ab-angle->ABCF B

Percentage Accurate: 53.6% → 67.5%

Time: 30.9s

Alternatives: 15

Speedup: 5.0×

• 33.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.6% 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: 67.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+27}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left|angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right|\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+169}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+273}:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a \cdot \left(-\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -5e+27)
   (*
    (+ a b)
    (* (- b a) (sin (* 2.0 (fabs (* angle (* PI 0.005555555555555556)))))))
   (if (<= (/ angle 180.0) 5e+169)
     (* (+ a b) (* (- b a) (sin (* 2.0 (* (/ angle 180.0) PI)))))
     (if (<= (/ angle 180.0) 1e+273)
       (* (* 2.0 (sin (/ (* angle PI) 180.0))) (* (+ a b) (- b a)))
       (* (+ a b) (* a (- (sin (* PI (* angle 0.011111111111111112))))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -5e+27) {
		tmp = (a + b) * ((b - a) * sin((2.0 * fabs((angle * (((double) M_PI) * 0.005555555555555556))))));
	} else if ((angle / 180.0) <= 5e+169) {
		tmp = (a + b) * ((b - a) * sin((2.0 * ((angle / 180.0) * ((double) M_PI)))));
	} else if ((angle / 180.0) <= 1e+273) {
		tmp = (2.0 * sin(((angle * ((double) M_PI)) / 180.0))) * ((a + b) * (b - a));
	} else {
		tmp = (a + b) * (a * -sin((((double) M_PI) * (angle * 0.011111111111111112))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -5e+27) {
		tmp = (a + b) * ((b - a) * Math.sin((2.0 * Math.abs((angle * (Math.PI * 0.005555555555555556))))));
	} else if ((angle / 180.0) <= 5e+169) {
		tmp = (a + b) * ((b - a) * Math.sin((2.0 * ((angle / 180.0) * Math.PI))));
	} else if ((angle / 180.0) <= 1e+273) {
		tmp = (2.0 * Math.sin(((angle * Math.PI) / 180.0))) * ((a + b) * (b - a));
	} else {
		tmp = (a + b) * (a * -Math.sin((Math.PI * (angle * 0.011111111111111112))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= -5e+27:
		tmp = (a + b) * ((b - a) * math.sin((2.0 * math.fabs((angle * (math.pi * 0.005555555555555556))))))
	elif (angle / 180.0) <= 5e+169:
		tmp = (a + b) * ((b - a) * math.sin((2.0 * ((angle / 180.0) * math.pi))))
	elif (angle / 180.0) <= 1e+273:
		tmp = (2.0 * math.sin(((angle * math.pi) / 180.0))) * ((a + b) * (b - a))
	else:
		tmp = (a + b) * (a * -math.sin((math.pi * (angle * 0.011111111111111112))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+27)
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(2.0 * abs(Float64(angle * Float64(pi * 0.005555555555555556)))))));
	elseif (Float64(angle / 180.0) <= 5e+169)
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(Float64(angle / 180.0) * pi)))));
	elseif (Float64(angle / 180.0) <= 1e+273)
		tmp = Float64(Float64(2.0 * sin(Float64(Float64(angle * pi) / 180.0))) * Float64(Float64(a + b) * Float64(b - a)));
	else
		tmp = Float64(Float64(a + b) * Float64(a * Float64(-sin(Float64(pi * Float64(angle * 0.011111111111111112))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= -5e+27)
		tmp = (a + b) * ((b - a) * sin((2.0 * abs((angle * (pi * 0.005555555555555556))))));
	elseif ((angle / 180.0) <= 5e+169)
		tmp = (a + b) * ((b - a) * sin((2.0 * ((angle / 180.0) * pi))));
	elseif ((angle / 180.0) <= 1e+273)
		tmp = (2.0 * sin(((angle * pi) / 180.0))) * ((a + b) * (b - a));
	else
		tmp = (a + b) * (a * -sin((pi * (angle * 0.011111111111111112))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+27], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[Abs[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+169], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+273], N[(N[(2.0 * N[Sin[N[(N[(angle * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(a * (-N[Sin[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < -4.99999999999999979e27

   1. Initial program 27.5%

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

      1. *-commutative 27.5%

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

      2. associate-*l* 27.5%

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

      3. associate-*l* 27.5%

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

      4. unpow2 27.5%

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

      5. unpow2 27.5%

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

      6. difference-of-squares 27.5%

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

   3. Simplified 27.5%

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

      1. difference-of-squares 27.5%

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

      2. *-commutative 27.5%

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

      3. prod-diff 18.7%

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

      4. fma-neg 18.7%

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

      5. distribute-lft-in 18.7%

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

   5. Applied egg-rr 17.8%

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

      1. *-commutative 17.8%

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

      2. distribute-rgt-out 17.8%

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

      3. *-commutative 17.8%

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

   7. Simplified 25.2%

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

      1. rem-cube-cbrt 29.5%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{{\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}}\right)\right) \]

   9. Applied egg-rr 29.5%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{{\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}}\right)\right) \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 25.2%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 20.8%

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

       4. associate-*r* 20.3%

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

       5. associate-*r* 19.2%

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

       6. swap-sqr 17.1%

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

       7. pow2 17.1%

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

       8. metadata-eval 17.1%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \sqrt{{\left(\pi \cdot angle\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}}}\right)\right) \]

   11. Applied egg-rr 17.1%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\sqrt{{\left(\pi \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}}}\right)\right) \]
   12. Step-by-step derivation

       1. unpow2 17.1%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \sqrt{\color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)\right)} \cdot 3.08641975308642 \cdot 10^{-5}}\right)\right) \]

       2. metadata-eval 17.1%

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

       3. swap-sqr 19.2%

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

       4. associate-*r* 20.3%

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

       5. associate-*r* 20.8%

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

       6. rem-sqrt-square 43.9%

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

       7. *-commutative 43.9%

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

       8. associate-*l* 41.6%

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

       9. *-commutative 41.6%

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

   13. Simplified 41.6%

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

   if -4.99999999999999979e27 < (/.f64 angle 180) < 5.00000000000000017e169

   1. Initial program 74.5%

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

      1. *-commutative 74.5%

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

      2. associate-*l* 74.5%

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

      3. associate-*l* 74.5%

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

      4. unpow2 74.5%

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

      5. unpow2 74.5%

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

      6. difference-of-squares 78.8%

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

   3. Simplified 78.8%

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

      1. difference-of-squares 74.5%

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

      2. *-commutative 74.5%

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

      3. prod-diff 58.9%

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

      4. fma-neg 58.9%

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

      5. distribute-lft-in 58.9%

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

   5. Applied egg-rr 61.8%

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

      1. *-commutative 61.8%

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

      2. distribute-rgt-out 61.8%

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

      3. *-commutative 61.8%

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

   7. Simplified 90.4%

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

      1. metadata-eval 90.4%

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

      2. div-inv 91.1%

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

   9. Applied egg-rr 91.1%

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

   if 5.00000000000000017e169 < (/.f64 angle 180) < 9.99999999999999945e272

   1. Initial program 17.7%

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

      1. *-commutative 17.7%

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

      2. associate-*l* 17.7%

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

      3. associate-*l* 17.7%

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

      4. unpow2 17.7%

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

      5. unpow2 17.7%

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

      6. difference-of-squares 17.7%

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

   3. Simplified 17.7%

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

      1. associate-*r/ 23.1%

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

   5. Applied egg-rr 23.1%

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

   6. Taylor expanded in angle around 0 40.4%

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

   if 9.99999999999999945e272 < (/.f64 angle 180)

   1. Initial program 44.1%

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

      1. *-commutative 44.1%

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

      2. associate-*l* 44.1%

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

      3. associate-*l* 44.1%

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

      4. unpow2 44.1%

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

      5. unpow2 44.1%

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

      6. difference-of-squares 44.1%

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

   3. Simplified 44.1%

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

      1. difference-of-squares 44.1%

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

      2. *-commutative 44.1%

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

      3. prod-diff 4.1%

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

      4. fma-neg 4.1%

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

      5. distribute-lft-in 4.1%

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

   5. Applied egg-rr 4.1%

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

      1. *-commutative 4.1%

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

      2. distribute-rgt-out 4.1%

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

      3. *-commutative 4.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in b around 0 48.0%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(-1 \cdot \left(a \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 48.0%

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

      2. *-commutative 48.0%

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

      3. distribute-rgt-neg-in 48.0%

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

      4. associate-*r* 64.1%

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

      5. *-commutative 64.1%

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

   10. Simplified 64.1%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+27}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left|angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right|\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+169}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+273}:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a \cdot \left(-\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 67.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+79}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left|angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot {\left(\sqrt[3]{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}\right)}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -5e+79)
   (*
    (+ a b)
    (* (- b a) (sin (* 2.0 (fabs (* angle (* PI 0.005555555555555556)))))))
   (*
    (+ a b)
    (*
     (- b a)
     (sin (* 2.0 (pow (cbrt (* 0.005555555555555556 (* angle PI))) 3.0)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -5e+79) {
		tmp = (a + b) * ((b - a) * sin((2.0 * fabs((angle * (((double) M_PI) * 0.005555555555555556))))));
	} else {
		tmp = (a + b) * ((b - a) * sin((2.0 * pow(cbrt((0.005555555555555556 * (angle * ((double) M_PI)))), 3.0))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -5e+79) {
		tmp = (a + b) * ((b - a) * Math.sin((2.0 * Math.abs((angle * (Math.PI * 0.005555555555555556))))));
	} else {
		tmp = (a + b) * ((b - a) * Math.sin((2.0 * Math.pow(Math.cbrt((0.005555555555555556 * (angle * Math.PI))), 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+79)
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(2.0 * abs(Float64(angle * Float64(pi * 0.005555555555555556)))))));
	else
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(2.0 * (cbrt(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+79], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[Abs[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[Power[N[Power[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -5e79

   1. Initial program 27.5%

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

      1. *-commutative 27.5%

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

      2. associate-*l* 27.5%

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

      3. associate-*l* 27.5%

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

      4. unpow2 27.5%

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

      5. unpow2 27.5%

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

      6. difference-of-squares 27.5%

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

   3. Simplified 27.5%

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

      1. difference-of-squares 27.5%

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

      2. *-commutative 27.5%

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

      3. prod-diff 18.6%

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

      4. fma-neg 18.6%

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

      5. distribute-lft-in 18.6%

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

   5. Applied egg-rr 17.5%

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

      1. *-commutative 17.5%

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

      2. distribute-rgt-out 17.5%

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

      3. *-commutative 17.5%

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

   7. Simplified 26.4%

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

      1. rem-cube-cbrt 27.4%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{{\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}}\right)\right) \]

   9. Applied egg-rr 27.4%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{{\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}}\right)\right) \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 26.4%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 16.2%

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

       4. associate-*r* 17.5%

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

       5. associate-*r* 16.1%

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

       6. swap-sqr 14.2%

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

       7. pow2 14.2%

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

       8. metadata-eval 14.2%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \sqrt{{\left(\pi \cdot angle\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}}}\right)\right) \]

   11. Applied egg-rr 14.2%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\sqrt{{\left(\pi \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}}}\right)\right) \]
   12. Step-by-step derivation

       1. unpow2 14.2%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \sqrt{\color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)\right)} \cdot 3.08641975308642 \cdot 10^{-5}}\right)\right) \]

       2. metadata-eval 14.2%

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

       3. swap-sqr 16.1%

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

       4. associate-*r* 17.5%

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

       5. associate-*r* 16.2%

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

       6. rem-sqrt-square 44.3%

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

       7. *-commutative 44.3%

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

       8. associate-*l* 43.6%

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

       9. *-commutative 43.6%

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

   13. Simplified 43.6%

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

   if -5e79 < (/.f64 angle 180)

   1. Initial program 65.8%

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

      1. *-commutative 65.8%

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

      2. associate-*l* 65.8%

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

      3. associate-*l* 65.8%

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

      4. unpow2 65.8%

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

      5. unpow2 65.8%

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

      6. difference-of-squares 69.3%

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

   3. Simplified 69.3%

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

      1. difference-of-squares 65.8%

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

      2. *-commutative 65.8%

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

      3. prod-diff 50.9%

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

      4. fma-neg 50.9%

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

      5. distribute-lft-in 50.9%

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

   5. Applied egg-rr 53.3%

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

      1. *-commutative 53.3%

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

      2. distribute-rgt-out 53.3%

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

      3. *-commutative 53.3%

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

   7. Simplified 78.9%

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

      1. rem-cube-cbrt 80.1%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{{\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}}\right)\right) \]

   9. Applied egg-rr 80.1%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{{\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}}\right)\right) \]

   10. Taylor expanded in angle around 0 80.4%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+79}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left|angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot {\left(\sqrt[3]{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}\right)}^{3}\right)\right)\\ \end{array} \]

Alternative 3: 66.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{+153}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.9e+153)
   (* (+ a b) (* (- b a) (sin (* 2.0 (/ 1.0 (/ 180.0 (* angle PI)))))))
   (* (+ a b) (* (- b a) (* PI (* angle 0.011111111111111112))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.9e+153) {
		tmp = (a + b) * ((b - a) * sin((2.0 * (1.0 / (180.0 / (angle * ((double) M_PI)))))));
	} else {
		tmp = (a + b) * ((b - a) * (((double) M_PI) * (angle * 0.011111111111111112)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.9e+153) {
		tmp = (a + b) * ((b - a) * Math.sin((2.0 * (1.0 / (180.0 / (angle * Math.PI))))));
	} else {
		tmp = (a + b) * ((b - a) * (Math.PI * (angle * 0.011111111111111112)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.9e+153:
		tmp = (a + b) * ((b - a) * math.sin((2.0 * (1.0 / (180.0 / (angle * math.pi))))))
	else:
		tmp = (a + b) * ((b - a) * (math.pi * (angle * 0.011111111111111112)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.9e+153)
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(1.0 / Float64(180.0 / Float64(angle * pi)))))));
	else
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(pi * Float64(angle * 0.011111111111111112))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.9e+153)
		tmp = (a + b) * ((b - a) * sin((2.0 * (1.0 / (180.0 / (angle * pi))))));
	else
		tmp = (a + b) * ((b - a) * (pi * (angle * 0.011111111111111112)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.9e+153], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(1.0 / N[(180.0 / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.89999999999999983e153

   1. Initial program 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-*l* 59.4%

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

      3. associate-*l* 59.4%

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

      4. unpow2 59.4%

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

      5. unpow2 59.4%

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

      6. difference-of-squares 61.1%

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

   3. Simplified 61.1%

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

      1. difference-of-squares 59.4%

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

      2. *-commutative 59.4%

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

      3. prod-diff 44.3%

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

      4. fma-neg 44.3%

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

      5. distribute-lft-in 44.3%

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

   5. Applied egg-rr 44.8%

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

      1. *-commutative 44.8%

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

      2. distribute-rgt-out 44.8%

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

      3. *-commutative 44.8%

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

   7. Simplified 66.4%

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

      1. associate-*r* 66.0%

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

      2. metadata-eval 66.0%

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

      3. div-inv 66.7%

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

      4. clear-num 67.4%

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

   9. Applied egg-rr 67.4%

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

   if 1.89999999999999983e153 < b

   1. Initial program 40.1%

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

      1. *-commutative 40.1%

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

      2. associate-*l* 40.1%

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

      3. associate-*l* 40.1%

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

      4. unpow2 40.1%

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

      5. unpow2 40.1%

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

      6. difference-of-squares 51.8%

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

   3. Simplified 51.8%

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

      1. difference-of-squares 40.1%

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

      2. *-commutative 40.1%

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

      3. prod-diff 40.1%

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

      4. fma-neg 40.1%

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

      5. distribute-lft-in 40.1%

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

   5. Applied egg-rr 51.8%

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

      1. *-commutative 51.8%

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

      2. distribute-rgt-out 51.8%

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

      3. *-commutative 51.8%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in angle around 0 80.6%

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

      1. associate-*r* 80.7%

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

   10. Simplified 80.7%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{+153}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 4: 67.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.9 \cdot 10^{+245}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 6.9e+245)
   (* (+ a b) (* (- b a) (sin (* 2.0 (* (/ angle 180.0) PI)))))
   (* 0.011111111111111112 (* (+ a b) (* (- b a) (* angle PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.9e+245) {
		tmp = (a + b) * ((b - a) * sin((2.0 * ((angle / 180.0) * ((double) M_PI)))));
	} else {
		tmp = 0.011111111111111112 * ((a + b) * ((b - a) * (angle * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.9e+245) {
		tmp = (a + b) * ((b - a) * Math.sin((2.0 * ((angle / 180.0) * Math.PI))));
	} else {
		tmp = 0.011111111111111112 * ((a + b) * ((b - a) * (angle * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 6.9e+245:
		tmp = (a + b) * ((b - a) * math.sin((2.0 * ((angle / 180.0) * math.pi))))
	else:
		tmp = 0.011111111111111112 * ((a + b) * ((b - a) * (angle * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 6.9e+245)
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(Float64(angle / 180.0) * pi)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(angle * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 6.9e+245)
		tmp = (a + b) * ((b - a) * sin((2.0 * ((angle / 180.0) * pi))));
	else
		tmp = 0.011111111111111112 * ((a + b) * ((b - a) * (angle * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 6.9e+245], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.89999999999999983e245

   1. Initial program 58.2%

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

      1. *-commutative 58.2%

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

      2. associate-*l* 58.2%

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

      3. associate-*l* 58.2%

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

      4. unpow2 58.2%

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

      5. unpow2 58.2%

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

      6. difference-of-squares 59.9%

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

   3. Simplified 59.9%

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

      1. difference-of-squares 58.2%

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

      2. *-commutative 58.2%

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

      3. prod-diff 43.9%

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

      4. fma-neg 43.9%

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

      5. distribute-lft-in 43.9%

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

   5. Applied egg-rr 44.4%

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

      1. *-commutative 44.4%

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

      2. distribute-rgt-out 44.4%

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

      3. *-commutative 44.4%

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

   7. Simplified 67.2%

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

      1. metadata-eval 67.2%

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

      2. div-inv 68.0%

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

   9. Applied egg-rr 68.0%

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

   if 6.89999999999999983e245 < b

   1. Initial program 43.2%

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

      1. *-commutative 43.2%

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

      2. associate-*l* 43.2%

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

      3. associate-*l* 43.2%

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

      4. unpow2 43.2%

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

      5. unpow2 43.2%

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

      6. difference-of-squares 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in angle around 0 79.0%

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

      1. add-sqr-sqrt 35.7%

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

      2. sqrt-unprod 42.9%

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

      3. pow2 42.9%

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

      4. associate-*r* 42.9%

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

      5. *-commutative 42.9%

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

      6. +-commutative 42.9%

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

      7. *-commutative 42.9%

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

   6. Applied egg-rr 42.9%

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

      1. sqrt-pow1 79.0%

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

      2. metadata-eval 79.0%

         \[\leadsto 0.011111111111111112 \cdot {\left(\left(\pi \cdot angle\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)}^{\color{blue}{1}} \]

      3. pow1 79.0%

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

      4. associate-*r* 85.7%

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

   8. Applied egg-rr 85.7%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.9 \cdot 10^{+245}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 5: 66.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{+23}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 7.6e+23)
   (* (+ a b) (* (- b a) (sin (* angle (* PI 0.011111111111111112)))))
   (* (+ a b) (* 0.011111111111111112 (* (- b a) (* angle PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 7.6e+23) {
		tmp = (a + b) * ((b - a) * sin((angle * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = (a + b) * (0.011111111111111112 * ((b - a) * (angle * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 7.6e+23) {
		tmp = (a + b) * ((b - a) * Math.sin((angle * (Math.PI * 0.011111111111111112))));
	} else {
		tmp = (a + b) * (0.011111111111111112 * ((b - a) * (angle * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 7.6e+23:
		tmp = (a + b) * ((b - a) * math.sin((angle * (math.pi * 0.011111111111111112))))
	else:
		tmp = (a + b) * (0.011111111111111112 * ((b - a) * (angle * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 7.6e+23)
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(angle * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(a + b) * Float64(0.011111111111111112 * Float64(Float64(b - a) * Float64(angle * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 7.6e+23)
		tmp = (a + b) * ((b - a) * sin((angle * (pi * 0.011111111111111112))));
	else
		tmp = (a + b) * (0.011111111111111112 * ((b - a) * (angle * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 7.6e+23], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(0.011111111111111112 * N[(N[(b - a), $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.5999999999999995e23

   1. Initial program 59.6%

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

      1. *-commutative 59.6%

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

      2. associate-*l* 59.6%

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

      3. associate-*l* 59.6%

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

      4. unpow2 59.6%

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

      5. unpow2 59.6%

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

      6. difference-of-squares 61.6%

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

   3. Simplified 61.6%

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

      1. difference-of-squares 59.6%

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

      2. *-commutative 59.6%

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

      3. prod-diff 43.8%

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

      4. fma-neg 43.8%

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

      5. distribute-lft-in 43.8%

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

   5. Applied egg-rr 44.6%

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

      1. *-commutative 44.6%

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

      2. distribute-rgt-out 44.6%

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

      3. *-commutative 44.6%

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

   7. Simplified 67.4%

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

      1. rem-cube-cbrt 67.5%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{{\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}}\right)\right) \]

   9. Applied egg-rr 67.5%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{{\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}}\right)\right) \]

   10. Taylor expanded in angle around 0 69.2%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot {\left(\sqrt[3]{\color{blue}{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}}\right)}^{3}\right)\right) \]

   11. Taylor expanded in angle around inf 66.9%

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

       1. *-commutative 66.9%

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

       2. associate-*l* 67.9%

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

   13. Simplified 67.9%

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

   if 7.5999999999999995e23 < b

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

      2. associate-*l* 49.5%

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

      3. associate-*l* 49.5%

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

      4. unpow2 49.5%

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

      5. unpow2 49.5%

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

      6. difference-of-squares 55.0%

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

   3. Simplified 55.0%

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

      1. difference-of-squares 49.5%

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

      2. *-commutative 49.5%

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

      3. prod-diff 44.0%

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

      4. fma-neg 44.0%

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

      5. distribute-lft-in 44.0%

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

   5. Applied egg-rr 48.8%

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

      1. *-commutative 48.8%

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

      2. distribute-rgt-out 48.8%

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

      3. *-commutative 48.8%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in angle around 0 69.0%

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

      1. associate-*r* 68.9%

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

   10. Simplified 68.9%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{+23}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 6: 53.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-122}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(-a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 9e-122)
   (* (+ a b) (* (- a) (sin (* (* angle PI) 0.011111111111111112))))
   (* (+ a b) (* (* angle 0.011111111111111112) (* (- b a) PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9e-122) {
		tmp = (a + b) * (-a * sin(((angle * ((double) M_PI)) * 0.011111111111111112)));
	} else {
		tmp = (a + b) * ((angle * 0.011111111111111112) * ((b - a) * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9e-122) {
		tmp = (a + b) * (-a * Math.sin(((angle * Math.PI) * 0.011111111111111112)));
	} else {
		tmp = (a + b) * ((angle * 0.011111111111111112) * ((b - a) * Math.PI));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 9e-122:
		tmp = (a + b) * (-a * math.sin(((angle * math.pi) * 0.011111111111111112)))
	else:
		tmp = (a + b) * ((angle * 0.011111111111111112) * ((b - a) * math.pi))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 9e-122)
		tmp = Float64(Float64(a + b) * Float64(Float64(-a) * sin(Float64(Float64(angle * pi) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(a + b) * Float64(Float64(angle * 0.011111111111111112) * Float64(Float64(b - a) * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 9e-122)
		tmp = (a + b) * (-a * sin(((angle * pi) * 0.011111111111111112)));
	else
		tmp = (a + b) * ((angle * 0.011111111111111112) * ((b - a) * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 9e-122], N[(N[(a + b), $MachinePrecision] * N[((-a) * N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.99999999999999959e-122

   1. Initial program 57.7%

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

      1. *-commutative 57.7%

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

      2. associate-*l* 57.7%

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

      3. associate-*l* 57.7%

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

      4. unpow2 57.7%

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

      5. unpow2 57.7%

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

      6. difference-of-squares 60.2%

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

   3. Simplified 60.2%

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

      1. difference-of-squares 57.7%

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

      2. *-commutative 57.7%

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

      3. prod-diff 43.1%

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

      4. fma-neg 43.1%

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

      5. distribute-lft-in 43.1%

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

   5. Applied egg-rr 44.0%

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

      1. *-commutative 44.0%

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

      2. distribute-rgt-out 44.0%

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

      3. *-commutative 44.0%

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

   7. Simplified 66.2%

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

      1. add-cube-cbrt 65.7%

         \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot \sqrt[3]{\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right) \cdot \sqrt[3]{\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)} \]

      2. pow3 65.7%

         \[\leadsto \left(a + b\right) \cdot \color{blue}{{\left(\sqrt[3]{\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}} \]

   9. Applied egg-rr 65.7%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{{\left(\sqrt[3]{\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}} \]

   10. Taylor expanded in b around 0 50.6%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(-1 \cdot \left({1}^{0.3333333333333333} \cdot \left(a \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 50.6%

          \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(-{1}^{0.3333333333333333} \cdot \left(a \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]

       2. pow-base-1 50.6%

          \[\leadsto \left(a + b\right) \cdot \left(-\color{blue}{1} \cdot \left(a \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       3. *-lft-identity 50.6%

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

       4. *-commutative 50.6%

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

   12. Simplified 50.6%

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

   if 8.99999999999999959e-122 < b

   1. Initial program 56.9%

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

      1. *-commutative 56.9%

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

      2. associate-*l* 56.9%

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

      3. associate-*l* 56.9%

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

      4. unpow2 56.9%

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

      5. unpow2 56.9%

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

      6. difference-of-squares 60.1%

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

   3. Simplified 60.1%

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

      1. difference-of-squares 56.9%

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

      2. *-commutative 56.9%

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

      3. prod-diff 45.1%

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

      4. fma-neg 45.1%

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

      5. distribute-lft-in 45.1%

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

   5. Applied egg-rr 47.9%

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

      1. *-commutative 47.9%

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

      2. distribute-rgt-out 47.9%

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

      3. *-commutative 47.9%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in angle around 0 69.3%

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

      1. associate-*r* 69.4%

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

      2. *-commutative 69.4%

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

   10. Simplified 69.4%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-122}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(-a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \end{array} \]

Alternative 7: 53.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-62}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a \cdot \left(-\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.2e-62)
   (* (+ a b) (* a (- (sin (* PI (* angle 0.011111111111111112))))))
   (* (+ a b) (* 0.011111111111111112 (* (- b a) (* angle PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.2e-62) {
		tmp = (a + b) * (a * -sin((((double) M_PI) * (angle * 0.011111111111111112))));
	} else {
		tmp = (a + b) * (0.011111111111111112 * ((b - a) * (angle * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.2e-62) {
		tmp = (a + b) * (a * -Math.sin((Math.PI * (angle * 0.011111111111111112))));
	} else {
		tmp = (a + b) * (0.011111111111111112 * ((b - a) * (angle * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2.2e-62:
		tmp = (a + b) * (a * -math.sin((math.pi * (angle * 0.011111111111111112))))
	else:
		tmp = (a + b) * (0.011111111111111112 * ((b - a) * (angle * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.2e-62)
		tmp = Float64(Float64(a + b) * Float64(a * Float64(-sin(Float64(pi * Float64(angle * 0.011111111111111112))))));
	else
		tmp = Float64(Float64(a + b) * Float64(0.011111111111111112 * Float64(Float64(b - a) * Float64(angle * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2.2e-62)
		tmp = (a + b) * (a * -sin((pi * (angle * 0.011111111111111112))));
	else
		tmp = (a + b) * (0.011111111111111112 * ((b - a) * (angle * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2.2e-62], N[(N[(a + b), $MachinePrecision] * N[(a * (-N[Sin[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(0.011111111111111112 * N[(N[(b - a), $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.20000000000000017e-62

   1. Initial program 58.1%

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

      1. *-commutative 58.1%

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

      2. associate-*l* 58.1%

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

      3. associate-*l* 58.1%

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

      4. unpow2 58.1%

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

      5. unpow2 58.1%

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

      6. difference-of-squares 60.3%

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

   3. Simplified 60.3%

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

      1. difference-of-squares 58.1%

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

      2. *-commutative 58.1%

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

      3. prod-diff 43.7%

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

      4. fma-neg 43.7%

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

      5. distribute-lft-in 43.7%

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

   5. Applied egg-rr 44.6%

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

      1. *-commutative 44.6%

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

      2. distribute-rgt-out 44.6%

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

      3. *-commutative 44.6%

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

   7. Simplified 66.8%

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

   8. Taylor expanded in b around 0 51.9%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(-1 \cdot \left(a \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 51.9%

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

      2. *-commutative 51.9%

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

      3. distribute-rgt-neg-in 51.9%

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

      4. associate-*r* 51.8%

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

      5. *-commutative 51.8%

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

   10. Simplified 51.8%

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

   if 2.20000000000000017e-62 < b

   1. Initial program 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-*l* 56.0%

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

      3. associate-*l* 56.0%

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

      4. unpow2 56.0%

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

      5. unpow2 56.0%

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

      6. difference-of-squares 59.9%

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

   3. Simplified 59.9%

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

      1. difference-of-squares 56.0%

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

      2. *-commutative 56.0%

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

      3. prod-diff 44.1%

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

      4. fma-neg 44.1%

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

      5. distribute-lft-in 44.1%

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

   5. Applied egg-rr 47.6%

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

      1. *-commutative 47.6%

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

      2. distribute-rgt-out 47.6%

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

      3. *-commutative 47.6%

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

   7. Simplified 69.1%

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

   8. Taylor expanded in angle around 0 71.1%

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

      1. associate-*r* 71.1%

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

   10. Simplified 71.1%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-62}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a \cdot \left(-\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 8: 49.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-147}:\\ \;\;\;\;\left(-2 \cdot \left(a \cdot a\right)\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 6.2e-147)
   (* (* -2.0 (* a a)) (sin (* angle (* PI 0.005555555555555556))))
   (* (+ a b) (* (* angle 0.011111111111111112) (* (- b a) PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.2e-147) {
		tmp = (-2.0 * (a * a)) * sin((angle * (((double) M_PI) * 0.005555555555555556)));
	} else {
		tmp = (a + b) * ((angle * 0.011111111111111112) * ((b - a) * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.2e-147) {
		tmp = (-2.0 * (a * a)) * Math.sin((angle * (Math.PI * 0.005555555555555556)));
	} else {
		tmp = (a + b) * ((angle * 0.011111111111111112) * ((b - a) * Math.PI));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 6.2e-147:
		tmp = (-2.0 * (a * a)) * math.sin((angle * (math.pi * 0.005555555555555556)))
	else:
		tmp = (a + b) * ((angle * 0.011111111111111112) * ((b - a) * math.pi))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 6.2e-147)
		tmp = Float64(Float64(-2.0 * Float64(a * a)) * sin(Float64(angle * Float64(pi * 0.005555555555555556))));
	else
		tmp = Float64(Float64(a + b) * Float64(Float64(angle * 0.011111111111111112) * Float64(Float64(b - a) * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 6.2e-147)
		tmp = (-2.0 * (a * a)) * sin((angle * (pi * 0.005555555555555556)));
	else
		tmp = (a + b) * ((angle * 0.011111111111111112) * ((b - a) * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 6.2e-147], N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.2000000000000005e-147

   1. Initial program 58.4%

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

      1. *-commutative 58.4%

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

      2. associate-*l* 58.4%

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

      3. associate-*l* 58.4%

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

      4. unpow2 58.4%

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

      5. unpow2 58.4%

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

      6. difference-of-squares 61.0%

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

   3. Simplified 61.0%

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

      1. associate-*r/ 60.7%

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

   5. Applied egg-rr 60.7%

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

   6. Taylor expanded in angle around 0 59.4%

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

   7. Taylor expanded in b around 0 40.6%

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

      1. associate-*r* 40.6%

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

      2. unpow2 40.6%

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

      3. *-commutative 40.6%

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

      4. associate-*l* 42.6%

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

   9. Simplified 42.6%

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

   if 6.2000000000000005e-147 < b

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*l* 55.9%

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

      3. associate-*l* 55.9%

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

      4. unpow2 55.9%

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

      5. unpow2 55.9%

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

      6. difference-of-squares 59.0%

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

   3. Simplified 59.0%

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

      1. difference-of-squares 55.9%

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

      2. *-commutative 55.9%

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

      3. prod-diff 43.9%

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

      4. fma-neg 43.9%

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

      5. distribute-lft-in 43.9%

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

   5. Applied egg-rr 46.3%

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

      1. *-commutative 46.3%

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

      2. distribute-rgt-out 46.3%

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

      3. *-commutative 46.3%

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

   7. Simplified 68.6%

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

   8. Taylor expanded in angle around 0 67.3%

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

      1. associate-*r* 67.3%

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

      2. *-commutative 67.3%

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

   10. Simplified 67.3%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-147}:\\ \;\;\;\;\left(-2 \cdot \left(a \cdot a\right)\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \end{array} \]

Alternative 9: 54.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\\ \mathbf{if}\;angle \leq -3.05 \cdot 10^{+237} \lor \neg \left(angle \leq -2 \cdot 10^{+173}\right) \land \left(angle \leq -2.2 \cdot 10^{+69} \lor \neg \left(angle \leq -6600\right) \land angle \leq 5.4 \cdot 10^{+169}\right):\\ \;\;\;\;0.011111111111111112 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot -0.011111111111111112\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI (* (+ a b) (- b a))))))
   (if (or (<= angle -3.05e+237)
           (and (not (<= angle -2e+173))
                (or (<= angle -2.2e+69)
                    (and (not (<= angle -6600.0)) (<= angle 5.4e+169)))))
     (* 0.011111111111111112 t_0)
     (* t_0 -0.011111111111111112))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * ((a + b) * (b - a)));
	double tmp;
	if ((angle <= -3.05e+237) || (!(angle <= -2e+173) && ((angle <= -2.2e+69) || (!(angle <= -6600.0) && (angle <= 5.4e+169))))) {
		tmp = 0.011111111111111112 * t_0;
	} else {
		tmp = t_0 * -0.011111111111111112;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (Math.PI * ((a + b) * (b - a)));
	double tmp;
	if ((angle <= -3.05e+237) || (!(angle <= -2e+173) && ((angle <= -2.2e+69) || (!(angle <= -6600.0) && (angle <= 5.4e+169))))) {
		tmp = 0.011111111111111112 * t_0;
	} else {
		tmp = t_0 * -0.011111111111111112;
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = angle * (math.pi * ((a + b) * (b - a)))
	tmp = 0
	if (angle <= -3.05e+237) or (not (angle <= -2e+173) and ((angle <= -2.2e+69) or (not (angle <= -6600.0) and (angle <= 5.4e+169)))):
		tmp = 0.011111111111111112 * t_0
	else:
		tmp = t_0 * -0.011111111111111112
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi * Float64(Float64(a + b) * Float64(b - a))))
	tmp = 0.0
	if ((angle <= -3.05e+237) || (!(angle <= -2e+173) && ((angle <= -2.2e+69) || (!(angle <= -6600.0) && (angle <= 5.4e+169)))))
		tmp = Float64(0.011111111111111112 * t_0);
	else
		tmp = Float64(t_0 * -0.011111111111111112);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = angle * (pi * ((a + b) * (b - a)));
	tmp = 0.0;
	if ((angle <= -3.05e+237) || (~((angle <= -2e+173)) && ((angle <= -2.2e+69) || (~((angle <= -6600.0)) && (angle <= 5.4e+169)))))
		tmp = 0.011111111111111112 * t_0;
	else
		tmp = t_0 * -0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[angle, -3.05e+237], And[N[Not[LessEqual[angle, -2e+173]], $MachinePrecision], Or[LessEqual[angle, -2.2e+69], And[N[Not[LessEqual[angle, -6600.0]], $MachinePrecision], LessEqual[angle, 5.4e+169]]]]], N[(0.011111111111111112 * t$95$0), $MachinePrecision], N[(t$95$0 * -0.011111111111111112), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < -3.0500000000000001e237 or -2e173 < angle < -2.2000000000000002e69 or -6600 < angle < 5.39999999999999981e169

   1. Initial program 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-*l* 65.1%

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

      3. associate-*l* 65.1%

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

      4. unpow2 65.1%

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

      5. unpow2 65.1%

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

      6. difference-of-squares 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in angle around 0 70.6%

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

   if -3.0500000000000001e237 < angle < -2e173 or -2.2000000000000002e69 < angle < -6600 or 5.39999999999999981e169 < angle

   1. Initial program 28.6%

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

      1. *-commutative 28.6%

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

      2. associate-*l* 28.6%

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

      3. associate-*l* 28.6%

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

      4. unpow2 28.6%

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

      5. unpow2 28.6%

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

      6. difference-of-squares 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in angle around 0 11.3%

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

      1. add-sqr-sqrt 6.7%

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

      2. sqrt-unprod 35.1%

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

      3. pow2 35.1%

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

      4. associate-*r* 35.1%

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

      5. *-commutative 35.1%

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

      6. +-commutative 35.1%

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

      7. *-commutative 35.1%

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

   6. Applied egg-rr 35.1%

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

   7. Taylor expanded in angle around -inf 47.5%

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

      1. +-commutative 47.5%

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

      2. *-commutative 47.5%

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

   9. Simplified 47.5%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -3.05 \cdot 10^{+237} \lor \neg \left(angle \leq -2 \cdot 10^{+173}\right) \land \left(angle \leq -2.2 \cdot 10^{+69} \lor \neg \left(angle \leq -6600\right) \land angle \leq 5.4 \cdot 10^{+169}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot -0.011111111111111112\\ \end{array} \]

Alternative 10: 63.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\\ t_1 := t_0 \cdot -0.011111111111111112\\ t_2 := 0.011111111111111112 \cdot t_0\\ \mathbf{if}\;angle \leq -3.05 \cdot 10^{+237}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;angle \leq -2 \cdot 10^{+173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;angle \leq -2.2 \cdot 10^{+69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;angle \leq -6600 \lor \neg \left(angle \leq 5.4 \cdot 10^{+169}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI (* (+ a b) (- b a)))))
        (t_1 (* t_0 -0.011111111111111112))
        (t_2 (* 0.011111111111111112 t_0)))
   (if (<= angle -3.05e+237)
     t_2
     (if (<= angle -2e+173)
       t_1
       (if (<= angle -2.2e+69)
         t_2
         (if (or (<= angle -6600.0) (not (<= angle 5.4e+169)))
           t_1
           (* 0.011111111111111112 (* (+ a b) (* (- b a) (* angle PI))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * ((a + b) * (b - a)));
	double t_1 = t_0 * -0.011111111111111112;
	double t_2 = 0.011111111111111112 * t_0;
	double tmp;
	if (angle <= -3.05e+237) {
		tmp = t_2;
	} else if (angle <= -2e+173) {
		tmp = t_1;
	} else if (angle <= -2.2e+69) {
		tmp = t_2;
	} else if ((angle <= -6600.0) || !(angle <= 5.4e+169)) {
		tmp = t_1;
	} else {
		tmp = 0.011111111111111112 * ((a + b) * ((b - a) * (angle * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (Math.PI * ((a + b) * (b - a)));
	double t_1 = t_0 * -0.011111111111111112;
	double t_2 = 0.011111111111111112 * t_0;
	double tmp;
	if (angle <= -3.05e+237) {
		tmp = t_2;
	} else if (angle <= -2e+173) {
		tmp = t_1;
	} else if (angle <= -2.2e+69) {
		tmp = t_2;
	} else if ((angle <= -6600.0) || !(angle <= 5.4e+169)) {
		tmp = t_1;
	} else {
		tmp = 0.011111111111111112 * ((a + b) * ((b - a) * (angle * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = angle * (math.pi * ((a + b) * (b - a)))
	t_1 = t_0 * -0.011111111111111112
	t_2 = 0.011111111111111112 * t_0
	tmp = 0
	if angle <= -3.05e+237:
		tmp = t_2
	elif angle <= -2e+173:
		tmp = t_1
	elif angle <= -2.2e+69:
		tmp = t_2
	elif (angle <= -6600.0) or not (angle <= 5.4e+169):
		tmp = t_1
	else:
		tmp = 0.011111111111111112 * ((a + b) * ((b - a) * (angle * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi * Float64(Float64(a + b) * Float64(b - a))))
	t_1 = Float64(t_0 * -0.011111111111111112)
	t_2 = Float64(0.011111111111111112 * t_0)
	tmp = 0.0
	if (angle <= -3.05e+237)
		tmp = t_2;
	elseif (angle <= -2e+173)
		tmp = t_1;
	elseif (angle <= -2.2e+69)
		tmp = t_2;
	elseif ((angle <= -6600.0) || !(angle <= 5.4e+169))
		tmp = t_1;
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(angle * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = angle * (pi * ((a + b) * (b - a)));
	t_1 = t_0 * -0.011111111111111112;
	t_2 = 0.011111111111111112 * t_0;
	tmp = 0.0;
	if (angle <= -3.05e+237)
		tmp = t_2;
	elseif (angle <= -2e+173)
		tmp = t_1;
	elseif (angle <= -2.2e+69)
		tmp = t_2;
	elseif ((angle <= -6600.0) || ~((angle <= 5.4e+169)))
		tmp = t_1;
	else
		tmp = 0.011111111111111112 * ((a + b) * ((b - a) * (angle * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * -0.011111111111111112), $MachinePrecision]}, Block[{t$95$2 = N[(0.011111111111111112 * t$95$0), $MachinePrecision]}, If[LessEqual[angle, -3.05e+237], t$95$2, If[LessEqual[angle, -2e+173], t$95$1, If[LessEqual[angle, -2.2e+69], t$95$2, If[Or[LessEqual[angle, -6600.0], N[Not[LessEqual[angle, 5.4e+169]], $MachinePrecision]], t$95$1, N[(0.011111111111111112 * N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if angle < -3.0500000000000001e237 or -2e173 < angle < -2.2000000000000002e69

   1. Initial program 31.8%

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

      1. *-commutative 31.8%

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

      2. associate-*l* 31.8%

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

      3. associate-*l* 31.8%

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

      4. unpow2 31.8%

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

      5. unpow2 31.8%

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

      6. difference-of-squares 31.8%

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

   3. Simplified 31.8%

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

   4. Taylor expanded in angle around 0 43.7%

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

   if -3.0500000000000001e237 < angle < -2e173 or -2.2000000000000002e69 < angle < -6600 or 5.39999999999999981e169 < angle

   1. Initial program 28.6%

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

      1. *-commutative 28.6%

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

      2. associate-*l* 28.6%

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

      3. associate-*l* 28.6%

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

      4. unpow2 28.6%

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

      5. unpow2 28.6%

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

      6. difference-of-squares 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in angle around 0 11.3%

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

      1. add-sqr-sqrt 6.7%

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

      2. sqrt-unprod 35.1%

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

      3. pow2 35.1%

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

      4. associate-*r* 35.1%

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

      5. *-commutative 35.1%

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

      6. +-commutative 35.1%

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

      7. *-commutative 35.1%

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

   6. Applied egg-rr 35.1%

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

   7. Taylor expanded in angle around -inf 47.5%

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

      1. +-commutative 47.5%

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

      2. *-commutative 47.5%

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

   9. Simplified 47.5%

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

   if -6600 < angle < 5.39999999999999981e169

   1. Initial program 74.2%

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

      1. *-commutative 74.2%

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

      2. associate-*l* 74.2%

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

      3. associate-*l* 74.2%

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

      4. unpow2 74.2%

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

      5. unpow2 74.2%

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

      6. difference-of-squares 78.0%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in angle around 0 77.9%

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

      1. add-sqr-sqrt 42.5%

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

      2. sqrt-unprod 38.4%

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

      3. pow2 38.4%

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

      4. associate-*r* 38.4%

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

      5. *-commutative 38.4%

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

      6. +-commutative 38.4%

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

      7. *-commutative 38.4%

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

   6. Applied egg-rr 38.4%

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

      1. sqrt-pow1 77.8%

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

      2. metadata-eval 77.8%

         \[\leadsto 0.011111111111111112 \cdot {\left(\left(\pi \cdot angle\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)}^{\color{blue}{1}} \]

      3. pow1 77.8%

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

      4. associate-*r* 90.6%

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

   8. Applied egg-rr 90.6%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -3.05 \cdot 10^{+237}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq -2 \cdot 10^{+173}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot -0.011111111111111112\\ \mathbf{elif}\;angle \leq -2.2 \cdot 10^{+69}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq -6600 \lor \neg \left(angle \leq 5.4 \cdot 10^{+169}\right):\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 11: 63.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\\ t_1 := t_0 \cdot -0.011111111111111112\\ t_2 := 0.011111111111111112 \cdot t_0\\ \mathbf{if}\;angle \leq -3.05 \cdot 10^{+237}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;angle \leq -2 \cdot 10^{+173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;angle \leq -2.2 \cdot 10^{+69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;angle \leq -6600 \lor \neg \left(angle \leq 5.4 \cdot 10^{+169}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI (* (+ a b) (- b a)))))
        (t_1 (* t_0 -0.011111111111111112))
        (t_2 (* 0.011111111111111112 t_0)))
   (if (<= angle -3.05e+237)
     t_2
     (if (<= angle -2e+173)
       t_1
       (if (<= angle -2.2e+69)
         t_2
         (if (or (<= angle -6600.0) (not (<= angle 5.4e+169)))
           t_1
           (* (+ a b) (* 0.011111111111111112 (* (- b a) (* angle PI))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * ((a + b) * (b - a)));
	double t_1 = t_0 * -0.011111111111111112;
	double t_2 = 0.011111111111111112 * t_0;
	double tmp;
	if (angle <= -3.05e+237) {
		tmp = t_2;
	} else if (angle <= -2e+173) {
		tmp = t_1;
	} else if (angle <= -2.2e+69) {
		tmp = t_2;
	} else if ((angle <= -6600.0) || !(angle <= 5.4e+169)) {
		tmp = t_1;
	} else {
		tmp = (a + b) * (0.011111111111111112 * ((b - a) * (angle * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (Math.PI * ((a + b) * (b - a)));
	double t_1 = t_0 * -0.011111111111111112;
	double t_2 = 0.011111111111111112 * t_0;
	double tmp;
	if (angle <= -3.05e+237) {
		tmp = t_2;
	} else if (angle <= -2e+173) {
		tmp = t_1;
	} else if (angle <= -2.2e+69) {
		tmp = t_2;
	} else if ((angle <= -6600.0) || !(angle <= 5.4e+169)) {
		tmp = t_1;
	} else {
		tmp = (a + b) * (0.011111111111111112 * ((b - a) * (angle * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = angle * (math.pi * ((a + b) * (b - a)))
	t_1 = t_0 * -0.011111111111111112
	t_2 = 0.011111111111111112 * t_0
	tmp = 0
	if angle <= -3.05e+237:
		tmp = t_2
	elif angle <= -2e+173:
		tmp = t_1
	elif angle <= -2.2e+69:
		tmp = t_2
	elif (angle <= -6600.0) or not (angle <= 5.4e+169):
		tmp = t_1
	else:
		tmp = (a + b) * (0.011111111111111112 * ((b - a) * (angle * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi * Float64(Float64(a + b) * Float64(b - a))))
	t_1 = Float64(t_0 * -0.011111111111111112)
	t_2 = Float64(0.011111111111111112 * t_0)
	tmp = 0.0
	if (angle <= -3.05e+237)
		tmp = t_2;
	elseif (angle <= -2e+173)
		tmp = t_1;
	elseif (angle <= -2.2e+69)
		tmp = t_2;
	elseif ((angle <= -6600.0) || !(angle <= 5.4e+169))
		tmp = t_1;
	else
		tmp = Float64(Float64(a + b) * Float64(0.011111111111111112 * Float64(Float64(b - a) * Float64(angle * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = angle * (pi * ((a + b) * (b - a)));
	t_1 = t_0 * -0.011111111111111112;
	t_2 = 0.011111111111111112 * t_0;
	tmp = 0.0;
	if (angle <= -3.05e+237)
		tmp = t_2;
	elseif (angle <= -2e+173)
		tmp = t_1;
	elseif (angle <= -2.2e+69)
		tmp = t_2;
	elseif ((angle <= -6600.0) || ~((angle <= 5.4e+169)))
		tmp = t_1;
	else
		tmp = (a + b) * (0.011111111111111112 * ((b - a) * (angle * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * -0.011111111111111112), $MachinePrecision]}, Block[{t$95$2 = N[(0.011111111111111112 * t$95$0), $MachinePrecision]}, If[LessEqual[angle, -3.05e+237], t$95$2, If[LessEqual[angle, -2e+173], t$95$1, If[LessEqual[angle, -2.2e+69], t$95$2, If[Or[LessEqual[angle, -6600.0], N[Not[LessEqual[angle, 5.4e+169]], $MachinePrecision]], t$95$1, N[(N[(a + b), $MachinePrecision] * N[(0.011111111111111112 * N[(N[(b - a), $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if angle < -3.0500000000000001e237 or -2e173 < angle < -2.2000000000000002e69

   1. Initial program 31.8%

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

      1. *-commutative 31.8%

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

      2. associate-*l* 31.8%

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

      3. associate-*l* 31.8%

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

      4. unpow2 31.8%

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

      5. unpow2 31.8%

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

      6. difference-of-squares 31.8%

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

   3. Simplified 31.8%

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

   4. Taylor expanded in angle around 0 43.7%

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

   if -3.0500000000000001e237 < angle < -2e173 or -2.2000000000000002e69 < angle < -6600 or 5.39999999999999981e169 < angle

   1. Initial program 28.6%

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

      1. *-commutative 28.6%

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

      2. associate-*l* 28.6%

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

      3. associate-*l* 28.6%

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

      4. unpow2 28.6%

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

      5. unpow2 28.6%

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

      6. difference-of-squares 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in angle around 0 11.3%

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

      1. add-sqr-sqrt 6.7%

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

      2. sqrt-unprod 35.1%

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

      3. pow2 35.1%

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

      4. associate-*r* 35.1%

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

      5. *-commutative 35.1%

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

      6. +-commutative 35.1%

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

      7. *-commutative 35.1%

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

   6. Applied egg-rr 35.1%

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

   7. Taylor expanded in angle around -inf 47.5%

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

      1. +-commutative 47.5%

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

      2. *-commutative 47.5%

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

   9. Simplified 47.5%

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

   if -6600 < angle < 5.39999999999999981e169

   1. Initial program 74.2%

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

      1. *-commutative 74.2%

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

      2. associate-*l* 74.2%

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

      3. associate-*l* 74.2%

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

      4. unpow2 74.2%

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

      5. unpow2 74.2%

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

      6. difference-of-squares 78.0%

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

   3. Simplified 78.0%

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

      1. difference-of-squares 74.2%

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

      2. *-commutative 74.2%

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

      3. prod-diff 59.9%

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

      4. fma-neg 59.9%

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

      5. distribute-lft-in 59.9%

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

   5. Applied egg-rr 62.9%

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

      1. *-commutative 62.9%

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

      2. distribute-rgt-out 62.9%

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

      3. *-commutative 62.9%

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

   7. Simplified 90.1%

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

   8. Taylor expanded in angle around 0 90.6%

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

      1. associate-*r* 90.6%

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

   10. Simplified 90.6%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -3.05 \cdot 10^{+237}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq -2 \cdot 10^{+173}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot -0.011111111111111112\\ \mathbf{elif}\;angle \leq -2.2 \cdot 10^{+69}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq -6600 \lor \neg \left(angle \leq 5.4 \cdot 10^{+169}\right):\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 12: 40.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 80000:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+208} \lor \neg \left(a \leq 2.3 \cdot 10^{+233}\right):\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot -0.011111111111111112\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 80000.0)
   (* 0.011111111111111112 (* angle (* PI (* b b))))
   (if (or (<= a 5.8e+208) (not (<= a 2.3e+233)))
     (* -0.011111111111111112 (* PI (* angle (* a a))))
     (* (* angle (* PI (* (+ a b) (- b a)))) -0.011111111111111112))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 80000.0) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	} else if ((a <= 5.8e+208) || !(a <= 2.3e+233)) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * (a * a)));
	} else {
		tmp = (angle * (((double) M_PI) * ((a + b) * (b - a)))) * -0.011111111111111112;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 80000.0) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	} else if ((a <= 5.8e+208) || !(a <= 2.3e+233)) {
		tmp = -0.011111111111111112 * (Math.PI * (angle * (a * a)));
	} else {
		tmp = (angle * (Math.PI * ((a + b) * (b - a)))) * -0.011111111111111112;
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 80000.0:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	elif (a <= 5.8e+208) or not (a <= 2.3e+233):
		tmp = -0.011111111111111112 * (math.pi * (angle * (a * a)))
	else:
		tmp = (angle * (math.pi * ((a + b) * (b - a)))) * -0.011111111111111112
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 80000.0)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	elseif ((a <= 5.8e+208) || !(a <= 2.3e+233))
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a * a))));
	else
		tmp = Float64(Float64(angle * Float64(pi * Float64(Float64(a + b) * Float64(b - a)))) * -0.011111111111111112);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 80000.0)
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	elseif ((a <= 5.8e+208) || ~((a <= 2.3e+233)))
		tmp = -0.011111111111111112 * (pi * (angle * (a * a)));
	else
		tmp = (angle * (pi * ((a + b) * (b - a)))) * -0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 80000.0], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a, 5.8e+208], N[Not[LessEqual[a, 2.3e+233]], $MachinePrecision]], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle * N[(Pi * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 8e4

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-*l* 58.9%

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

      3. associate-*l* 58.9%

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

      4. unpow2 58.9%

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

      5. unpow2 58.9%

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

      6. difference-of-squares 59.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in angle around 0 56.7%

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

   5. Taylor expanded in a around 0 42.3%

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

      1. *-commutative 42.3%

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

      2. unpow2 42.3%

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

   7. Simplified 42.3%

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

   if 8e4 < a < 5.80000000000000017e208 or 2.30000000000000001e233 < a

   1. Initial program 53.1%

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

      1. *-commutative 53.1%

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

      2. associate-*l* 53.1%

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

      3. associate-*l* 53.1%

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

      4. unpow2 53.1%

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

      5. unpow2 53.1%

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

      6. difference-of-squares 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in angle around 0 66.7%

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

   5. Taylor expanded in a around inf 52.9%

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

      1. *-commutative 52.9%

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

      2. associate-*r* 53.0%

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

      3. *-commutative 53.0%

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

      4. *-commutative 53.0%

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

      5. unpow2 53.0%

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

   7. Simplified 53.0%

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

   if 5.80000000000000017e208 < a < 2.30000000000000001e233

   1. Initial program 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-*l* 50.0%

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

      3. associate-*l* 50.0%

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

      4. unpow2 50.0%

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

      5. unpow2 50.0%

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

      6. difference-of-squares 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in angle around 0 37.5%

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

      1. add-sqr-sqrt 37.5%

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

      2. sqrt-unprod 62.5%

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

      3. pow2 62.5%

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

      4. associate-*r* 62.5%

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

      5. *-commutative 62.5%

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

      6. +-commutative 62.5%

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

      7. *-commutative 62.5%

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

   6. Applied egg-rr 62.5%

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

   7. Taylor expanded in angle around -inf 62.5%

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

      1. +-commutative 62.5%

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

      2. *-commutative 62.5%

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

   9. Simplified 62.5%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 80000:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+208} \lor \neg \left(a \leq 2.3 \cdot 10^{+233}\right):\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot -0.011111111111111112\\ \end{array} \]

Alternative 13: 34.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;angle \leq -7.2 \cdot 10^{+106}:\\ \;\;\;\;-0.011111111111111112 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI (* b b)))))
   (if (<= angle -7.2e+106)
     (* -0.011111111111111112 t_0)
     (* 0.011111111111111112 t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * (b * b));
	double tmp;
	if (angle <= -7.2e+106) {
		tmp = -0.011111111111111112 * t_0;
	} else {
		tmp = 0.011111111111111112 * t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (Math.PI * (b * b));
	double tmp;
	if (angle <= -7.2e+106) {
		tmp = -0.011111111111111112 * t_0;
	} else {
		tmp = 0.011111111111111112 * t_0;
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = angle * (math.pi * (b * b))
	tmp = 0
	if angle <= -7.2e+106:
		tmp = -0.011111111111111112 * t_0
	else:
		tmp = 0.011111111111111112 * t_0
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi * Float64(b * b)))
	tmp = 0.0
	if (angle <= -7.2e+106)
		tmp = Float64(-0.011111111111111112 * t_0);
	else
		tmp = Float64(0.011111111111111112 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = angle * (pi * (b * b));
	tmp = 0.0;
	if (angle <= -7.2e+106)
		tmp = -0.011111111111111112 * t_0;
	else
		tmp = 0.011111111111111112 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, -7.2e+106], N[(-0.011111111111111112 * t$95$0), $MachinePrecision], N[(0.011111111111111112 * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < -7.2000000000000002e106

   1. Initial program 27.0%

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

      1. *-commutative 27.0%

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

      2. associate-*l* 27.0%

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

      3. associate-*l* 27.0%

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

      4. unpow2 27.0%

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

      5. unpow2 27.0%

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

      6. difference-of-squares 27.0%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in angle around 0 27.1%

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

      1. add-sqr-sqrt 15.5%

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

      2. sqrt-unprod 30.7%

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

      3. pow2 30.7%

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

      4. associate-*r* 30.7%

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

      5. *-commutative 30.7%

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

      6. +-commutative 30.7%

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

      7. *-commutative 30.7%

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

   6. Applied egg-rr 30.7%

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

   7. Taylor expanded in b around -inf 28.4%

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

      1. *-commutative 28.4%

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

      2. *-commutative 28.4%

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

      3. unpow2 28.4%

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

   9. Simplified 28.4%

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

   if -7.2000000000000002e106 < angle

   1. Initial program 64.4%

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

      1. *-commutative 64.4%

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

      2. associate-*l* 64.4%

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

      3. associate-*l* 64.4%

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

      4. unpow2 64.4%

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

      5. unpow2 64.4%

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

      6. difference-of-squares 67.8%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in angle around 0 65.3%

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

   5. Taylor expanded in a around 0 40.1%

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

      1. *-commutative 40.1%

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

      2. unpow2 40.1%

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

   7. Simplified 40.1%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -7.2 \cdot 10^{+106}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 14: 40.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 31000:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 31000.0)
   (* 0.011111111111111112 (* angle (* PI (* b b))))
   (* -0.011111111111111112 (* PI (* angle (* a a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 31000.0) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	} else {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * (a * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 31000.0) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	} else {
		tmp = -0.011111111111111112 * (Math.PI * (angle * (a * a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 31000.0:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	else:
		tmp = -0.011111111111111112 * (math.pi * (angle * (a * a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 31000.0)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 31000.0)
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	else
		tmp = -0.011111111111111112 * (pi * (angle * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 31000.0], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 31000

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-*l* 58.9%

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

      3. associate-*l* 58.9%

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

      4. unpow2 58.9%

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

      5. unpow2 58.9%

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

      6. difference-of-squares 59.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in angle around 0 56.7%

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

   5. Taylor expanded in a around 0 42.3%

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

      1. *-commutative 42.3%

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

      2. unpow2 42.3%

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

   7. Simplified 42.3%

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

   if 31000 < a

   1. Initial program 52.7%

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

      1. *-commutative 52.7%

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

      2. associate-*l* 52.7%

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

      3. associate-*l* 52.7%

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

      4. unpow2 52.7%

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

      5. unpow2 52.7%

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

      6. difference-of-squares 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in angle around 0 62.8%

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

   5. Taylor expanded in a around inf 50.8%

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

      1. *-commutative 50.8%

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

      2. associate-*r* 50.9%

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

      3. *-commutative 50.9%

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

      4. *-commutative 50.9%

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

      5. unpow2 50.9%

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

   7. Simplified 50.9%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 31000:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 15: 34.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.4%

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

   1. *-commutative 57.4%

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

   2. associate-*l* 57.4%

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

   3. associate-*l* 57.4%

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

   4. unpow2 57.4%

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

   5. unpow2 57.4%

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

   6. difference-of-squares 60.2%

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

3. Simplified 60.2%

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

4. Taylor expanded in angle around 0 58.1%

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

5. Taylor expanded in a around 0 36.0%

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

   1. *-commutative 36.0%

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

   2. unpow2 36.0%

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

7. Simplified 36.0%

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

8. Final simplification 36.0%

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

Reproduce

herbie shell --seed 2023293 
(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)))))