fabs(a);
	double t_2 = fabs(a) + fabs(b);
	double tmp;
	if (fabs(angle) <= 7e+129) {
		tmp = ((t_1 * cos((-0.005555555555555556 * (fabs(angle) * ((double) M_PI))))) * 2.0) * (sin(((1.0 / (180.0 / fabs(angle))) * ((double) M_PI))) * t_2);
	} else if (fabs(angle) <= 2.2e+287) {
		tmp = ((sin(t_0) * t_2) * t_1) * (sin(fma(0.5, ((double) M_PI), t_0)) * 2.0);
	} else {
		tmp = t_1 * ((fabs(angle) * fma(-2.2862368541380886e-7, (pow(fabs(angle), 2.0) * pow(((double) M_PI), 3.0)), (0.011111111111111112 * ((double) M_PI)))) * fabs(b));
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(0.005555555555555556 * abs(angle)) * pi)
	t_1 = Float64(abs(b) - abs(a))
	t_2 = Float64(abs(a) + abs(b))
	tmp = 0.0
	if (abs(angle) <= 7e+129)
		tmp = Float64(Float64(Float64(t_1 * cos(Float64(-0.005555555555555556 * Float64(abs(angle) * pi)))) * 2.0) * Float64(sin(Float64(Float64(1.0 / Float64(180.0 / abs(angle))) * pi)) * t_2));
	elseif (abs(angle) <= 2.2e+287)
		tmp = Float64(Float64(Float64(sin(t_0) * t_2) * t_1) * Float64(sin(fma(0.5, pi, t_0)) * 2.0));
	else
		tmp = Float64(t_1 * Float64(Float64(abs(angle) * fma(-2.2862368541380886e-7, Float64((abs(angle) ^ 2.0) * (pi ^ 3.0)), Float64(0.011111111111111112 * pi))) * abs(b)));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 7e+129], N[(N[(N[(t$95$1 * N[Cos[N[(-0.005555555555555556 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[N[(N[(1.0 / N[(180.0 / N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 2.2e+287], N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sin[N[(0.5 * Pi + t$95$0), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[Abs[angle], $MachinePrecision] * N[(-2.2862368541380886e-7 * N[(N[Power[N[Abs[angle], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if angle < 6.9999999999999997e129

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 68.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip-rev N/A

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

      5. div-flip N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-/.f64 67.5

         \[\leadsto \left(\left(\left(b - a\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot 2\right) \cdot \left(\sin \left(\frac{1}{\color{blue}{\frac{180}{angle}}} \cdot \pi\right) \cdot \left(a + b\right)\right) \]

   7. Applied rewrites 67.5%

      \[\leadsto \left(\left(\left(b - a\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot 2\right) \cdot \left(\sin \left(\color{blue}{\frac{1}{\frac{180}{angle}}} \cdot \pi\right) \cdot \left(a + b\right)\right) \]

   if 6.9999999999999997e129 < angle < 2.20000000000000003e287

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 68.2%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. mult-flip N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 66.9

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

   5. Applied rewrites 67.2%

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

   if 2.20000000000000003e287 < angle

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 42.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 42.4

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

   8. Applied rewrites 42.4%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \left(b - a\right) \cdot \left(\color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot b\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(b - a\right) \cdot \left(\left(angle \cdot \color{blue}{\left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot b\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(b - a\right) \cdot \left(\left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{{angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \left(b - a\right) \cdot \left(\left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, {angle}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \left(b - a\right) \cdot \left(\left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, {angle}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto \left(b - a\right) \cdot \left(\left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{3}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\right) \]

       6. lower-PI.f64 N/A

          \[\leadsto \left(b - a\right) \cdot \left(\left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, {angle}^{2} \cdot {\pi}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(b - a\right) \cdot \left(\left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, {angle}^{2} \cdot {\pi}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\right) \]

       8. lower-PI.f64 40.2

          \[\leadsto \left(b - a\right) \cdot \left(\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, {angle}^{2} \cdot {\pi}^{3}, 0.011111111111111112 \cdot \pi\right)\right) \cdot b\right) \]

   11. Applied rewrites 40.2%

       \[\leadsto \left(b - a\right) \cdot \left(\color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, {angle}^{2} \cdot {\pi}^{3}, 0.011111111111111112 \cdot \pi\right)\right)} \cdot b\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 67.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left(0.005555555555555556 \cdot \left|angle\right|\right) \cdot \pi\right) \cdot \left(a + b\right)\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 9.6 \cdot 10^{+144}:\\ \;\;\;\;\left(t\_0 \cdot \left(b - a\right)\right) \cdot \left(\left(-\sin \left(-0.5 \cdot \pi - \left|\left|angle\right|\right| \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot \left|angle\right|, 0.005555555555555556, 0.5 \cdot \pi\right)\right)\right) \cdot 2\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (sin (* (* 0.005555555555555556 (fabs angle)) PI)) (+ a b))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 9.6e+144)
      (*
       (* t_0 (- b a))
       (*
        (-
         (sin
          (-
           (* -0.5 PI)
           (* (fabs (fabs angle)) (* PI -0.005555555555555556)))))
        2.0))
      (*
       (*
        (*
         (- b a)
         (sin (fma (* PI (fabs angle)) 0.005555555555555556 (* 0.5 PI))))
        2.0)
       t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = sin(((0.005555555555555556 * fabs(angle)) * ((double) M_PI))) * (a + b);
	double tmp;
	if (fabs(angle) <= 9.6e+144) {
		tmp = (t_0 * (b - a)) * (-sin(((-0.5 * ((double) M_PI)) - (fabs(fabs(angle)) * (((double) M_PI) * -0.005555555555555556)))) * 2.0);
	} else {
		tmp = (((b - a) * sin(fma((((double) M_PI) * fabs(angle)), 0.005555555555555556, (0.5 * ((double) M_PI))))) * 2.0) * t_0;
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(sin(Float64(Float64(0.005555555555555556 * abs(angle)) * pi)) * Float64(a + b))
	tmp = 0.0
	if (abs(angle) <= 9.6e+144)
		tmp = Float64(Float64(t_0 * Float64(b - a)) * Float64(Float64(-sin(Float64(Float64(-0.5 * pi) - Float64(abs(abs(angle)) * Float64(pi * -0.005555555555555556))))) * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(b - a) * sin(fma(Float64(pi * abs(angle)), 0.005555555555555556, Float64(0.5 * pi)))) * 2.0) * t_0);
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Sin[N[(N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 9.6e+144], N[(N[(t$95$0 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[((-N[Sin[N[(N[(-0.5 * Pi), $MachinePrecision] - N[(N[Abs[N[Abs[angle], $MachinePrecision]], $MachinePrecision] * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * 0.005555555555555556 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 9.6000000000000002e144

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 68.2%

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

   5. Applied rewrites 67.2%

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

   if 9.6000000000000002e144 < angle

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 68.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. cos-neg-rev N/A

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

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

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

      11. lower-sin.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-neg-in N/A

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

      18. lift-*.f64 N/A

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

   7. Applied rewrites 67.3%

      \[\leadsto \left(\left(\left(b - a\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 0.5 \cdot \pi\right)\right)}\right) \cdot 2\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 67.3% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 7 \cdot 10^{+129}:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\left|angle\right| \cdot \pi\right)\right)\right) \cdot 2\right) \cdot \left(\sin \left(\frac{1}{\frac{180}{\left|angle\right|}} \cdot \pi\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot \left|angle\right|, 0.005555555555555556, 0.5 \cdot \pi\right)\right)\right) \cdot 2\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot \left|angle\right|\right) \cdot \pi\right) \cdot \left(a + b\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 7e+129)
    (*
     (* (* (- b a) (cos (* -0.005555555555555556 (* (fabs angle) PI)))) 2.0)
     (* (sin (* (/ 1.0 (/ 180.0 (fabs angle))) PI)) (+ a b)))
    (*
     (*
      (*
       (- b a)
       (sin (fma (* PI (fabs angle)) 0.005555555555555556 (* 0.5 PI))))
      2.0)
     (* (sin (* (* 0.005555555555555556 (fabs angle)) PI)) (+ a b))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 7e+129) {
		tmp = (((b - a) * cos((-0.005555555555555556 * (fabs(angle) * ((double) M_PI))))) * 2.0) * (sin(((1.0 / (180.0 / fabs(angle))) * ((double) M_PI))) * (a + b));
	} else {
		tmp = (((b - a) * sin(fma((((double) M_PI) * fabs(angle)), 0.005555555555555556, (0.5 * ((double) M_PI))))) * 2.0) * (sin(((0.005555555555555556 * fabs(angle)) * ((double) M_PI))) * (a + b));
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 7e+129)
		tmp = Float64(Float64(Float64(Float64(b - a) * cos(Float64(-0.005555555555555556 * Float64(abs(angle) * pi)))) * 2.0) * Float64(sin(Float64(Float64(1.0 / Float64(180.0 / abs(angle))) * pi)) * Float64(a + b)));
	else
		tmp = Float64(Float64(Float64(Float64(b - a) * sin(fma(Float64(pi * abs(angle)), 0.005555555555555556, Float64(0.5 * pi)))) * 2.0) * Float64(sin(Float64(Float64(0.005555555555555556 * abs(angle)) * pi)) * Float64(a + b)));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 7e+129], N[(N[(N[(N[(b - a), $MachinePrecision] * N[Cos[N[(-0.005555555555555556 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[N[(N[(1.0 / N[(180.0 / N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * 0.005555555555555556 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[N[(N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 6.9999999999999997e129

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 68.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip-rev N/A

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

      5. div-flip N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-/.f64 67.5

         \[\leadsto \left(\left(\left(b - a\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot 2\right) \cdot \left(\sin \left(\frac{1}{\color{blue}{\frac{180}{angle}}} \cdot \pi\right) \cdot \left(a + b\right)\right) \]

   7. Applied rewrites 67.5%

      \[\leadsto \left(\left(\left(b - a\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot 2\right) \cdot \left(\sin \left(\color{blue}{\frac{1}{\frac{180}{angle}}} \cdot \pi\right) \cdot \left(a + b\right)\right) \]

   if 6.9999999999999997e129 < angle

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 68.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. cos-neg-rev N/A

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

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

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

      11. lower-sin.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-neg-in N/A

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

      18. lift-*.f64 N/A

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

   7. Applied rewrites 67.3%

      \[\leadsto \left(\left(\left(b - a\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 0.5 \cdot \pi\right)\right)}\right) \cdot 2\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.9% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 9.2 \cdot 10^{+142}:\\ \;\;\;\;\left(\left(\left(b - \left|a\right|\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\left|angle\right| \cdot \pi\right)\right)\right) \cdot 2\right) \cdot \left(\sin \left(\frac{\left|angle\right|}{180} \cdot \pi\right) \cdot \left(\left|a\right| + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left|angle\right| \cdot \mathsf{fma}\left(b, b, \left(-\left|a\right|\right) \cdot \left|a\right|\right)\right) \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 9.2e+142)
    (*
     (*
      (* (- b (fabs a)) (cos (* -0.005555555555555556 (* (fabs angle) PI))))
      2.0)
     (* (sin (* (/ (fabs angle) 180.0) PI)) (+ (fabs a) b)))
    (*
     0.011111111111111112
     (* (* (fabs angle) (fma b b (* (- (fabs a)) (fabs a)))) PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 9.2e+142) {
		tmp = (((b - fabs(a)) * cos((-0.005555555555555556 * (fabs(angle) * ((double) M_PI))))) * 2.0) * (sin(((fabs(angle) / 180.0) * ((double) M_PI))) * (fabs(a) + b));
	} else {
		tmp = 0.011111111111111112 * ((fabs(angle) * fma(b, b, (-fabs(a) * fabs(a)))) * ((double) M_PI));
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 9.2e+142)
		tmp = Float64(Float64(Float64(Float64(b - abs(a)) * cos(Float64(-0.005555555555555556 * Float64(abs(angle) * pi)))) * 2.0) * Float64(sin(Float64(Float64(abs(angle) / 180.0) * pi)) * Float64(abs(a) + b)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(abs(angle) * fma(b, b, Float64(Float64(-abs(a)) * abs(a)))) * pi));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 9.2e+142], N[(N[(N[(N[(b - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.005555555555555556 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[N[(N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[Abs[angle], $MachinePrecision] * N[(b * b + N[((-N[Abs[a], $MachinePrecision]) * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 9.20000000000000009e142

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 68.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 67.9

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

   7. Applied rewrites 67.9%

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

   if 9.20000000000000009e142 < angle

   1. Initial program 54.4%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 51.1

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 51.1

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. difference-of-squares-rev N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 55.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. difference-of-squares-rev N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

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

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 53.9

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \left(-a\right) \cdot a\right)\right) \cdot \pi\right) \]

   8. Applied rewrites 53.9%

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

Alternative 6: 66.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ t_1 := \left|a\right| + \left|b\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t\_1 \cdot \left(0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot t\_0\right)\right)\right)\\ \mathbf{elif}\;\left|angle\right| \leq 9.6 \cdot 10^{+144}:\\ \;\;\;\;\left(t\_0 \cdot t\_1\right) \cdot \sin \left(\left(\left|angle\right| \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left|angle\right| \cdot \mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(-\left|a\right|\right) \cdot \left|a\right|\right)\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (fabs b) (fabs a))) (t_1 (+ (fabs a) (fabs b))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 5e-11)
      (* t_1 (* 0.011111111111111112 (* (fabs angle) (* PI t_0))))
      (if (<= (fabs angle) 9.6e+144)
        (* (* t_0 t_1) (sin (* (* (fabs angle) PI) 0.011111111111111112)))
        (*
         0.011111111111111112
         (*
          (* (fabs angle) (fma (fabs b) (fabs b) (* (- (fabs a)) (fabs a))))
          PI)))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double t_1 = fabs(a) + fabs(b);
	double tmp;
	if (fabs(angle) <= 5e-11) {
		tmp = t_1 * (0.011111111111111112 * (fabs(angle) * (((double) M_PI) * t_0)));
	} else if (fabs(angle) <= 9.6e+144) {
		tmp = (t_0 * t_1) * sin(((fabs(angle) * ((double) M_PI)) * 0.011111111111111112));
	} else {
		tmp = 0.011111111111111112 * ((fabs(angle) * fma(fabs(b), fabs(b), (-fabs(a) * fabs(a)))) * ((double) M_PI));
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	t_1 = Float64(abs(a) + abs(b))
	tmp = 0.0
	if (abs(angle) <= 5e-11)
		tmp = Float64(t_1 * Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * t_0))));
	elseif (abs(angle) <= 9.6e+144)
		tmp = Float64(Float64(t_0 * t_1) * sin(Float64(Float64(abs(angle) * pi) * 0.011111111111111112)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(abs(angle) * fma(abs(b), abs(b), Float64(Float64(-abs(a)) * abs(a)))) * pi));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 5e-11], N[(t$95$1 * N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 9.6e+144], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[Sin[N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[Abs[angle], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision] + N[((-N[Abs[a], $MachinePrecision]) * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 5.00000000000000018e-11

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 62.8

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

   6. Applied rewrites 62.8%

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

   if 5.00000000000000018e-11 < angle < 9.6000000000000002e144

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 58.6%

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

   if 9.6000000000000002e144 < angle

   1. Initial program 54.4%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 51.1

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 51.1

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. difference-of-squares-rev N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 55.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. difference-of-squares-rev N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

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

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 53.9

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \left(-a\right) \cdot a\right)\right) \cdot \pi\right) \]

   8. Applied rewrites 53.9%

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

Alternative 7: 66.9% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 9.6 \cdot 10^{+144}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\sin \left(0.011111111111111112 \cdot \left(\pi \cdot \left|angle\right|\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left|angle\right| \cdot \mathsf{fma}\left(b, b, \left(-a\right) \cdot a\right)\right) \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 9.6e+144)
    (* (- b a) (* (sin (* 0.011111111111111112 (* PI (fabs angle)))) (+ b a)))
    (* 0.011111111111111112 (* (* (fabs angle) (fma b b (* (- a) a))) PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 9.6e+144) {
		tmp = (b - a) * (sin((0.011111111111111112 * (((double) M_PI) * fabs(angle)))) * (b + a));
	} else {
		tmp = 0.011111111111111112 * ((fabs(angle) * fma(b, b, (-a * a))) * ((double) M_PI));
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 9.6e+144)
		tmp = Float64(Float64(b - a) * Float64(sin(Float64(0.011111111111111112 * Float64(pi * abs(angle)))) * Float64(b + a)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(abs(angle) * fma(b, b, Float64(Float64(-a) * a))) * pi));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 9.6e+144], N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(0.011111111111111112 * N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[Abs[angle], $MachinePrecision] * N[(b * b + N[((-a) * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 9.6000000000000002e144

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 68.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 68.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 68.0

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

   5. Applied rewrites 68.0%

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

   if 9.6000000000000002e144 < angle

   1. Initial program 54.4%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 51.1

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 51.1

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. difference-of-squares-rev N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 55.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. difference-of-squares-rev N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

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

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 53.9

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \left(-a\right) \cdot a\right)\right) \cdot \pi\right) \]

   8. Applied rewrites 53.9%

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

Alternative 8: 66.2% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 3.1 \cdot 10^{+25}:\\ \;\;\;\;\left(\left|a\right| + \left|b\right|\right) \cdot \left(0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(\left|b\right| - \left|a\right|\right)\right)\right)\right)\\ \mathbf{elif}\;\left|angle\right| \leq 1.4 \cdot 10^{+50}:\\ \;\;\;\;\left|b\right| \cdot \left(\left|b\right| \cdot \sin \left(\left(\left|angle\right| \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left|angle\right| \cdot \mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(-\left|a\right|\right) \cdot \left|a\right|\right)\right) \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 3.1e+25)
    (*
     (+ (fabs a) (fabs b))
     (* 0.011111111111111112 (* (fabs angle) (* PI (- (fabs b) (fabs a))))))
    (if (<= (fabs angle) 1.4e+50)
      (*
       (fabs b)
       (* (fabs b) (sin (* (* (fabs angle) PI) 0.011111111111111112))))
      (*
       0.011111111111111112
       (*
        (* (fabs angle) (fma (fabs b) (fabs b) (* (- (fabs a)) (fabs a))))
        PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 3.1e+25) {
		tmp = (fabs(a) + fabs(b)) * (0.011111111111111112 * (fabs(angle) * (((double) M_PI) * (fabs(b) - fabs(a)))));
	} else if (fabs(angle) <= 1.4e+50) {
		tmp = fabs(b) * (fabs(b) * sin(((fabs(angle) * ((double) M_PI)) * 0.011111111111111112)));
	} else {
		tmp = 0.011111111111111112 * ((fabs(angle) * fma(fabs(b), fabs(b), (-fabs(a) * fabs(a)))) * ((double) M_PI));
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 3.1e+25)
		tmp = Float64(Float64(abs(a) + abs(b)) * Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * Float64(abs(b) - abs(a))))));
	elseif (abs(angle) <= 1.4e+50)
		tmp = Float64(abs(b) * Float64(abs(b) * sin(Float64(Float64(abs(angle) * pi) * 0.011111111111111112))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(abs(angle) * fma(abs(b), abs(b), Float64(Float64(-abs(a)) * abs(a)))) * pi));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 3.1e+25], N[(N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 1.4e+50], N[(N[Abs[b], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[Sin[N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[Abs[angle], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision] + N[((-N[Abs[a], $MachinePrecision]) * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 3.0999999999999998e25

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 62.8

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

   6. Applied rewrites 62.8%

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

   if 3.0999999999999998e25 < angle < 1.3999999999999999e50

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 42.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto b \cdot \left(\color{blue}{b} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 40.9%

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

   if 1.3999999999999999e50 < angle

   1. Initial program 54.4%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 51.1

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 51.1

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. difference-of-squares-rev N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 55.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. difference-of-squares-rev N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

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

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 53.9

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \left(-a\right) \cdot a\right)\right) \cdot \pi\right) \]

   8. Applied rewrites 53.9%

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

Alternative 9: 64.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ \mathbf{if}\;2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right) \leq 5 \cdot 10^{-229}:\\ \;\;\;\;t\_0 \cdot \left(\left|a\right| \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left|a\right| + \left|b\right|\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot t\_0\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (fabs b) (fabs a))))
   (if (<= (* 2.0 (- (pow (fabs b) 2.0) (pow (fabs a) 2.0))) 5e-229)
     (* t_0 (* (fabs a) (sin (* 0.011111111111111112 (* angle PI)))))
     (* (+ (fabs a) (fabs b)) (* 0.011111111111111112 (* angle (* PI t_0)))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double tmp;
	if ((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) <= 5e-229) {
		tmp = t_0 * (fabs(a) * sin((0.011111111111111112 * (angle * ((double) M_PI)))));
	} else {
		tmp = (fabs(a) + fabs(b)) * (0.011111111111111112 * (angle * (((double) M_PI) * t_0)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - Math.abs(a);
	double tmp;
	if ((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) <= 5e-229) {
		tmp = t_0 * (Math.abs(a) * Math.sin((0.011111111111111112 * (angle * Math.PI))));
	} else {
		tmp = (Math.abs(a) + Math.abs(b)) * (0.011111111111111112 * (angle * (Math.PI * t_0)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	tmp = 0
	if (2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) <= 5e-229:
		tmp = t_0 * (math.fabs(a) * math.sin((0.011111111111111112 * (angle * math.pi))))
	else:
		tmp = (math.fabs(a) + math.fabs(b)) * (0.011111111111111112 * (angle * (math.pi * t_0)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	tmp = 0.0
	if (Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) <= 5e-229)
		tmp = Float64(t_0 * Float64(abs(a) * sin(Float64(0.011111111111111112 * Float64(angle * pi)))));
	else
		tmp = Float64(Float64(abs(a) + abs(b)) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	tmp = 0.0;
	if ((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) <= 5e-229)
		tmp = t_0 * (abs(a) * sin((0.011111111111111112 * (angle * pi))));
	else
		tmp = (abs(a) + abs(b)) * (0.011111111111111112 * (angle * (pi * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-229], N[(t$95$0 * N[(N[Abs[a], $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * t$95$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)))) < 5.00000000000000016e-229

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 68.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 68.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 42.6

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

   8. Applied rewrites 42.6%

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

   if 5.00000000000000016e-229 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 62.8

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

   6. Applied rewrites 62.8%

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

Alternative 10: 62.8% accurate, 5.2× speedup

Math

\[\left(\left|a\right| + \left|b\right|\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left|b\right| - \left|a\right|\right)\right)\right)\right) \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (+ (fabs a) (fabs b))
  (* 0.011111111111111112 (* angle (* PI (- (fabs b) (fabs a)))))))

C

double code(double a, double b, double angle) {
	return (fabs(a) + fabs(b)) * (0.011111111111111112 * (angle * (((double) M_PI) * (fabs(b) - fabs(a)))));
}

Java

public static double code(double a, double b, double angle) {
	return (Math.abs(a) + Math.abs(b)) * (0.011111111111111112 * (angle * (Math.PI * (Math.abs(b) - Math.abs(a)))));
}

Python

def code(a, b, angle):
	return (math.fabs(a) + math.fabs(b)) * (0.011111111111111112 * (angle * (math.pi * (math.fabs(b) - math.fabs(a)))))

Julia

function code(a, b, angle)
	return Float64(Float64(abs(a) + abs(b)) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(abs(b) - abs(a))))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (abs(a) + abs(b)) * (0.011111111111111112 * (angle * (pi * (abs(b) - abs(a)))));
end

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. 2-sin N/A

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

   10. count-2 N/A

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

3. Applied rewrites 68.0%

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

4. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower--.f64 62.8

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

6. Applied rewrites 62.8%

   \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
7. Add Preprocessing

Alternative 11: 62.8% accurate, 5.2× speedup

Math

\[\left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left|b\right| - \left|a\right|\right)\right)\right) \cdot \left(\left|a\right| + \left|b\right|\right)\right) \cdot \pi \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (*
   (* 0.011111111111111112 (* angle (- (fabs b) (fabs a))))
   (+ (fabs a) (fabs b)))
  PI))

C

double code(double a, double b, double angle) {
	return ((0.011111111111111112 * (angle * (fabs(b) - fabs(a)))) * (fabs(a) + fabs(b))) * ((double) M_PI);
}

Java

public static double code(double a, double b, double angle) {
	return ((0.011111111111111112 * (angle * (Math.abs(b) - Math.abs(a)))) * (Math.abs(a) + Math.abs(b))) * Math.PI;
}

Python

def code(a, b, angle):
	return ((0.011111111111111112 * (angle * (math.fabs(b) - math.fabs(a)))) * (math.fabs(a) + math.fabs(b))) * math.pi

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(0.011111111111111112 * Float64(angle * Float64(abs(b) - abs(a)))) * Float64(abs(a) + abs(b))) * pi)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((0.011111111111111112 * (angle * (abs(b) - abs(a)))) * (abs(a) + abs(b))) * pi;
end

Wolfram

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

Derivation

1. Initial program 54.4%

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

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. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 51.1

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 51.1%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 51.1

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. pow2 N/A

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

   12. difference-of-squares-rev N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 55.3

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 55.3

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

6. Applied rewrites 55.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 62.8

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 62.8

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

8. Applied rewrites 62.8%

   \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(a + b\right)\right) \cdot \color{blue}{\pi} \]
9. Add Preprocessing

Alternative 12: 58.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 1.56 \cdot 10^{+162}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\right) \cdot \left(\left(\left|a\right| + \left|b\right|\right) \cdot \left(\left(\left|b\right| - \left|a\right|\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|b\right| \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\left|b\right| \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (fabs b) 1.56e+162)
   (*
    (* 0.011111111111111112 angle)
    (* (+ (fabs a) (fabs b)) (* (- (fabs b) (fabs a)) PI)))
   (* (fabs b) (* 0.011111111111111112 (* angle (* (fabs b) PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(b) <= 1.56e+162) {
		tmp = (0.011111111111111112 * angle) * ((fabs(a) + fabs(b)) * ((fabs(b) - fabs(a)) * ((double) M_PI)));
	} else {
		tmp = fabs(b) * (0.011111111111111112 * (angle * (fabs(b) * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(b) <= 1.56e+162) {
		tmp = (0.011111111111111112 * angle) * ((Math.abs(a) + Math.abs(b)) * ((Math.abs(b) - Math.abs(a)) * Math.PI));
	} else {
		tmp = Math.abs(b) * (0.011111111111111112 * (angle * (Math.abs(b) * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(b) <= 1.56e+162:
		tmp = (0.011111111111111112 * angle) * ((math.fabs(a) + math.fabs(b)) * ((math.fabs(b) - math.fabs(a)) * math.pi))
	else:
		tmp = math.fabs(b) * (0.011111111111111112 * (angle * (math.fabs(b) * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(b) <= 1.56e+162)
		tmp = Float64(Float64(0.011111111111111112 * angle) * Float64(Float64(abs(a) + abs(b)) * Float64(Float64(abs(b) - abs(a)) * pi)));
	else
		tmp = Float64(abs(b) * Float64(0.011111111111111112 * Float64(angle * Float64(abs(b) * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(b) <= 1.56e+162)
		tmp = (0.011111111111111112 * angle) * ((abs(a) + abs(b)) * ((abs(b) - abs(a)) * pi));
	else
		tmp = abs(b) * (0.011111111111111112 * (angle * (abs(b) * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[b], $MachinePrecision], 1.56e+162], N[(N[(0.011111111111111112 * angle), $MachinePrecision] * N[(N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[b], $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5600000000000001e162

   1. Initial program 54.4%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 51.1

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 51.1

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. difference-of-squares-rev N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 55.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-*.f64 55.3

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

   8. Applied rewrites 55.3%

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

   if 1.5600000000000001e162 < b

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 42.4%

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

   7. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 41.7

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

   9. Applied rewrites 41.7%

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

   10. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-PI.f64 38.3

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

   12. Applied rewrites 38.3%

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

Alternative 13: 58.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 1.56 \cdot 10^{+162}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(\left|b\right| - a\right) \cdot \left(\left|b\right| + a\right)\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left|b\right| \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\left|b\right| \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (fabs b) 1.56e+162)
   (* 0.011111111111111112 (* (* angle (* (- (fabs b) a) (+ (fabs b) a))) PI))
   (* (fabs b) (* 0.011111111111111112 (* angle (* (fabs b) PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(b) <= 1.56e+162) {
		tmp = 0.011111111111111112 * ((angle * ((fabs(b) - a) * (fabs(b) + a))) * ((double) M_PI));
	} else {
		tmp = fabs(b) * (0.011111111111111112 * (angle * (fabs(b) * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(b) <= 1.56e+162) {
		tmp = 0.011111111111111112 * ((angle * ((Math.abs(b) - a) * (Math.abs(b) + a))) * Math.PI);
	} else {
		tmp = Math.abs(b) * (0.011111111111111112 * (angle * (Math.abs(b) * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(b) <= 1.56e+162:
		tmp = 0.011111111111111112 * ((angle * ((math.fabs(b) - a) * (math.fabs(b) + a))) * math.pi)
	else:
		tmp = math.fabs(b) * (0.011111111111111112 * (angle * (math.fabs(b) * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(b) <= 1.56e+162)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle * Float64(Float64(abs(b) - a) * Float64(abs(b) + a))) * pi));
	else
		tmp = Float64(abs(b) * Float64(0.011111111111111112 * Float64(angle * Float64(abs(b) * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(b) <= 1.56e+162)
		tmp = 0.011111111111111112 * ((angle * ((abs(b) - a) * (abs(b) + a))) * pi);
	else
		tmp = abs(b) * (0.011111111111111112 * (angle * (abs(b) * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[b], $MachinePrecision], 1.56e+162], N[(0.011111111111111112 * N[(N[(angle * N[(N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[Abs[b], $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5600000000000001e162

   1. Initial program 54.4%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 51.1

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 51.1

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. difference-of-squares-rev N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 55.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 55.3

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

   6. Applied rewrites 55.3%

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

   if 1.5600000000000001e162 < b

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 42.4%

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

   7. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 41.7

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

   9. Applied rewrites 41.7%

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

   10. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-PI.f64 38.3

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

   12. Applied rewrites 38.3%

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

Alternative 14: 41.7% accurate, 6.8× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (* (fabs b) (* 0.011111111111111112 (* angle (* PI (- (fabs b) a))))))

C

double code(double a, double b, double angle) {
	return fabs(b) * (0.011111111111111112 * (angle * (((double) M_PI) * (fabs(b) - a))));
}

Java

public static double code(double a, double b, double angle) {
	return Math.abs(b) * (0.011111111111111112 * (angle * (Math.PI * (Math.abs(b) - a))));
}

Python

def code(a, b, angle):
	return math.fabs(b) * (0.011111111111111112 * (angle * (math.pi * (math.fabs(b) - a))))

Julia

function code(a, b, angle)
	return Float64(abs(b) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(abs(b) - a)))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = abs(b) * (0.011111111111111112 * (angle * (pi * (abs(b) - a))));
end

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. 2-sin N/A

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

   10. count-2 N/A

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

3. Applied rewrites 68.0%

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

4. Taylor expanded in a around 0

   \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 42.4%

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

7. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower--.f64 41.7

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

9. Applied rewrites 41.7%

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

Alternative 15: 41.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;2 \cdot \left({\left(\left|b\right|\right)}^{2} - {a}^{2}\right) \leq -2 \cdot 10^{-206}:\\ \;\;\;\;\left|b\right| \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|b\right| \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\left|b\right| \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow (fabs b) 2.0) (pow a 2.0))) -2e-206)
   (* (fabs b) (* -0.011111111111111112 (* a (* angle PI))))
   (* (fabs b) (* 0.011111111111111112 (* angle (* (fabs b) PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((2.0 * (pow(fabs(b), 2.0) - pow(a, 2.0))) <= -2e-206) {
		tmp = fabs(b) * (-0.011111111111111112 * (a * (angle * ((double) M_PI))));
	} else {
		tmp = fabs(b) * (0.011111111111111112 * (angle * (fabs(b) * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(a, 2.0))) <= -2e-206) {
		tmp = Math.abs(b) * (-0.011111111111111112 * (a * (angle * Math.PI)));
	} else {
		tmp = Math.abs(b) * (0.011111111111111112 * (angle * (Math.abs(b) * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(a, 2.0))) <= -2e-206:
		tmp = math.fabs(b) * (-0.011111111111111112 * (a * (angle * math.pi)))
	else:
		tmp = math.fabs(b) * (0.011111111111111112 * (angle * (math.fabs(b) * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((abs(b) ^ 2.0) - (a ^ 2.0))) <= -2e-206)
		tmp = Float64(abs(b) * Float64(-0.011111111111111112 * Float64(a * Float64(angle * pi))));
	else
		tmp = Float64(abs(b) * Float64(0.011111111111111112 * Float64(angle * Float64(abs(b) * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((2.0 * ((abs(b) ^ 2.0) - (a ^ 2.0))) <= -2e-206)
		tmp = abs(b) * (-0.011111111111111112 * (a * (angle * pi)));
	else
		tmp = abs(b) * (0.011111111111111112 * (angle * (abs(b) * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-206], N[(N[Abs[b], $MachinePrecision] * N[(-0.011111111111111112 * N[(a * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[b], $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(N[Abs[b], $MachinePrecision] * Pi), $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)))) < -2.00000000000000006e-206

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 42.4%

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

   7. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 41.7

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

   9. Applied rewrites 41.7%

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

   10. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-PI.f64 20.4

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

   12. Applied rewrites 20.4%

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

   if -2.00000000000000006e-206 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift-sin.f64 N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-cos.f64 N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

      9. 2-sin N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      10. count-2 N/A

          \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)} \]

   3. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 42.4%

      \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \]

   7. Taylor expanded in angle around 0

      \[\leadsto b \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto b \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      5. lower--.f64 41.7

         \[\leadsto b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - \color{blue}{a}\right)\right)\right)\right) \]

   9. Applied rewrites 41.7%

      \[\leadsto b \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]

   10. Taylor expanded in a around 0

       \[\leadsto b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       2. lower-PI.f64 38.3

          \[\leadsto b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

   12. Applied rewrites 38.3%

       \[\leadsto b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \color{blue}{\pi}\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 20.2% accurate, 8.6× speedup

Math

\[\left|b\right| \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* (fabs b) (* -0.011111111111111112 (* a (* angle PI)))))

C

double code(double a, double b, double angle) {
	return fabs(b) * (-0.011111111111111112 * (a * (angle * ((double) M_PI))));
}

Java

public static double code(double a, double b, double angle) {
	return Math.abs(b) * (-0.011111111111111112 * (a * (angle * Math.PI)));
}

Python

def code(a, b, angle):
	return math.fabs(b) * (-0.011111111111111112 * (a * (angle * math.pi)))

Julia

function code(a, b, angle)
	return Float64(abs(b) * Float64(-0.011111111111111112 * Float64(a * Float64(angle * pi))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = abs(b) * (-0.011111111111111112 * (a * (angle * pi)));
end

Wolfram

code[a_, b_, angle_] := N[(N[Abs[b], $MachinePrecision] * N[(-0.011111111111111112 * N[(a * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   7. lift-sin.f64 N/A

      \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   8. lift-cos.f64 N/A

      \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

   9. 2-sin N/A

      \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   10. count-2 N/A

       \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)} \]

3. Applied rewrites 68.0%

   \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

4. Taylor expanded in a around 0

   \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 42.4%

   \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \]

7. Taylor expanded in angle around 0

   \[\leadsto b \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto b \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

   5. lower--.f64 41.7

      \[\leadsto b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - \color{blue}{a}\right)\right)\right)\right) \]

9. Applied rewrites 41.7%

   \[\leadsto b \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]

10. Taylor expanded in a around inf

    \[\leadsto b \cdot \left(\frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto b \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

    2. lower-*.f64 N/A

       \[\leadsto b \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

    3. lower-*.f64 N/A

       \[\leadsto b \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    4. lower-PI.f64 20.4

       \[\leadsto b \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \]

12. Applied rewrites 20.4%

    \[\leadsto b \cdot \left(-0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025171 
(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)))))