ab-angle->ABCF B

Percentage Accurate: 53.3% → 67.3%

Time: 9.5s

Alternatives: 20

Speedup: 5.5×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right)\\ t_1 := 2 \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.8 \cdot 10^{+142}:\\ \;\;\;\;\left(a + b\right) \cdot \left(t\_1 \cdot \frac{t\_0 + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle\_m, -0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)}{2}\right)\\ \mathbf{elif}\;angle\_m \leq 2.1 \cdot 10^{+286}:\\ \;\;\;\;\left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(b, b, a \cdot a\right)\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(t\_1 \cdot \frac{t\_0 + \sin \left(-1 \cdot \left(angle\_m \cdot \mathsf{fma}\left(-0.005555555555555556, \pi, 0.005555555555555556 \cdot \pi\right)\right)\right)}{2}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* (* 0.011111111111111112 angle_m) PI)))
        (t_1 (* 2.0 (- b a))))
   (*
    angle_s
    (if (<= angle_m 2.8e+142)
      (*
       (+ a b)
       (*
        t_1
        (/
         (+
          t_0
          (sin
           (fma
            (* PI 0.005555555555555556)
            angle_m
            (* -0.005555555555555556 (* PI angle_m)))))
         2.0)))
      (if (<= angle_m 2.1e+286)
        (*
         (*
          (* (sin (* PI (* 0.005555555555555556 angle_m))) 2.0)
          (fma b b (* a a)))
         (cos (/ PI (/ 180.0 angle_m))))
        (*
         (+ a b)
         (*
          t_1
          (/
           (+
            t_0
            (sin
             (*
              -1.0
              (*
               angle_m
               (fma -0.005555555555555556 PI (* 0.005555555555555556 PI))))))
           2.0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin(((0.011111111111111112 * angle_m) * ((double) M_PI)));
	double t_1 = 2.0 * (b - a);
	double tmp;
	if (angle_m <= 2.8e+142) {
		tmp = (a + b) * (t_1 * ((t_0 + sin(fma((((double) M_PI) * 0.005555555555555556), angle_m, (-0.005555555555555556 * (((double) M_PI) * angle_m))))) / 2.0));
	} else if (angle_m <= 2.1e+286) {
		tmp = ((sin((((double) M_PI) * (0.005555555555555556 * angle_m))) * 2.0) * fma(b, b, (a * a))) * cos((((double) M_PI) / (180.0 / angle_m)));
	} else {
		tmp = (a + b) * (t_1 * ((t_0 + sin((-1.0 * (angle_m * fma(-0.005555555555555556, ((double) M_PI), (0.005555555555555556 * ((double) M_PI))))))) / 2.0));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(Float64(0.011111111111111112 * angle_m) * pi))
	t_1 = Float64(2.0 * Float64(b - a))
	tmp = 0.0
	if (angle_m <= 2.8e+142)
		tmp = Float64(Float64(a + b) * Float64(t_1 * Float64(Float64(t_0 + sin(fma(Float64(pi * 0.005555555555555556), angle_m, Float64(-0.005555555555555556 * Float64(pi * angle_m))))) / 2.0)));
	elseif (angle_m <= 2.1e+286)
		tmp = Float64(Float64(Float64(sin(Float64(pi * Float64(0.005555555555555556 * angle_m))) * 2.0) * fma(b, b, Float64(a * a))) * cos(Float64(pi / Float64(180.0 / angle_m))));
	else
		tmp = Float64(Float64(a + b) * Float64(t_1 * Float64(Float64(t_0 + sin(Float64(-1.0 * Float64(angle_m * fma(-0.005555555555555556, pi, Float64(0.005555555555555556 * pi)))))) / 2.0)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 2.8e+142], N[(N[(a + b), $MachinePrecision] * N[(t$95$1 * N[(N[(t$95$0 + N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m + N[(-0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 2.1e+286], N[(N[(N[(N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(t$95$1 * N[(N[(t$95$0 + N[Sin[N[(-1.0 * N[(angle$95$m * N[(-0.005555555555555556 * Pi + N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 2.8e142

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \color{blue}{\frac{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle, -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{2}}\right) \]

   if 2.8e142 < angle < 2.1000000000000001e286

   1. Initial program 53.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 53.8

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

   3. Applied rewrites 53.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 53.5

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

   5. Applied rewrites 53.5%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. sqr-neg-rev N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. pow2 N/A

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

      7. pow-to-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-log.f64 26.2

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

   7. Applied rewrites 26.2%

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

   9. Applied rewrites 40.1%

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

   if 2.1000000000000001e286 < angle

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \color{blue}{\frac{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle, -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{2}}\right) \]

   8. Taylor expanded in angle around -inf

      \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \frac{\sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) + \sin \color{blue}{\left(-1 \cdot \left(angle \cdot \left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}{2}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \frac{\sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) + \sin \left(-1 \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \frac{\sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) + \sin \left(-1 \cdot \left(angle \cdot \color{blue}{\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{2}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \frac{\sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) + \sin \left(-1 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{180}, \color{blue}{\mathsf{PI}\left(\right)}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \frac{\sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) + \sin \left(-1 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{180}, \pi, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \frac{\sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) + \sin \left(-1 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{180}, \pi, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2}\right) \]

      6. lower-PI.f64 66.7

         \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \frac{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) + \sin \left(-1 \cdot \left(angle \cdot \mathsf{fma}\left(-0.005555555555555556, \pi, 0.005555555555555556 \cdot \pi\right)\right)\right)}{2}\right) \]

   10. Applied rewrites 66.7%

       \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \frac{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) + \sin \color{blue}{\left(-1 \cdot \left(angle \cdot \mathsf{fma}\left(-0.005555555555555556, \pi, 0.005555555555555556 \cdot \pi\right)\right)\right)}}{2}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 67.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\pi \cdot angle\_m, 0.011111111111111112, 0\right) \cdot 0.5\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 8.5 \cdot 10^{+158}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \frac{\mathsf{fma}\left(2, \left(\cos t\_0 \cdot \sin t\_0\right) \cdot 0.5, 0.5 \cdot \left(\sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right) - \sin 0\right)\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle\_m, -0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\mathsf{fma}\left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \sqrt{\pi}, \sqrt{\pi}, 0.5 \cdot \pi\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (fma (* PI angle_m) 0.011111111111111112 0.0) 0.5)))
   (*
    angle_s
    (if (<= angle_m 8.5e+158)
      (*
       (+ a b)
       (*
        (* 2.0 (- b a))
        (/
         (+
          (fma
           2.0
           (* (* (cos t_0) (sin t_0)) 0.5)
           (* 0.5 (- (sin (* PI (* angle_m 0.011111111111111112))) (sin 0.0))))
          (sin
           (fma
            (* PI 0.005555555555555556)
            angle_m
            (* -0.005555555555555556 (* PI angle_m)))))
         2.0)))
      (*
       (*
        (sin
         (fma
          (* (* 0.005555555555555556 angle_m) (sqrt PI))
          (sqrt PI)
          (* 0.5 PI)))
        (* (* (+ b a) (- b a)) 2.0))
       (sin (* (* 0.005555555555555556 angle_m) PI)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = fma((((double) M_PI) * angle_m), 0.011111111111111112, 0.0) * 0.5;
	double tmp;
	if (angle_m <= 8.5e+158) {
		tmp = (a + b) * ((2.0 * (b - a)) * ((fma(2.0, ((cos(t_0) * sin(t_0)) * 0.5), (0.5 * (sin((((double) M_PI) * (angle_m * 0.011111111111111112))) - sin(0.0)))) + sin(fma((((double) M_PI) * 0.005555555555555556), angle_m, (-0.005555555555555556 * (((double) M_PI) * angle_m))))) / 2.0));
	} else {
		tmp = (sin(fma(((0.005555555555555556 * angle_m) * sqrt(((double) M_PI))), sqrt(((double) M_PI)), (0.5 * ((double) M_PI)))) * (((b + a) * (b - a)) * 2.0)) * sin(((0.005555555555555556 * angle_m) * ((double) M_PI)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(fma(Float64(pi * angle_m), 0.011111111111111112, 0.0) * 0.5)
	tmp = 0.0
	if (angle_m <= 8.5e+158)
		tmp = Float64(Float64(a + b) * Float64(Float64(2.0 * Float64(b - a)) * Float64(Float64(fma(2.0, Float64(Float64(cos(t_0) * sin(t_0)) * 0.5), Float64(0.5 * Float64(sin(Float64(pi * Float64(angle_m * 0.011111111111111112))) - sin(0.0)))) + sin(fma(Float64(pi * 0.005555555555555556), angle_m, Float64(-0.005555555555555556 * Float64(pi * angle_m))))) / 2.0)));
	else
		tmp = Float64(Float64(sin(fma(Float64(Float64(0.005555555555555556 * angle_m) * sqrt(pi)), sqrt(pi), Float64(0.5 * pi))) * Float64(Float64(Float64(b + a) * Float64(b - a)) * 2.0)) * sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112 + 0.0), $MachinePrecision] * 0.5), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 8.5e+158], N[(N[(a + b), $MachinePrecision] * N[(N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * N[(N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + N[(0.5 * N[(N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Sin[0.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m + N[(-0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[N[(N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 8.49999999999999978e158

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \color{blue}{\frac{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle, -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{2}}\right) \]

   9. Applied rewrites 66.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \left(\cos \left(\mathsf{fma}\left(\pi \cdot angle, 0.011111111111111112, 0\right) \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.011111111111111112, 0\right) \cdot 0.5\right)\right) \cdot 0.5, 0.5 \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) - \sin 0\right)\right)} + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle, -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{2}\right) \]

   if 8.49999999999999978e158 < angle

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. cos-fabs-rev N/A

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

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

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

      12. lower-sin.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. fabs-mul N/A

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

      16. lift-PI.f64 N/A

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

      17. add-exp-log N/A

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

      18. exp-fabs N/A

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

      19. add-exp-log N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-fma.f64 N/A

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

   5. Applied rewrites 57.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

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

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

      8. sqrt-prod N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lower-sqrt.f64 57.0

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 57.0

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

   7. Applied rewrites 57.0%

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

Alternative 3: 67.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 8.5 \cdot 10^{+158}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \frac{\sin \left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle\_m, -0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\mathsf{fma}\left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \sqrt{\pi}, \sqrt{\pi}, 0.5 \cdot \pi\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 8.5e+158)
    (*
     (+ a b)
     (*
      (* 2.0 (- b a))
      (/
       (+
        (sin (* (* 0.011111111111111112 angle_m) PI))
        (sin
         (fma
          (* PI 0.005555555555555556)
          angle_m
          (* -0.005555555555555556 (* PI angle_m)))))
       2.0)))
    (*
     (*
      (sin
       (fma
        (* (* 0.005555555555555556 angle_m) (sqrt PI))
        (sqrt PI)
        (* 0.5 PI)))
      (* (* (+ b a) (- b a)) 2.0))
     (sin (* (* 0.005555555555555556 angle_m) PI))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 8.5e+158) {
		tmp = (a + b) * ((2.0 * (b - a)) * ((sin(((0.011111111111111112 * angle_m) * ((double) M_PI))) + sin(fma((((double) M_PI) * 0.005555555555555556), angle_m, (-0.005555555555555556 * (((double) M_PI) * angle_m))))) / 2.0));
	} else {
		tmp = (sin(fma(((0.005555555555555556 * angle_m) * sqrt(((double) M_PI))), sqrt(((double) M_PI)), (0.5 * ((double) M_PI)))) * (((b + a) * (b - a)) * 2.0)) * sin(((0.005555555555555556 * angle_m) * ((double) M_PI)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 8.5e+158)
		tmp = Float64(Float64(a + b) * Float64(Float64(2.0 * Float64(b - a)) * Float64(Float64(sin(Float64(Float64(0.011111111111111112 * angle_m) * pi)) + sin(fma(Float64(pi * 0.005555555555555556), angle_m, Float64(-0.005555555555555556 * Float64(pi * angle_m))))) / 2.0)));
	else
		tmp = Float64(Float64(sin(fma(Float64(Float64(0.005555555555555556 * angle_m) * sqrt(pi)), sqrt(pi), Float64(0.5 * pi))) * Float64(Float64(Float64(b + a) * Float64(b - a)) * 2.0)) * sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 8.5e+158], N[(N[(a + b), $MachinePrecision] * N[(N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m + N[(-0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[N[(N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 8.49999999999999978e158

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \color{blue}{\frac{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle, -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{2}}\right) \]

   if 8.49999999999999978e158 < angle

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. cos-fabs-rev N/A

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

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

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

      12. lower-sin.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. fabs-mul N/A

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

      16. lift-PI.f64 N/A

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

      17. add-exp-log N/A

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

      18. exp-fabs N/A

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

      19. add-exp-log N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-fma.f64 N/A

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

   5. Applied rewrites 57.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

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

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

      8. sqrt-prod N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lower-sqrt.f64 57.0

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 57.0

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

   7. Applied rewrites 57.0%

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

Alternative 4: 67.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.7 \cdot 10^{+102}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) + \sin \left(\mathsf{fma}\left(-0.005555555555555556, angle\_m \cdot \pi, 0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, a \cdot a\right) \cdot \sin \left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 1.7e+102)
    (*
     (+ a b)
     (*
      (+
       (sin (* 0.011111111111111112 (* angle_m PI)))
       (sin
        (fma
         -0.005555555555555556
         (* angle_m PI)
         (* 0.005555555555555556 (* angle_m PI)))))
      (- b a)))
    (* (fma b b (* a a)) (sin (* (* 0.011111111111111112 angle_m) PI))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.7e+102) {
		tmp = (a + b) * ((sin((0.011111111111111112 * (angle_m * ((double) M_PI)))) + sin(fma(-0.005555555555555556, (angle_m * ((double) M_PI)), (0.005555555555555556 * (angle_m * ((double) M_PI)))))) * (b - a));
	} else {
		tmp = fma(b, b, (a * a)) * sin(((0.011111111111111112 * angle_m) * ((double) M_PI)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 1.7e+102)
		tmp = Float64(Float64(a + b) * Float64(Float64(sin(Float64(0.011111111111111112 * Float64(angle_m * pi))) + sin(fma(-0.005555555555555556, Float64(angle_m * pi), Float64(0.005555555555555556 * Float64(angle_m * pi))))) * Float64(b - a)));
	else
		tmp = Float64(fma(b, b, Float64(a * a)) * sin(Float64(Float64(0.011111111111111112 * angle_m) * pi)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1.7e+102], N[(N[(a + b), $MachinePrecision] * N[(N[(N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(-0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision] + N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.7e102

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \color{blue}{\frac{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle, -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{2}}\right) \]

   8. Taylor expanded in angle around inf

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \sin \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(b - a\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \sin \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   10. Applied rewrites 67.0%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\left(\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) + \sin \left(\mathsf{fma}\left(-0.005555555555555556, angle \cdot \pi, 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \left(b - a\right)\right)} \]

   if 1.7e102 < angle

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 39.9%

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

Alternative 5: 67.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.15 \cdot 10^{+71}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right) \cdot \sin \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot 2\right) \cdot \left(\left(\left(b - a\right) \cdot \cos t\_0\right) \cdot \sin t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle_m))))
   (*
    angle_s
    (if (<= angle_m 1.15e+71)
      (*
       (+ a b)
       (*
        (* 2.0 (- b a))
        (*
         (cos (* -0.005555555555555556 (* PI angle_m)))
         (sin (* (* angle_m 0.005555555555555556) PI)))))
      (* (* (- b a) 2.0) (* (* (- b a) (cos t_0)) (sin t_0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle_m);
	double tmp;
	if (angle_m <= 1.15e+71) {
		tmp = (a + b) * ((2.0 * (b - a)) * (cos((-0.005555555555555556 * (((double) M_PI) * angle_m))) * sin(((angle_m * 0.005555555555555556) * ((double) M_PI)))));
	} else {
		tmp = ((b - a) * 2.0) * (((b - a) * cos(t_0)) * sin(t_0));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (0.005555555555555556 * angle_m);
	double tmp;
	if (angle_m <= 1.15e+71) {
		tmp = (a + b) * ((2.0 * (b - a)) * (Math.cos((-0.005555555555555556 * (Math.PI * angle_m))) * Math.sin(((angle_m * 0.005555555555555556) * Math.PI))));
	} else {
		tmp = ((b - a) * 2.0) * (((b - a) * Math.cos(t_0)) * Math.sin(t_0));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (0.005555555555555556 * angle_m)
	tmp = 0
	if angle_m <= 1.15e+71:
		tmp = (a + b) * ((2.0 * (b - a)) * (math.cos((-0.005555555555555556 * (math.pi * angle_m))) * math.sin(((angle_m * 0.005555555555555556) * math.pi))))
	else:
		tmp = ((b - a) * 2.0) * (((b - a) * math.cos(t_0)) * math.sin(t_0))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle_m))
	tmp = 0.0
	if (angle_m <= 1.15e+71)
		tmp = Float64(Float64(a + b) * Float64(Float64(2.0 * Float64(b - a)) * Float64(cos(Float64(-0.005555555555555556 * Float64(pi * angle_m))) * sin(Float64(Float64(angle_m * 0.005555555555555556) * pi)))));
	else
		tmp = Float64(Float64(Float64(b - a) * 2.0) * Float64(Float64(Float64(b - a) * cos(t_0)) * sin(t_0)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (0.005555555555555556 * angle_m);
	tmp = 0.0;
	if (angle_m <= 1.15e+71)
		tmp = (a + b) * ((2.0 * (b - a)) * (cos((-0.005555555555555556 * (pi * angle_m))) * sin(((angle_m * 0.005555555555555556) * pi))));
	else
		tmp = ((b - a) * 2.0) * (((b - a) * cos(t_0)) * sin(t_0));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 1.15e+71], N[(N[(a + b), $MachinePrecision] * N[(N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(-0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.1500000000000001e71

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. lower-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. mult-flip N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. div-flip N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/r/ N/A

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

      18. associate-*l/ N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. *-commutative N/A

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

      22. *-commutative N/A

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

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

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

      24. *-commutative N/A

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

   7. Applied rewrites 67.0%

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

   if 1.1500000000000001e71 < angle

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 44.8%

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

Alternative 6: 67.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;\left(a + b\right) \cdot \left(angle\_m \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right)\\ \mathbf{elif}\;angle\_m \leq 8.5 \cdot 10^{+69}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(\left(\pi \cdot 0.011111111111111112\right) \cdot angle\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, a \cdot a\right) \cdot \sin \left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 9.5e-7)
    (*
     (+ a b)
     (*
      angle_m
      (*
       (fma
        -0.005555555555555556
        PI
        (fma 0.005555555555555556 PI (* 0.011111111111111112 PI)))
       (- b a))))
    (if (<= angle_m 8.5e+69)
      (* (* (- b a) (+ a b)) (sin (* (* PI 0.011111111111111112) angle_m)))
      (* (fma b b (* a a)) (sin (* (* 0.011111111111111112 angle_m) PI)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 9.5e-7) {
		tmp = (a + b) * (angle_m * (fma(-0.005555555555555556, ((double) M_PI), fma(0.005555555555555556, ((double) M_PI), (0.011111111111111112 * ((double) M_PI)))) * (b - a)));
	} else if (angle_m <= 8.5e+69) {
		tmp = ((b - a) * (a + b)) * sin(((((double) M_PI) * 0.011111111111111112) * angle_m));
	} else {
		tmp = fma(b, b, (a * a)) * sin(((0.011111111111111112 * angle_m) * ((double) M_PI)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 9.5e-7)
		tmp = Float64(Float64(a + b) * Float64(angle_m * Float64(fma(-0.005555555555555556, pi, fma(0.005555555555555556, pi, Float64(0.011111111111111112 * pi))) * Float64(b - a))));
	elseif (angle_m <= 8.5e+69)
		tmp = Float64(Float64(Float64(b - a) * Float64(a + b)) * sin(Float64(Float64(pi * 0.011111111111111112) * angle_m)));
	else
		tmp = Float64(fma(b, b, Float64(a * a)) * sin(Float64(Float64(0.011111111111111112 * angle_m) * pi)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 9.5e-7], N[(N[(a + b), $MachinePrecision] * N[(angle$95$m * N[(N[(-0.005555555555555556 * Pi + N[(0.005555555555555556 * Pi + N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 8.5e+69], N[(N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(Pi * 0.011111111111111112), $MachinePrecision] * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 9.5000000000000001e-7

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \color{blue}{\frac{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle, -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{2}}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \color{blue}{\left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \mathsf{PI}\left(\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b - a\right)\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      6. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      8. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right) \]

      9. lower--.f64 62.2

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   10. Applied rewrites 62.2%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right)} \]

   if 9.5000000000000001e-7 < angle < 8.5000000000000002e69

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 57.1

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

   7. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 57.2

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

   9. Applied rewrites 57.2%

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

   if 8.5000000000000002e69 < angle

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 39.9%

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

Alternative 7: 67.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\left(a + b\right) \cdot \left(angle\_m \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right)\\ \mathbf{elif}\;angle\_m \leq 9 \cdot 10^{+69}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, a \cdot a\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* (* 0.011111111111111112 angle_m) PI))))
   (*
    angle_s
    (if (<= angle_m 8e-7)
      (*
       (+ a b)
       (*
        angle_m
        (*
         (fma
          -0.005555555555555556
          PI
          (fma 0.005555555555555556 PI (* 0.011111111111111112 PI)))
         (- b a))))
      (if (<= angle_m 9e+69)
        (* (* (- b a) (+ a b)) t_0)
        (* (fma b b (* a a)) t_0))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin(((0.011111111111111112 * angle_m) * ((double) M_PI)));
	double tmp;
	if (angle_m <= 8e-7) {
		tmp = (a + b) * (angle_m * (fma(-0.005555555555555556, ((double) M_PI), fma(0.005555555555555556, ((double) M_PI), (0.011111111111111112 * ((double) M_PI)))) * (b - a)));
	} else if (angle_m <= 9e+69) {
		tmp = ((b - a) * (a + b)) * t_0;
	} else {
		tmp = fma(b, b, (a * a)) * t_0;
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(Float64(0.011111111111111112 * angle_m) * pi))
	tmp = 0.0
	if (angle_m <= 8e-7)
		tmp = Float64(Float64(a + b) * Float64(angle_m * Float64(fma(-0.005555555555555556, pi, fma(0.005555555555555556, pi, Float64(0.011111111111111112 * pi))) * Float64(b - a))));
	elseif (angle_m <= 9e+69)
		tmp = Float64(Float64(Float64(b - a) * Float64(a + b)) * t_0);
	else
		tmp = Float64(fma(b, b, Float64(a * a)) * t_0);
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 8e-7], N[(N[(a + b), $MachinePrecision] * N[(angle$95$m * N[(N[(-0.005555555555555556 * Pi + N[(0.005555555555555556 * Pi + N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 9e+69], N[(N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if angle < 7.9999999999999996e-7

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \color{blue}{\frac{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle, -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{2}}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \color{blue}{\left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \mathsf{PI}\left(\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b - a\right)\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      6. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      8. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right) \]

      9. lower--.f64 62.2

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   10. Applied rewrites 62.2%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right)} \]

   if 7.9999999999999996e-7 < angle < 8.9999999999999999e69

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 57.1

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

   7. Applied rewrites 57.1%

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

   if 8.9999999999999999e69 < angle

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 39.9%

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

Alternative 8: 67.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 9 \cdot 10^{+69}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, a \cdot a\right) \cdot \sin \left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 9e+69)
    (* (- b a) (* (sin (* PI (* angle_m 0.011111111111111112))) (+ b a)))
    (* (fma b b (* a a)) (sin (* (* 0.011111111111111112 angle_m) PI))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 9e+69) {
		tmp = (b - a) * (sin((((double) M_PI) * (angle_m * 0.011111111111111112))) * (b + a));
	} else {
		tmp = fma(b, b, (a * a)) * sin(((0.011111111111111112 * angle_m) * ((double) M_PI)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 9e+69)
		tmp = Float64(Float64(b - a) * Float64(sin(Float64(pi * Float64(angle_m * 0.011111111111111112))) * Float64(b + a)));
	else
		tmp = Float64(fma(b, b, Float64(a * a)) * sin(Float64(Float64(0.011111111111111112 * angle_m) * pi)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 9e+69], N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 8.9999999999999999e69

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \color{blue}{\frac{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle, -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{2}}\right) \]

   9. Applied rewrites 66.9%

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

   if 8.9999999999999999e69 < angle

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 39.9%

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

Alternative 9: 66.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 9 \cdot 10^{+69}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot 2\right) \cdot \left(\left(\left(b - a\right) \cdot \cos t\_0\right) \cdot \sin t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle_m))))
   (*
    angle_s
    (if (<= angle_m 9e+69)
      (* (- b a) (* (sin (* PI (* angle_m 0.011111111111111112))) (+ b a)))
      (* (* (- b a) 2.0) (* (* (- b a) (cos t_0)) (sin t_0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle_m);
	double tmp;
	if (angle_m <= 9e+69) {
		tmp = (b - a) * (sin((((double) M_PI) * (angle_m * 0.011111111111111112))) * (b + a));
	} else {
		tmp = ((b - a) * 2.0) * (((b - a) * cos(t_0)) * sin(t_0));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (0.005555555555555556 * angle_m);
	double tmp;
	if (angle_m <= 9e+69) {
		tmp = (b - a) * (Math.sin((Math.PI * (angle_m * 0.011111111111111112))) * (b + a));
	} else {
		tmp = ((b - a) * 2.0) * (((b - a) * Math.cos(t_0)) * Math.sin(t_0));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (0.005555555555555556 * angle_m)
	tmp = 0
	if angle_m <= 9e+69:
		tmp = (b - a) * (math.sin((math.pi * (angle_m * 0.011111111111111112))) * (b + a))
	else:
		tmp = ((b - a) * 2.0) * (((b - a) * math.cos(t_0)) * math.sin(t_0))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle_m))
	tmp = 0.0
	if (angle_m <= 9e+69)
		tmp = Float64(Float64(b - a) * Float64(sin(Float64(pi * Float64(angle_m * 0.011111111111111112))) * Float64(b + a)));
	else
		tmp = Float64(Float64(Float64(b - a) * 2.0) * Float64(Float64(Float64(b - a) * cos(t_0)) * sin(t_0)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (0.005555555555555556 * angle_m);
	tmp = 0.0;
	if (angle_m <= 9e+69)
		tmp = (b - a) * (sin((pi * (angle_m * 0.011111111111111112))) * (b + a));
	else
		tmp = ((b - a) * 2.0) * (((b - a) * cos(t_0)) * sin(t_0));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 9e+69], N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 8.9999999999999999e69

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \color{blue}{\frac{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle, -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{2}}\right) \]

   9. Applied rewrites 66.9%

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

   if 8.9999999999999999e69 < angle

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 44.8%

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

Alternative 10: 66.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 9 \cdot 10^{+69}:\\ \;\;\;\;\left(a + b\right) \cdot \left(t\_0 \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, a \cdot a\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* (* 0.011111111111111112 angle_m) PI))))
   (*
    angle_s
    (if (<= angle_m 9e+69)
      (* (+ a b) (* t_0 (- b a)))
      (* (fma b b (* a a)) t_0)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin(((0.011111111111111112 * angle_m) * ((double) M_PI)));
	double tmp;
	if (angle_m <= 9e+69) {
		tmp = (a + b) * (t_0 * (b - a));
	} else {
		tmp = fma(b, b, (a * a)) * t_0;
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(Float64(0.011111111111111112 * angle_m) * pi))
	tmp = 0.0
	if (angle_m <= 9e+69)
		tmp = Float64(Float64(a + b) * Float64(t_0 * Float64(b - a)));
	else
		tmp = Float64(fma(b, b, Float64(a * a)) * t_0);
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 9e+69], N[(N[(a + b), $MachinePrecision] * N[(t$95$0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 8.9999999999999999e69

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. sin-2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-*.f64 66.9

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

   7. Applied rewrites 66.9%

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

   if 8.9999999999999999e69 < angle

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 39.9%

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

Alternative 11: 66.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 0.01:\\ \;\;\;\;\left(a + b\right) \cdot \left(angle\_m \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(b - a\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 0.01)
    (*
     (+ a b)
     (*
      angle_m
      (*
       (fma
        -0.005555555555555556
        PI
        (fma 0.005555555555555556 PI (* 0.011111111111111112 PI)))
       (- b a))))
    (* (* (sin (* (* 0.011111111111111112 angle_m) PI)) (- b a)) (- b a)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 0.01) {
		tmp = (a + b) * (angle_m * (fma(-0.005555555555555556, ((double) M_PI), fma(0.005555555555555556, ((double) M_PI), (0.011111111111111112 * ((double) M_PI)))) * (b - a)));
	} else {
		tmp = (sin(((0.011111111111111112 * angle_m) * ((double) M_PI))) * (b - a)) * (b - a);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 0.01)
		tmp = Float64(Float64(a + b) * Float64(angle_m * Float64(fma(-0.005555555555555556, pi, fma(0.005555555555555556, pi, Float64(0.011111111111111112 * pi))) * Float64(b - a))));
	else
		tmp = Float64(Float64(sin(Float64(Float64(0.011111111111111112 * angle_m) * pi)) * Float64(b - a)) * Float64(b - a));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 0.01], N[(N[(a + b), $MachinePrecision] * N[(angle$95$m * N[(N[(-0.005555555555555556 * Pi + N[(0.005555555555555556 * Pi + N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 0.0100000000000000002

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \color{blue}{\frac{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle, -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{2}}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \color{blue}{\left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \mathsf{PI}\left(\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b - a\right)\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      6. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      8. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right) \]

      9. lower--.f64 62.2

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   10. Applied rewrites 62.2%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right)} \]

   if 0.0100000000000000002 < angle

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. lift-neg.f64 N/A

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

      7. rem-exp-log N/A

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

      8. lift-log.f64 N/A

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

      9. exp-fabs N/A

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

      10. lift-log.f64 N/A

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

      11. rem-exp-log N/A

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

      12. lift-neg.f64 N/A

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

      13. neg-fabs N/A

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

      14. rem-exp-log N/A

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

      15. exp-fabs N/A

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

      16. rem-exp-log N/A

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

      17. lift--.f64 N/A

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

      18. lower-*.f64 44.8

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

   7. Applied rewrites 44.8%

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

Alternative 12: 66.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 0.01:\\ \;\;\;\;\left(a + b\right) \cdot \left(angle\_m \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, a \cdot a\right) \cdot \sin \left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 0.01)
    (*
     (+ a b)
     (*
      angle_m
      (*
       (fma
        -0.005555555555555556
        PI
        (fma 0.005555555555555556 PI (* 0.011111111111111112 PI)))
       (- b a))))
    (* (fma b b (* a a)) (sin (* (* 0.011111111111111112 angle_m) PI))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 0.01) {
		tmp = (a + b) * (angle_m * (fma(-0.005555555555555556, ((double) M_PI), fma(0.005555555555555556, ((double) M_PI), (0.011111111111111112 * ((double) M_PI)))) * (b - a)));
	} else {
		tmp = fma(b, b, (a * a)) * sin(((0.011111111111111112 * angle_m) * ((double) M_PI)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 0.01)
		tmp = Float64(Float64(a + b) * Float64(angle_m * Float64(fma(-0.005555555555555556, pi, fma(0.005555555555555556, pi, Float64(0.011111111111111112 * pi))) * Float64(b - a))));
	else
		tmp = Float64(fma(b, b, Float64(a * a)) * sin(Float64(Float64(0.011111111111111112 * angle_m) * pi)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 0.01], N[(N[(a + b), $MachinePrecision] * N[(angle$95$m * N[(N[(-0.005555555555555556 * Pi + N[(0.005555555555555556 * Pi + N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 0.0100000000000000002

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \color{blue}{\frac{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle, -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{2}}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \color{blue}{\left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \mathsf{PI}\left(\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b - a\right)\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      6. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      8. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right) \]

      9. lower--.f64 62.2

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   10. Applied rewrites 62.2%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right)} \]

   if 0.0100000000000000002 < angle

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 39.9%

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

Alternative 13: 62.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.5 \cdot 10^{+269}:\\ \;\;\;\;\left(a + b\right) \cdot \left(angle\_m \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(\left(b - a\right) \cdot \left(a - b\right)\right) \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 1.5e+269)
    (*
     (+ a b)
     (*
      angle_m
      (*
       (fma
        -0.005555555555555556
        PI
        (fma 0.005555555555555556 PI (* 0.011111111111111112 PI)))
       (- b a))))
    (* 0.011111111111111112 (* angle_m (* (* (- b a) (- a b)) PI))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.5e+269) {
		tmp = (a + b) * (angle_m * (fma(-0.005555555555555556, ((double) M_PI), fma(0.005555555555555556, ((double) M_PI), (0.011111111111111112 * ((double) M_PI)))) * (b - a)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((b - a) * (a - b)) * ((double) M_PI)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 1.5e+269)
		tmp = Float64(Float64(a + b) * Float64(angle_m * Float64(fma(-0.005555555555555556, pi, fma(0.005555555555555556, pi, Float64(0.011111111111111112 * pi))) * Float64(b - a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(Float64(b - a) * Float64(a - b)) * pi)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1.5e+269], N[(N[(a + b), $MachinePrecision] * N[(angle$95$m * N[(N[(-0.005555555555555556 * Pi + N[(0.005555555555555556 * Pi + N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(N[(b - a), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.5000000000000001e269

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 57.1%

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.6%

      \[\leadsto \left(a + b\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \color{blue}{\frac{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle, -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{2}}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \color{blue}{\left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\left(\frac{-1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \mathsf{PI}\left(\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b - a\right)\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      6. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b - a\right)\right)\right) \]

      8. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(\frac{-1}{180}, \pi, \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{90} \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right) \]

      9. lower--.f64 62.2

         \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   10. Applied rewrites 62.2%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\mathsf{fma}\left(-0.005555555555555556, \pi, \mathsf{fma}\left(0.005555555555555556, \pi, 0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\right)} \]

   if 1.5000000000000001e269 < angle

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.4

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 54.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-to-mult N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

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

      6. rem-exp-log N/A

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

      7. neg-fabs N/A

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

      8. lift-neg.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lift-log.f64 N/A

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

      11. exp-fabs N/A

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

      12. lift-log.f64 N/A

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

      13. rem-exp-log N/A

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

      14. lift-neg.f64 N/A

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

      15. frac-2neg N/A

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

      16. sub-to-mult N/A

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

      17. lower--.f64 38.1

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

   8. Applied rewrites 38.1%

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

Alternative 14: 62.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.5 \cdot 10^{+269}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\_m\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(\left(b - a\right) \cdot \left(a - b\right)\right) \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 1.5e+269)
    (* (* (* (* PI angle_m) (- b a)) (+ a b)) 0.011111111111111112)
    (* 0.011111111111111112 (* angle_m (* (* (- b a) (- a b)) PI))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.5e+269) {
		tmp = (((((double) M_PI) * angle_m) * (b - a)) * (a + b)) * 0.011111111111111112;
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((b - a) * (a - b)) * ((double) M_PI)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.5e+269) {
		tmp = (((Math.PI * angle_m) * (b - a)) * (a + b)) * 0.011111111111111112;
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((b - a) * (a - b)) * Math.PI));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 1.5e+269:
		tmp = (((math.pi * angle_m) * (b - a)) * (a + b)) * 0.011111111111111112
	else:
		tmp = 0.011111111111111112 * (angle_m * (((b - a) * (a - b)) * math.pi))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 1.5e+269)
		tmp = Float64(Float64(Float64(Float64(pi * angle_m) * Float64(b - a)) * Float64(a + b)) * 0.011111111111111112);
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(Float64(b - a) * Float64(a - b)) * pi)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 1.5e+269)
		tmp = (((pi * angle_m) * (b - a)) * (a + b)) * 0.011111111111111112;
	else
		tmp = 0.011111111111111112 * (angle_m * (((b - a) * (a - b)) * pi));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1.5e+269], N[(N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(N[(b - a), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.5000000000000001e269

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.4

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

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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. lower-*.f64 50.4

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

   6. Applied rewrites 62.1%

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

   if 1.5000000000000001e269 < angle

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.4

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 54.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-to-mult N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

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

      6. rem-exp-log N/A

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

      7. neg-fabs N/A

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

      8. lift-neg.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lift-log.f64 N/A

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

      11. exp-fabs N/A

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

      12. lift-log.f64 N/A

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

      13. rem-exp-log N/A

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

      14. lift-neg.f64 N/A

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

      15. frac-2neg N/A

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

      16. sub-to-mult N/A

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

      17. lower--.f64 38.1

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

   8. Applied rewrites 38.1%

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

Alternative 15: 57.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq 0:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(\left(b - a\right) \cdot \left(a - b\right)\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(b - a\right)\right) \cdot \left(b \cdot \pi\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         0.0)
      (* 0.011111111111111112 (* angle_m (* (* (- b a) (- a b)) PI)))
      (* (* (* angle_m 0.011111111111111112) (- b a)) (* b PI))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= 0.0) {
		tmp = 0.011111111111111112 * (angle_m * (((b - a) * (a - b)) * ((double) M_PI)));
	} else {
		tmp = ((angle_m * 0.011111111111111112) * (b - a)) * (b * ((double) M_PI));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m / 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)) <= 0.0) {
		tmp = 0.011111111111111112 * (angle_m * (((b - a) * (a - b)) * Math.PI));
	} else {
		tmp = ((angle_m * 0.011111111111111112) * (b - a)) * (b * Math.PI);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (angle_m / 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)) <= 0.0:
		tmp = 0.011111111111111112 * (angle_m * (((b - a) * (a - b)) * math.pi))
	else:
		tmp = ((angle_m * 0.011111111111111112) * (b - a)) * (b * math.pi)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 0.0)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(Float64(b - a) * Float64(a - b)) * pi)));
	else
		tmp = Float64(Float64(Float64(angle_m * 0.011111111111111112) * Float64(b - a)) * Float64(b * pi));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (angle_m / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 0.0)
		tmp = 0.011111111111111112 * (angle_m * (((b - a) * (a - b)) * pi));
	else
		tmp = ((angle_m * 0.011111111111111112) * (b - a)) * (b * pi);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * 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], 0.0], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(N[(b - a), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b * 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))))) < 0.0

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.4

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 54.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-to-mult N/A

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

      4. rem-exp-log N/A

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

      5. exp-fabs N/A

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

      6. rem-exp-log N/A

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

      7. neg-fabs N/A

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

      8. lift-neg.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lift-log.f64 N/A

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

      11. exp-fabs N/A

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

      12. lift-log.f64 N/A

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

      13. rem-exp-log N/A

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

      14. lift-neg.f64 N/A

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

      15. frac-2neg N/A

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

      16. sub-to-mult N/A

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

      17. lower--.f64 38.1

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

   8. Applied rewrites 38.1%

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

   if 0.0 < (*.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 53.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.4

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 54.0%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 41.3

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

   11. Applied rewrites 41.3%

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

Alternative 16: 42.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq 10^{+138}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot angle\_m\right) \cdot \left(b \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(b - a\right)\right) \cdot \left(b \cdot \pi\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         1e+138)
      (* 0.011111111111111112 (* (* PI angle_m) (* b (- b a))))
      (* (* (* angle_m 0.011111111111111112) (- b a)) (* b PI))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+138) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * angle_m) * (b * (b - a)));
	} else {
		tmp = ((angle_m * 0.011111111111111112) * (b - a)) * (b * ((double) M_PI));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m / 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+138) {
		tmp = 0.011111111111111112 * ((Math.PI * angle_m) * (b * (b - a)));
	} else {
		tmp = ((angle_m * 0.011111111111111112) * (b - a)) * (b * Math.PI);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (angle_m / 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+138:
		tmp = 0.011111111111111112 * ((math.pi * angle_m) * (b * (b - a)))
	else:
		tmp = ((angle_m * 0.011111111111111112) * (b - a)) * (b * math.pi)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 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+138)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * angle_m) * Float64(b * Float64(b - a))));
	else
		tmp = Float64(Float64(Float64(angle_m * 0.011111111111111112) * Float64(b - a)) * Float64(b * pi));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (angle_m / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+138)
		tmp = 0.011111111111111112 * ((pi * angle_m) * (b * (b - a)));
	else
		tmp = ((angle_m * 0.011111111111111112) * (b - a)) * (b * pi);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * 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+138], N[(0.011111111111111112 * N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b * 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))))) < 1e138

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.4

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 54.0%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lower-*.f64 37.1

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 37.1

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

   11. Applied rewrites 37.1%

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

   if 1e138 < (*.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 53.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.4

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 54.0%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 41.3

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

   11. Applied rewrites 41.3%

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

Alternative 17: 41.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq 10^{+81}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot angle\_m\right) \cdot \left(b \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(\left(angle\_m \cdot b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         1e+81)
      (* 0.011111111111111112 (* (* PI angle_m) (* b (- b a))))
      (* (* 0.011111111111111112 (* (* angle_m b) (- b a))) PI)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+81) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * angle_m) * (b * (b - a)));
	} else {
		tmp = (0.011111111111111112 * ((angle_m * b) * (b - a))) * ((double) M_PI);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m / 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+81) {
		tmp = 0.011111111111111112 * ((Math.PI * angle_m) * (b * (b - a)));
	} else {
		tmp = (0.011111111111111112 * ((angle_m * b) * (b - a))) * Math.PI;
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (angle_m / 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+81:
		tmp = 0.011111111111111112 * ((math.pi * angle_m) * (b * (b - a)))
	else:
		tmp = (0.011111111111111112 * ((angle_m * b) * (b - a))) * math.pi
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 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+81)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * angle_m) * Float64(b * Float64(b - a))));
	else
		tmp = Float64(Float64(0.011111111111111112 * Float64(Float64(angle_m * b) * Float64(b - a))) * pi);
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (angle_m / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+81)
		tmp = 0.011111111111111112 * ((pi * angle_m) * (b * (b - a)));
	else
		tmp = (0.011111111111111112 * ((angle_m * b) * (b - a))) * pi;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * 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+81], N[(0.011111111111111112 * N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * N[(N[(angle$95$m * b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $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))))) < 9.99999999999999921e80

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.4

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 54.0%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lower-*.f64 37.1

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 37.1

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

   11. Applied rewrites 37.1%

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

   if 9.99999999999999921e80 < (*.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 53.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.4

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 54.0%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 39.9

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

   11. Applied rewrites 39.9%

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

Alternative 18: 41.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 2 \cdot 10^{+255}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot angle\_m\right) \cdot \left(b \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(angle\_m \cdot b\right) \cdot \left(b - a\right)\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 2e+255)
    (* 0.011111111111111112 (* (* PI angle_m) (* b (- b a))))
    (* 0.011111111111111112 (* (* (* angle_m b) (- b a)) PI)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 2e+255) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * angle_m) * (b * (b - a)));
	} else {
		tmp = 0.011111111111111112 * (((angle_m * b) * (b - a)) * ((double) M_PI));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= 2e+255) {
		tmp = 0.011111111111111112 * ((Math.PI * angle_m) * (b * (b - a)));
	} else {
		tmp = 0.011111111111111112 * (((angle_m * b) * (b - a)) * Math.PI);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= 2e+255:
		tmp = 0.011111111111111112 * ((math.pi * angle_m) * (b * (b - a)))
	else:
		tmp = 0.011111111111111112 * (((angle_m * b) * (b - a)) * math.pi)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 2e+255)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * angle_m) * Float64(b * Float64(b - a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(angle_m * b) * Float64(b - a)) * pi));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= 2e+255)
		tmp = 0.011111111111111112 * ((pi * angle_m) * (b * (b - a)));
	else
		tmp = 0.011111111111111112 * (((angle_m * b) * (b - a)) * pi);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+255], N[(0.011111111111111112 * N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[(angle$95$m * b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.99999999999999998e255

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.4

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 54.0%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lower-*.f64 37.1

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 37.1

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

   11. Applied rewrites 37.1%

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

   if 1.99999999999999998e255 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.4

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 54.0%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 39.9

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

   11. Applied rewrites 39.9%

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

Alternative 19: 37.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(\left(\pi \cdot angle\_m\right) \cdot \left(b \cdot \left(b - a\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* (* PI angle_m) (* b (- b a))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((((double) M_PI) * angle_m) * (b * (b - a))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((Math.PI * angle_m) * (b * (b - a))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * ((math.pi * angle_m) * (b * (b - a))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(Float64(pi * angle_m) * Float64(b * Float64(b - a)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * ((pi * angle_m) * (b * (b - a))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 50.4

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

4. Applied rewrites 50.4%

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

6. Applied rewrites 54.0%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 37.1%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. lift-*.f64 N/A

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

    7. lower-*.f64 37.1

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

    8. lift-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 37.1

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

11. Applied rewrites 37.1%

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

Alternative 20: 37.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(\left(b - a\right) \cdot b\right) \cdot \pi\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* angle_m (* (* (- b a) b) PI)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (((b - a) * b) * ((double) M_PI))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (((b - a) * b) * Math.PI)));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * (((b - a) * b) * math.pi)))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(Float64(b - a) * b) * pi))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * (((b - a) * b) * pi)));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(N[(N[(b - a), $MachinePrecision] * b), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 50.4

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

4. Applied rewrites 50.4%

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

6. Applied rewrites 54.0%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 37.1%

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

Reproduce

herbie shell --seed 2025140 
(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)))))