ab-angle->ABCF B

Percentage Accurate: 54.5% → 68.3%

Time: 7.6s

Alternatives: 15

Speedup: 5.9×

• 67.3% 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: 54.5% 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: 68.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 10^{+133}:\\ \;\;\;\;\left(\cos \left(\left(-0.005555555555555556 \cdot \left|angle\right|\right) \cdot \pi\right) \cdot \left(2 \cdot \left(\sin \left(\frac{\pi \cdot \left|angle\right|}{180}\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ (fabs angle) 180.0))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 1e+133)
      (*
       (*
        (cos (* (* -0.005555555555555556 (fabs angle)) PI))
        (* 2.0 (* (sin (/ (* PI (fabs angle)) 180.0)) (- b a))))
       (+ b a))
      (* (* (* 2.0 (fma (- a) a (* b b))) (sin t_0)) (cos t_0))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (fabs(angle) / 180.0);
	double tmp;
	if (fabs(angle) <= 1e+133) {
		tmp = (cos(((-0.005555555555555556 * fabs(angle)) * ((double) M_PI))) * (2.0 * (sin(((((double) M_PI) * fabs(angle)) / 180.0)) * (b - a)))) * (b + a);
	} else {
		tmp = ((2.0 * fma(-a, a, (b * b))) * sin(t_0)) * cos(t_0);
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(abs(angle) / 180.0))
	tmp = 0.0
	if (abs(angle) <= 1e+133)
		tmp = Float64(Float64(cos(Float64(Float64(-0.005555555555555556 * abs(angle)) * pi)) * Float64(2.0 * Float64(sin(Float64(Float64(pi * abs(angle)) / 180.0)) * Float64(b - a)))) * Float64(b + a));
	else
		tmp = Float64(Float64(Float64(2.0 * fma(Float64(-a), a, Float64(b * b))) * sin(t_0)) * cos(t_0));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $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], 1e+133], N[(N[(N[Cos[N[(N[(-0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[Sin[N[(N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[((-a) * a + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 1e133

   1. Initial program 54.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 68.3%

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

   5. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. mult-flip N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 67.8%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 67.8%

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

   7. Applied rewrites 67.8%

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

   if 1e133 < angle

   1. Initial program 54.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 57.3%

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 57.3%

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

   3. Applied rewrites 57.3%

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

Alternative 2: 68.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
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 68.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 68.3%

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

5. Applied rewrites 68.3%

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

Alternative 3: 68.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - a\\ t_1 := a + \left|b\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 10^{-24}:\\ \;\;\;\;\left(\left(\left|angle\right| \cdot t\_1\right) \cdot t\_0\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_1\right) \cdot \sin \left(\left(\left|angle\right| \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (fabs b) a)) (t_1 (+ a (fabs b))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 1e-24)
      (* (* (* (fabs angle) t_1) t_0) (* PI 0.011111111111111112))
      (* (* t_0 t_1) (sin (* (* (fabs angle) PI) 0.011111111111111112)))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - a;
	double t_1 = a + fabs(b);
	double tmp;
	if (fabs(angle) <= 1e-24) {
		tmp = ((fabs(angle) * t_1) * t_0) * (((double) M_PI) * 0.011111111111111112);
	} else {
		tmp = (t_0 * t_1) * sin(((fabs(angle) * ((double) M_PI)) * 0.011111111111111112));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - a;
	double t_1 = a + Math.abs(b);
	double tmp;
	if (Math.abs(angle) <= 1e-24) {
		tmp = ((Math.abs(angle) * t_1) * t_0) * (Math.PI * 0.011111111111111112);
	} else {
		tmp = (t_0 * t_1) * Math.sin(((Math.abs(angle) * Math.PI) * 0.011111111111111112));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - a
	t_1 = a + math.fabs(b)
	tmp = 0
	if math.fabs(angle) <= 1e-24:
		tmp = ((math.fabs(angle) * t_1) * t_0) * (math.pi * 0.011111111111111112)
	else:
		tmp = (t_0 * t_1) * math.sin(((math.fabs(angle) * math.pi) * 0.011111111111111112))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - a)
	t_1 = Float64(a + abs(b))
	tmp = 0.0
	if (abs(angle) <= 1e-24)
		tmp = Float64(Float64(Float64(abs(angle) * t_1) * t_0) * Float64(pi * 0.011111111111111112));
	else
		tmp = Float64(Float64(t_0 * t_1) * sin(Float64(Float64(abs(angle) * pi) * 0.011111111111111112)));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - a;
	t_1 = a + abs(b);
	tmp = 0.0;
	if (abs(angle) <= 1e-24)
		tmp = ((abs(angle) * t_1) * t_0) * (pi * 0.011111111111111112);
	else
		tmp = (t_0 * t_1) * sin(((abs(angle) * pi) * 0.011111111111111112));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision]}, Block[{t$95$1 = N[(a + 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], 1e-24], N[(N[(N[(N[Abs[angle], $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[Sin[N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 9.99999999999999924e-25

   1. Initial program 54.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

      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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

      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 54.8%

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

   6. Applied rewrites 54.8%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-*.f64 62.8%

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

   8. Applied rewrites 62.8%

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

   if 9.99999999999999924e-25 < angle

   1. Initial program 54.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 58.2%

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

Alternative 4: 68.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
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 68.1%

   \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]
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. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 68.1%

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 68.1%

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

5. Applied rewrites 68.1%

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

Alternative 5: 65.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| + \left|b\right|\\ \mathbf{if}\;\left|a\right| \leq 2.6 \cdot 10^{-122}:\\ \;\;\;\;t\_0 \cdot \left(\left|b\right| \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot t\_0\right) \cdot \left(\left|b\right| - \left|a\right|\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (+ (fabs a) (fabs b))))
   (if (<= (fabs a) 2.6e-122)
     (* t_0 (* (fabs b) (sin (* 0.011111111111111112 (* angle PI)))))
     (* (* (* angle t_0) (- (fabs b) (fabs a))) (* PI 0.011111111111111112)))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(a) + fabs(b);
	double tmp;
	if (fabs(a) <= 2.6e-122) {
		tmp = t_0 * (fabs(b) * sin((0.011111111111111112 * (angle * ((double) M_PI)))));
	} else {
		tmp = ((angle * t_0) * (fabs(b) - fabs(a))) * (((double) M_PI) * 0.011111111111111112);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(a) + Math.abs(b);
	double tmp;
	if (Math.abs(a) <= 2.6e-122) {
		tmp = t_0 * (Math.abs(b) * Math.sin((0.011111111111111112 * (angle * Math.PI))));
	} else {
		tmp = ((angle * t_0) * (Math.abs(b) - Math.abs(a))) * (Math.PI * 0.011111111111111112);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(a) + math.fabs(b)
	tmp = 0
	if math.fabs(a) <= 2.6e-122:
		tmp = t_0 * (math.fabs(b) * math.sin((0.011111111111111112 * (angle * math.pi))))
	else:
		tmp = ((angle * t_0) * (math.fabs(b) - math.fabs(a))) * (math.pi * 0.011111111111111112)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(a) + abs(b))
	tmp = 0.0
	if (abs(a) <= 2.6e-122)
		tmp = Float64(t_0 * Float64(abs(b) * sin(Float64(0.011111111111111112 * Float64(angle * pi)))));
	else
		tmp = Float64(Float64(Float64(angle * t_0) * Float64(abs(b) - abs(a))) * Float64(pi * 0.011111111111111112));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(a) + abs(b);
	tmp = 0.0;
	if (abs(a) <= 2.6e-122)
		tmp = t_0 * (abs(b) * sin((0.011111111111111112 * (angle * pi))));
	else
		tmp = ((angle * t_0) * (abs(b) - abs(a))) * (pi * 0.011111111111111112);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 2.6e-122], N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(angle * t$95$0), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.59999999999999975e-122

   1. Initial program 54.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 68.1%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 42.2%

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

   6. Applied rewrites 42.2%

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

   if 2.59999999999999975e-122 < a

   1. Initial program 54.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

      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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

      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 54.8%

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

   6. Applied rewrites 54.8%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-*.f64 62.8%

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

   8. Applied rewrites 62.8%

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

Alternative 6: 64.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ t_1 := \left|angle\right| \cdot \left(\left|a\right| + \left|b\right|\right)\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 5 \cdot 10^{+121}:\\ \;\;\;\;\left(t\_1 \cdot t\_0\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;\left|angle\right| \leq 1.85 \cdot 10^{+194}:\\ \;\;\;\;\log \left(e^{\left(0.011111111111111112 \cdot \pi\right) \cdot \left(t\_1 \cdot \left|b\right|\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left|angle\right| \cdot \log \left({\left(e^{\pi}\right)}^{\left(t\_0 \cdot \left(\left|b\right| + \left|a\right|\right)\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (fabs b) (fabs a)))
        (t_1 (* (fabs angle) (+ (fabs a) (fabs b)))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 5e+121)
      (* (* t_1 t_0) (* PI 0.011111111111111112))
      (if (<= (fabs angle) 1.85e+194)
        (log (exp (* (* 0.011111111111111112 PI) (* t_1 (fabs b)))))
        (*
         0.011111111111111112
         (*
          (fabs angle)
          (log (pow (exp PI) (* t_0 (+ (fabs b) (fabs a))))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double t_1 = fabs(angle) * (fabs(a) + fabs(b));
	double tmp;
	if (fabs(angle) <= 5e+121) {
		tmp = (t_1 * t_0) * (((double) M_PI) * 0.011111111111111112);
	} else if (fabs(angle) <= 1.85e+194) {
		tmp = log(exp(((0.011111111111111112 * ((double) M_PI)) * (t_1 * fabs(b)))));
	} else {
		tmp = 0.011111111111111112 * (fabs(angle) * log(pow(exp(((double) M_PI)), (t_0 * (fabs(b) + fabs(a))))));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - Math.abs(a);
	double t_1 = Math.abs(angle) * (Math.abs(a) + Math.abs(b));
	double tmp;
	if (Math.abs(angle) <= 5e+121) {
		tmp = (t_1 * t_0) * (Math.PI * 0.011111111111111112);
	} else if (Math.abs(angle) <= 1.85e+194) {
		tmp = Math.log(Math.exp(((0.011111111111111112 * Math.PI) * (t_1 * Math.abs(b)))));
	} else {
		tmp = 0.011111111111111112 * (Math.abs(angle) * Math.log(Math.pow(Math.exp(Math.PI), (t_0 * (Math.abs(b) + Math.abs(a))))));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	t_1 = math.fabs(angle) * (math.fabs(a) + math.fabs(b))
	tmp = 0
	if math.fabs(angle) <= 5e+121:
		tmp = (t_1 * t_0) * (math.pi * 0.011111111111111112)
	elif math.fabs(angle) <= 1.85e+194:
		tmp = math.log(math.exp(((0.011111111111111112 * math.pi) * (t_1 * math.fabs(b)))))
	else:
		tmp = 0.011111111111111112 * (math.fabs(angle) * math.log(math.pow(math.exp(math.pi), (t_0 * (math.fabs(b) + math.fabs(a))))))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	t_1 = Float64(abs(angle) * Float64(abs(a) + abs(b)))
	tmp = 0.0
	if (abs(angle) <= 5e+121)
		tmp = Float64(Float64(t_1 * t_0) * Float64(pi * 0.011111111111111112));
	elseif (abs(angle) <= 1.85e+194)
		tmp = log(exp(Float64(Float64(0.011111111111111112 * pi) * Float64(t_1 * abs(b)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(abs(angle) * log((exp(pi) ^ Float64(t_0 * Float64(abs(b) + abs(a)))))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	t_1 = abs(angle) * (abs(a) + abs(b));
	tmp = 0.0;
	if (abs(angle) <= 5e+121)
		tmp = (t_1 * t_0) * (pi * 0.011111111111111112);
	elseif (abs(angle) <= 1.85e+194)
		tmp = log(exp(((0.011111111111111112 * pi) * (t_1 * abs(b)))));
	else
		tmp = 0.011111111111111112 * (abs(angle) * log((exp(pi) ^ (t_0 * (abs(b) + abs(a))))));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * 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[angle], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $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+121], N[(N[(t$95$1 * t$95$0), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 1.85e+194], N[Log[N[Exp[N[(N[(0.011111111111111112 * Pi), $MachinePrecision] * N[(t$95$1 * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[Log[N[Power[N[Exp[Pi], $MachinePrecision], N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 5.00000000000000007e121

   1. Initial program 54.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

      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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

      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 54.8%

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

   6. Applied rewrites 54.8%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-*.f64 62.8%

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

   8. Applied rewrites 62.8%

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

   if 5.00000000000000007e121 < angle < 1.8500000000000001e194

   1. Initial program 54.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

      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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

      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 54.8%

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

   6. Applied rewrites 54.8%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lift-PI.f64 N/A

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

       5. add-log-exp N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \left(b + a\right)\right)\right)\right) \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right) \]

       6. log-pow-rev N/A

          \[\leadsto \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \left(b + a\right)\right)\right)\right)}\right) \]

       7. lower-log.f64 N/A

          \[\leadsto \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \left(b + a\right)\right)\right)\right)}\right) \]

       8. lift-PI.f64 N/A

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

       9. pow-exp N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lower-exp.f64 31.3%

           \[\leadsto \log \left(e^{0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot \left(b + a\right)\right)\right) \cdot \pi\right)}\right) \]

   11. Applied rewrites 31.3%

       \[\leadsto \log \left(e^{\left(0.011111111111111112 \cdot \pi\right) \cdot \left(\left(angle \cdot \left(a + b\right)\right) \cdot b\right)}\right) \]

   if 1.8500000000000001e194 < angle

   1. Initial program 54.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

   4. Applied rewrites 51.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-log-exp N/A

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

      5. log-pow-rev N/A

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

      6. lower-log.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-exp.f64 32.1%

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

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

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift--.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 35.7%

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 35.7%

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

   6. Applied rewrites 35.7%

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

Alternative 7: 64.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ t_1 := \left|angle\right| \cdot \left(\left|a\right| + \left|b\right|\right)\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 5 \cdot 10^{+121}:\\ \;\;\;\;\left(t\_1 \cdot t\_0\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;\left|angle\right| \leq 1.95 \cdot 10^{+194}:\\ \;\;\;\;\log \left(e^{\left(0.011111111111111112 \cdot \pi\right) \cdot \left(t\_1 \cdot \left|b\right|\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left|angle\right| \cdot \left(t\_0 \cdot \left(\left|b\right| + \left|a\right|\right)\right)\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (fabs b) (fabs a)))
        (t_1 (* (fabs angle) (+ (fabs a) (fabs b)))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 5e+121)
      (* (* t_1 t_0) (* PI 0.011111111111111112))
      (if (<= (fabs angle) 1.95e+194)
        (log (exp (* (* 0.011111111111111112 PI) (* t_1 (fabs b)))))
        (*
         0.011111111111111112
         (* (* (fabs angle) (* t_0 (+ (fabs b) (fabs a)))) PI)))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double t_1 = fabs(angle) * (fabs(a) + fabs(b));
	double tmp;
	if (fabs(angle) <= 5e+121) {
		tmp = (t_1 * t_0) * (((double) M_PI) * 0.011111111111111112);
	} else if (fabs(angle) <= 1.95e+194) {
		tmp = log(exp(((0.011111111111111112 * ((double) M_PI)) * (t_1 * fabs(b)))));
	} else {
		tmp = 0.011111111111111112 * ((fabs(angle) * (t_0 * (fabs(b) + fabs(a)))) * ((double) M_PI));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - Math.abs(a);
	double t_1 = Math.abs(angle) * (Math.abs(a) + Math.abs(b));
	double tmp;
	if (Math.abs(angle) <= 5e+121) {
		tmp = (t_1 * t_0) * (Math.PI * 0.011111111111111112);
	} else if (Math.abs(angle) <= 1.95e+194) {
		tmp = Math.log(Math.exp(((0.011111111111111112 * Math.PI) * (t_1 * Math.abs(b)))));
	} else {
		tmp = 0.011111111111111112 * ((Math.abs(angle) * (t_0 * (Math.abs(b) + Math.abs(a)))) * Math.PI);
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	t_1 = math.fabs(angle) * (math.fabs(a) + math.fabs(b))
	tmp = 0
	if math.fabs(angle) <= 5e+121:
		tmp = (t_1 * t_0) * (math.pi * 0.011111111111111112)
	elif math.fabs(angle) <= 1.95e+194:
		tmp = math.log(math.exp(((0.011111111111111112 * math.pi) * (t_1 * math.fabs(b)))))
	else:
		tmp = 0.011111111111111112 * ((math.fabs(angle) * (t_0 * (math.fabs(b) + math.fabs(a)))) * math.pi)
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	t_1 = Float64(abs(angle) * Float64(abs(a) + abs(b)))
	tmp = 0.0
	if (abs(angle) <= 5e+121)
		tmp = Float64(Float64(t_1 * t_0) * Float64(pi * 0.011111111111111112));
	elseif (abs(angle) <= 1.95e+194)
		tmp = log(exp(Float64(Float64(0.011111111111111112 * pi) * Float64(t_1 * abs(b)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(abs(angle) * Float64(t_0 * Float64(abs(b) + abs(a)))) * pi));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	t_1 = abs(angle) * (abs(a) + abs(b));
	tmp = 0.0;
	if (abs(angle) <= 5e+121)
		tmp = (t_1 * t_0) * (pi * 0.011111111111111112);
	elseif (abs(angle) <= 1.95e+194)
		tmp = log(exp(((0.011111111111111112 * pi) * (t_1 * abs(b)))));
	else
		tmp = 0.011111111111111112 * ((abs(angle) * (t_0 * (abs(b) + abs(a)))) * pi);
	end
	tmp_2 = (sign(angle) * abs(1.0)) * 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[angle], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $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+121], N[(N[(t$95$1 * t$95$0), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[angle], $MachinePrecision], 1.95e+194], N[Log[N[Exp[N[(N[(0.011111111111111112 * Pi), $MachinePrecision] * N[(t$95$1 * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[Abs[angle], $MachinePrecision] * N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 5.00000000000000007e121

   1. Initial program 54.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

      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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

      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 54.8%

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

   6. Applied rewrites 54.8%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-*.f64 62.8%

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

   8. Applied rewrites 62.8%

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

   if 5.00000000000000007e121 < angle < 1.95000000000000008e194

   1. Initial program 54.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

      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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

      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 54.8%

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

   6. Applied rewrites 54.8%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lift-PI.f64 N/A

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

       5. add-log-exp N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \left(b + a\right)\right)\right)\right) \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right) \]

       6. log-pow-rev N/A

          \[\leadsto \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \left(b + a\right)\right)\right)\right)}\right) \]

       7. lower-log.f64 N/A

          \[\leadsto \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \left(b + a\right)\right)\right)\right)}\right) \]

       8. lift-PI.f64 N/A

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

       9. pow-exp N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lower-exp.f64 31.3%

           \[\leadsto \log \left(e^{0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot \left(b + a\right)\right)\right) \cdot \pi\right)}\right) \]

   11. Applied rewrites 31.3%

       \[\leadsto \log \left(e^{\left(0.011111111111111112 \cdot \pi\right) \cdot \left(\left(angle \cdot \left(a + b\right)\right) \cdot b\right)}\right) \]

   if 1.95000000000000008e194 < angle

   1. Initial program 54.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

      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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

      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 54.8%

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

   6. Applied rewrites 54.8%

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

Alternative 8: 62.9% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 51.2%

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

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 51.2%

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

   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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

   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 54.8%

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

6. Applied rewrites 54.8%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lower-*.f64 62.8%

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

8. Applied rewrites 62.8%

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

Alternative 9: 62.8% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 51.2%

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

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 51.2%

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

   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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

   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 54.8%

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

6. Applied rewrites 54.8%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
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. *-commutative N/A

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

   12. lower-*.f64 62.9%

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 62.9%

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

8. Applied rewrites 62.9%

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

Alternative 10: 58.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq 10^{+213}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot \left|angle\right|\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|angle\right| \cdot \left(a + b\right)\right) \cdot b\right) \cdot \left(0.011111111111111112 \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ (fabs angle) 180.0))))
   (*
    (copysign 1.0 angle)
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         1e+213)
      (* (* (* 0.011111111111111112 (fabs angle)) (* (- b a) (+ b a))) PI)
      (* (* (* (fabs angle) (+ a b)) b) (* 0.011111111111111112 PI))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (fabs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+213) {
		tmp = ((0.011111111111111112 * fabs(angle)) * ((b - a) * (b + a))) * ((double) M_PI);
	} else {
		tmp = ((fabs(angle) * (a + b)) * b) * (0.011111111111111112 * ((double) M_PI));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (Math.abs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0)) <= 1e+213) {
		tmp = ((0.011111111111111112 * Math.abs(angle)) * ((b - a) * (b + a))) * Math.PI;
	} else {
		tmp = ((Math.abs(angle) * (a + b)) * b) * (0.011111111111111112 * Math.PI);
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (math.fabs(angle) / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)) <= 1e+213:
		tmp = ((0.011111111111111112 * math.fabs(angle)) * ((b - a) * (b + a))) * math.pi
	else:
		tmp = ((math.fabs(angle) * (a + b)) * b) * (0.011111111111111112 * math.pi)
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(abs(angle) / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+213)
		tmp = Float64(Float64(Float64(0.011111111111111112 * abs(angle)) * Float64(Float64(b - a) * Float64(b + a))) * pi);
	else
		tmp = Float64(Float64(Float64(abs(angle) * Float64(a + b)) * b) * Float64(0.011111111111111112 * pi));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (abs(angle) / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+213)
		tmp = ((0.011111111111111112 * abs(angle)) * ((b - a) * (b + a))) * pi;
	else
		tmp = ((abs(angle) * (a + b)) * b) * (0.011111111111111112 * pi);
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[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], 1e+213], N[(N[(N[(0.011111111111111112 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision], N[(N[(N[(N[Abs[angle], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 9.99999999999999984e212

   1. Initial program 54.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

   4. Applied rewrites 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 51.2%

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

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

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift--.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 54.8%

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 54.8%

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

   6. Applied rewrites 54.8%

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

   if 9.99999999999999984e212 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 54.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

      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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

      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 54.8%

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

   6. Applied rewrites 54.8%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-+.f64 N/A

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

       13. +-commutative N/A

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

       14. lift-+.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 41.5%

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

   11. Applied rewrites 41.5%

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

Alternative 11: 58.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq 10^{+213}:\\ \;\;\;\;\left(\pi \cdot \left|angle\right|\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|angle\right| \cdot \left(a + b\right)\right) \cdot b\right) \cdot \left(0.011111111111111112 \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ (fabs angle) 180.0))))
   (*
    (copysign 1.0 angle)
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         1e+213)
      (* (* PI (fabs angle)) (* (* (- b a) (+ b a)) 0.011111111111111112))
      (* (* (* (fabs angle) (+ a b)) b) (* 0.011111111111111112 PI))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (fabs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+213) {
		tmp = (((double) M_PI) * fabs(angle)) * (((b - a) * (b + a)) * 0.011111111111111112);
	} else {
		tmp = ((fabs(angle) * (a + b)) * b) * (0.011111111111111112 * ((double) M_PI));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (Math.abs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0)) <= 1e+213) {
		tmp = (Math.PI * Math.abs(angle)) * (((b - a) * (b + a)) * 0.011111111111111112);
	} else {
		tmp = ((Math.abs(angle) * (a + b)) * b) * (0.011111111111111112 * Math.PI);
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (math.fabs(angle) / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)) <= 1e+213:
		tmp = (math.pi * math.fabs(angle)) * (((b - a) * (b + a)) * 0.011111111111111112)
	else:
		tmp = ((math.fabs(angle) * (a + b)) * b) * (0.011111111111111112 * math.pi)
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(abs(angle) / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+213)
		tmp = Float64(Float64(pi * abs(angle)) * Float64(Float64(Float64(b - a) * Float64(b + a)) * 0.011111111111111112));
	else
		tmp = Float64(Float64(Float64(abs(angle) * Float64(a + b)) * b) * Float64(0.011111111111111112 * pi));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (abs(angle) / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+213)
		tmp = (pi * abs(angle)) * (((b - a) * (b + a)) * 0.011111111111111112);
	else
		tmp = ((abs(angle) * (a + b)) * b) * (0.011111111111111112 * pi);
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[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], 1e+213], N[(N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[angle], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 9.99999999999999984e212

   1. Initial program 54.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

   4. Applied rewrites 51.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 51.3%

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

   6. Applied rewrites 54.8%

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

   if 9.99999999999999984e212 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 54.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

      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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

      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 54.8%

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

   6. Applied rewrites 54.8%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-+.f64 N/A

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

       13. +-commutative N/A

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

       14. lift-+.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 41.5%

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

   11. Applied rewrites 41.5%

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

Alternative 12: 58.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq 10^{+213}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left|angle\right| \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|angle\right| \cdot \left(a + b\right)\right) \cdot b\right) \cdot \left(0.011111111111111112 \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ (fabs angle) 180.0))))
   (*
    (copysign 1.0 angle)
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         1e+213)
      (* 0.011111111111111112 (* (* (fabs angle) (* (- b a) (+ b a))) PI))
      (* (* (* (fabs angle) (+ a b)) b) (* 0.011111111111111112 PI))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (fabs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+213) {
		tmp = 0.011111111111111112 * ((fabs(angle) * ((b - a) * (b + a))) * ((double) M_PI));
	} else {
		tmp = ((fabs(angle) * (a + b)) * b) * (0.011111111111111112 * ((double) M_PI));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (Math.abs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0)) <= 1e+213) {
		tmp = 0.011111111111111112 * ((Math.abs(angle) * ((b - a) * (b + a))) * Math.PI);
	} else {
		tmp = ((Math.abs(angle) * (a + b)) * b) * (0.011111111111111112 * Math.PI);
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (math.fabs(angle) / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)) <= 1e+213:
		tmp = 0.011111111111111112 * ((math.fabs(angle) * ((b - a) * (b + a))) * math.pi)
	else:
		tmp = ((math.fabs(angle) * (a + b)) * b) * (0.011111111111111112 * math.pi)
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(abs(angle) / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+213)
		tmp = Float64(0.011111111111111112 * Float64(Float64(abs(angle) * Float64(Float64(b - a) * Float64(b + a))) * pi));
	else
		tmp = Float64(Float64(Float64(abs(angle) * Float64(a + b)) * b) * Float64(0.011111111111111112 * pi));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (abs(angle) / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+213)
		tmp = 0.011111111111111112 * ((abs(angle) * ((b - a) * (b + a))) * pi);
	else
		tmp = ((abs(angle) * (a + b)) * b) * (0.011111111111111112 * pi);
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[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], 1e+213], N[(0.011111111111111112 * N[(N[(N[Abs[angle], $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[angle], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 9.99999999999999984e212

   1. Initial program 54.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

      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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

      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 54.8%

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

   6. Applied rewrites 54.8%

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

   if 9.99999999999999984e212 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 54.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 51.2%

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

      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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

      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 54.8%

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

   6. Applied rewrites 54.8%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-+.f64 N/A

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

       13. +-commutative N/A

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

       14. lift-+.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 41.5%

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

   11. Applied rewrites 41.5%

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

Alternative 13: 41.5% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 51.2%

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

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 51.2%

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

   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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

   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 54.8%

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

6. Applied rewrites 54.8%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 37.5%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. associate-*l* N/A

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

    5. lower-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. associate-*r* N/A

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

    10. lower-*.f64 N/A

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

    11. lower-*.f64 N/A

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

    12. lift-+.f64 N/A

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

    13. +-commutative N/A

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

    14. lift-+.f64 N/A

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

    15. *-commutative N/A

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

    16. lower-*.f64 41.5%

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

11. Applied rewrites 41.5%

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

Alternative 14: 41.5% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 51.2%

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

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 51.2%

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

   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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

   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 54.8%

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

6. Applied rewrites 54.8%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 37.5%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. lower-*.f64 N/A

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

    6. lower-*.f64 41.5%

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

    7. lift-+.f64 N/A

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

    8. +-commutative N/A

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

    9. lift-+.f64 41.5%

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

11. Applied rewrites 41.5%

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

Alternative 15: 37.4% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 51.2%

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

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 51.2%

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

   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. unpow2 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. unpow2 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. lift-*.f64 54.8%

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

   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 54.8%

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

6. Applied rewrites 54.8%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 37.5%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025187 
(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)))))