ab-angle->ABCF B

Percentage Accurate: 54.7% → 64.5%

Time: 41.4s

Alternatives: 13

Speedup: 5.5×

• 32.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% 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: 64.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+22}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(a - b\right)\right) \cdot \left(\sin t_0 \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+114}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(a - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<= (/ angle 180.0) -1e+22)
     (*
      2.0
      (* (* (+ b a) (- a b)) (* (sin t_0) (cos (* angle (/ PI -180.0))))))
     (if (<= (/ angle 180.0) 2e+114)
       (* -0.011111111111111112 (* (* (* PI angle) (+ b a)) (- a b)))
       (*
        (cos t_0)
        (* 0.011111111111111112 (* angle (* PI (* (- b a) (+ b a))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((angle / 180.0) <= -1e+22) {
		tmp = 2.0 * (((b + a) * (a - b)) * (sin(t_0) * cos((angle * (((double) M_PI) / -180.0)))));
	} else if ((angle / 180.0) <= 2e+114) {
		tmp = -0.011111111111111112 * (((((double) M_PI) * angle) * (b + a)) * (a - b));
	} else {
		tmp = cos(t_0) * (0.011111111111111112 * (angle * (((double) M_PI) * ((b - a) * (b + a)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double tmp;
	if ((angle / 180.0) <= -1e+22) {
		tmp = 2.0 * (((b + a) * (a - b)) * (Math.sin(t_0) * Math.cos((angle * (Math.PI / -180.0)))));
	} else if ((angle / 180.0) <= 2e+114) {
		tmp = -0.011111111111111112 * (((Math.PI * angle) * (b + a)) * (a - b));
	} else {
		tmp = Math.cos(t_0) * (0.011111111111111112 * (angle * (Math.PI * ((b - a) * (b + a)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	tmp = 0
	if (angle / 180.0) <= -1e+22:
		tmp = 2.0 * (((b + a) * (a - b)) * (math.sin(t_0) * math.cos((angle * (math.pi / -180.0)))))
	elif (angle / 180.0) <= 2e+114:
		tmp = -0.011111111111111112 * (((math.pi * angle) * (b + a)) * (a - b))
	else:
		tmp = math.cos(t_0) * (0.011111111111111112 * (angle * (math.pi * ((b - a) * (b + a)))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -1e+22)
		tmp = Float64(2.0 * Float64(Float64(Float64(b + a) * Float64(a - b)) * Float64(sin(t_0) * cos(Float64(angle * Float64(pi / -180.0))))));
	elseif (Float64(angle / 180.0) <= 2e+114)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(Float64(pi * angle) * Float64(b + a)) * Float64(a - b)));
	else
		tmp = Float64(cos(t_0) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(b - a) * Float64(b + a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = 0.0;
	if ((angle / 180.0) <= -1e+22)
		tmp = 2.0 * (((b + a) * (a - b)) * (sin(t_0) * cos((angle * (pi / -180.0)))));
	elseif ((angle / 180.0) <= 2e+114)
		tmp = -0.011111111111111112 * (((pi * angle) * (b + a)) * (a - b));
	else
		tmp = cos(t_0) * (0.011111111111111112 * (angle * (pi * ((b - a) * (b + a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1e+22], N[(2.0 * N[(N[(N[(b + a), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+114], N[(-0.011111111111111112 * N[(N[(N[(Pi * angle), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[t$95$0], $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -1e22

   1. Initial program 37.2%

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

      1. associate-*l* 37.2%

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

      2. associate-*l* 37.2%

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

      3. cos-neg 37.2%

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

      4. distribute-rgt-neg-out 37.2%

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

      5. distribute-frac-neg 37.2%

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

      6. neg-mul-1 37.2%

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

      7. associate-/l* 32.5%

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

      8. associate-*r/ 37.8%

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

      9. associate-/r/ 27.0%

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

      10. associate-/l* 27.0%

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

      11. metadata-eval 27.0%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in b around 0 27.0%

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

      1. neg-mul-1 27.0%

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

      2. +-commutative 27.0%

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

      3. sub-neg 27.0%

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

      4. rem-square-sqrt 17.1%

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

      5. fabs-sqr 17.1%

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

      6. rem-square-sqrt 28.9%

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

      7. fabs-sub 28.9%

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

      8. rem-square-sqrt 19.0%

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

      9. fabs-sqr 19.0%

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

      10. rem-square-sqrt 35.9%

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

   6. Simplified 35.9%

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

      1. unpow2 35.9%

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

      2. unpow2 35.9%

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

      3. difference-of-squares 39.5%

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

   8. Applied egg-rr 39.5%

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

   if -1e22 < (/.f64 angle 180) < 2e114

   1. Initial program 60.5%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in angle around 0 60.1%

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

      1. unpow2 16.5%

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

      2. unpow2 16.5%

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

      3. difference-of-squares 17.7%

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

   6. Applied egg-rr 66.3%

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

      1. expm1-log1p-u 47.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-udef 31.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

      3. *-commutative 31.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

      4. associate-*r* 31.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

      5. +-commutative 31.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

   8. Applied egg-rr 31.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 47.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

      2. expm1-log1p 66.3%

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

      3. *-commutative 66.3%

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

      4. associate-*r* 83.9%

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

      5. +-commutative 83.9%

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

   10. Simplified 83.9%

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

   if 2e114 < (/.f64 angle 180)

   1. Initial program 35.8%

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

      1. unpow2 35.8%

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

      2. unpow2 35.8%

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

      3. difference-of-squares 41.1%

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

   3. Applied egg-rr 41.1%

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

   4. Taylor expanded in angle around 0 40.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+22}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \left(a - b\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+114}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(a - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - {a}^{2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t_0 \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (pow a 2.0))))
   (if (<= t_0 (- INFINITY))
     (* -0.011111111111111112 (* (- a b) (* a (* PI angle))))
     (if (<= t_0 5e+306)
       (* t_0 (sin (* PI (* 2.0 (* angle 0.005555555555555556)))))
       (* -0.011111111111111112 (* (- a b) (* angle (* PI (+ b a)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = pow(b, 2.0) - pow(a, 2.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -0.011111111111111112 * ((a - b) * (a * (((double) M_PI) * angle)));
	} else if (t_0 <= 5e+306) {
		tmp = t_0 * sin((((double) M_PI) * (2.0 * (angle * 0.005555555555555556))));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (((double) M_PI) * (b + a))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.pow(b, 2.0) - Math.pow(a, 2.0);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -0.011111111111111112 * ((a - b) * (a * (Math.PI * angle)));
	} else if (t_0 <= 5e+306) {
		tmp = t_0 * Math.sin((Math.PI * (2.0 * (angle * 0.005555555555555556))));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (Math.PI * (b + a))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pow(b, 2.0) - math.pow(a, 2.0)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -0.011111111111111112 * ((a - b) * (a * (math.pi * angle)))
	elif t_0 <= 5e+306:
		tmp = t_0 * math.sin((math.pi * (2.0 * (angle * 0.005555555555555556))))
	else:
		tmp = -0.011111111111111112 * ((a - b) * (angle * (math.pi * (b + a))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64((b ^ 2.0) - (a ^ 2.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(a * Float64(pi * angle))));
	elseif (t_0 <= 5e+306)
		tmp = Float64(t_0 * sin(Float64(pi * Float64(2.0 * Float64(angle * 0.005555555555555556)))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(angle * Float64(pi * Float64(b + a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (b ^ 2.0) - (a ^ 2.0);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -0.011111111111111112 * ((a - b) * (a * (pi * angle)));
	elseif (t_0 <= 5e+306)
		tmp = t_0 * sin((pi * (2.0 * (angle * 0.005555555555555556))));
	else
		tmp = -0.011111111111111112 * ((a - b) * (angle * (pi * (b + a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+306], N[(t$95$0 * N[Sin[N[(Pi * N[(2.0 * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(angle * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < -inf.0

   1. Initial program 56.8%

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

   3. Simplified 47.4%

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

   4. Taylor expanded in angle around 0 51.2%

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

      1. unpow2 28.4%

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

      2. unpow2 28.4%

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

      3. difference-of-squares 28.4%

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

   6. Applied egg-rr 51.2%

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

      1. expm1-log1p-u 18.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-udef 18.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

      3. *-commutative 18.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

      4. associate-*r* 18.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

      5. +-commutative 18.0%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

   8. Applied egg-rr 18.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 18.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

      2. expm1-log1p 51.2%

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

      3. *-commutative 51.2%

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

      4. associate-*r* 75.4%

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

      5. +-commutative 75.4%

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

   10. Simplified 75.4%

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

   11. Taylor expanded in a around inf 75.4%

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

   if -inf.0 < (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < 4.99999999999999993e306

   1. Initial program 56.2%

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

      1. *-commutative 56.2%

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

      2. associate-*l* 56.2%

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

      3. associate-*l* 56.2%

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

   3. Simplified 56.2%

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

      1. add-cbrt-cube 35.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}} \]

      2. pow1/3 24.0%

         \[\leadsto \color{blue}{{\left(\left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}^{0.3333333333333333}} \]

   5. Applied egg-rr 24.0%

      \[\leadsto \color{blue}{{\left({\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   6. Step-by-step derivation

      1. unpow1/3 34.9%

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

      2. rem-cbrt-cube 55.6%

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

      3. metadata-eval 55.6%

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

      4. div-inv 56.2%

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

      5. *-commutative 56.2%

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

      6. associate-*r* 56.2%

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

      7. div-inv 55.6%

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

      8. metadata-eval 55.6%

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

   7. Applied egg-rr 55.6%

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

   if 4.99999999999999993e306 < (-.f64 (pow.f64 b 2) (pow.f64 a 2))

   1. Initial program 40.2%

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

   3. Simplified 36.0%

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

   4. Taylor expanded in angle around 0 41.5%

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

      1. unpow2 20.9%

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

      2. unpow2 20.9%

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

      3. difference-of-squares 26.5%

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

   6. Applied egg-rr 61.1%

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

      1. expm1-log1p-u 36.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-udef 36.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

      3. *-commutative 36.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

      4. associate-*r* 36.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

      5. +-commutative 36.6%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

   8. Applied egg-rr 36.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 36.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

      2. expm1-log1p 61.1%

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

      3. *-commutative 61.1%

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

      4. associate-*r* 83.2%

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

      5. +-commutative 83.2%

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

   10. Simplified 83.2%

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

   11. Taylor expanded in angle around 0 83.2%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -\infty:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \end{array} \]

Alternative 3: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-224}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(a - b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (pow b 2.0) 5e-224)
   (*
    (* (* 2.0 (* (- b a) (+ b a))) (sin (* PI (/ angle 180.0))))
    (cos (* (/ angle 180.0) (pow (sqrt PI) 2.0))))
   (* -0.011111111111111112 (* (* (* PI angle) (+ b a)) (- a b)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (pow(b, 2.0) <= 5e-224) {
		tmp = ((2.0 * ((b - a) * (b + a))) * sin((((double) M_PI) * (angle / 180.0)))) * cos(((angle / 180.0) * pow(sqrt(((double) M_PI)), 2.0)));
	} else {
		tmp = -0.011111111111111112 * (((((double) M_PI) * angle) * (b + a)) * (a - b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.pow(b, 2.0) <= 5e-224) {
		tmp = ((2.0 * ((b - a) * (b + a))) * Math.sin((Math.PI * (angle / 180.0)))) * Math.cos(((angle / 180.0) * Math.pow(Math.sqrt(Math.PI), 2.0)));
	} else {
		tmp = -0.011111111111111112 * (((Math.PI * angle) * (b + a)) * (a - b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.pow(b, 2.0) <= 5e-224:
		tmp = ((2.0 * ((b - a) * (b + a))) * math.sin((math.pi * (angle / 180.0)))) * math.cos(((angle / 180.0) * math.pow(math.sqrt(math.pi), 2.0)))
	else:
		tmp = -0.011111111111111112 * (((math.pi * angle) * (b + a)) * (a - b))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((b ^ 2.0) <= 5e-224)
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a) * Float64(b + a))) * sin(Float64(pi * Float64(angle / 180.0)))) * cos(Float64(Float64(angle / 180.0) * (sqrt(pi) ^ 2.0))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(Float64(Float64(pi * angle) * Float64(b + a)) * Float64(a - b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((b ^ 2.0) <= 5e-224)
		tmp = ((2.0 * ((b - a) * (b + a))) * sin((pi * (angle / 180.0)))) * cos(((angle / 180.0) * (sqrt(pi) ^ 2.0)));
	else
		tmp = -0.011111111111111112 * (((pi * angle) * (b + a)) * (a - b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 5e-224], N[(N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(N[(N[(Pi * angle), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b 2) < 4.9999999999999999e-224

   1. Initial program 65.4%

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

      1. unpow2 65.4%

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

      2. unpow2 65.4%

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

      3. difference-of-squares 65.4%

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

   3. Applied egg-rr 65.4%

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

      1. add-sqr-sqrt 65.9%

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

      2. pow2 65.9%

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

   5. Applied egg-rr 65.9%

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

   if 4.9999999999999999e-224 < (pow.f64 b 2)

   1. Initial program 46.1%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in angle around 0 44.7%

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

      1. unpow2 16.4%

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

      2. unpow2 16.4%

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

      3. difference-of-squares 18.6%

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

   6. Applied egg-rr 52.5%

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

      1. expm1-log1p-u 32.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-udef 24.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

      3. *-commutative 24.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

      4. associate-*r* 24.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

      5. +-commutative 24.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

   8. Applied egg-rr 24.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 32.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

      2. expm1-log1p 52.5%

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

      3. *-commutative 52.5%

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

      4. associate-*r* 66.9%

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

      5. +-commutative 66.9%

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

   10. Simplified 66.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-224}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(a - b\right)\right)\\ \end{array} \]

Alternative 4: 64.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-224}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin t_0\right) \cdot \cos t_0\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(a - b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<= (pow b 2.0) 5e-224)
     (* (* (* 2.0 (* (- b a) (+ b a))) (sin t_0)) (cos t_0))
     (* -0.011111111111111112 (* (* (* PI angle) (+ b a)) (- a b))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if (pow(b, 2.0) <= 5e-224) {
		tmp = ((2.0 * ((b - a) * (b + a))) * sin(t_0)) * cos(t_0);
	} else {
		tmp = -0.011111111111111112 * (((((double) M_PI) * angle) * (b + a)) * (a - b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double tmp;
	if (Math.pow(b, 2.0) <= 5e-224) {
		tmp = ((2.0 * ((b - a) * (b + a))) * Math.sin(t_0)) * Math.cos(t_0);
	} else {
		tmp = -0.011111111111111112 * (((Math.PI * angle) * (b + a)) * (a - b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	tmp = 0
	if math.pow(b, 2.0) <= 5e-224:
		tmp = ((2.0 * ((b - a) * (b + a))) * math.sin(t_0)) * math.cos(t_0)
	else:
		tmp = -0.011111111111111112 * (((math.pi * angle) * (b + a)) * (a - b))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if ((b ^ 2.0) <= 5e-224)
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a) * Float64(b + a))) * sin(t_0)) * cos(t_0));
	else
		tmp = Float64(-0.011111111111111112 * Float64(Float64(Float64(pi * angle) * Float64(b + a)) * Float64(a - b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = 0.0;
	if ((b ^ 2.0) <= 5e-224)
		tmp = ((2.0 * ((b - a) * (b + a))) * sin(t_0)) * cos(t_0);
	else
		tmp = -0.011111111111111112 * (((pi * angle) * (b + a)) * (a - b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 5e-224], N[(N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(N[(N[(Pi * angle), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b 2) < 4.9999999999999999e-224

   1. Initial program 65.4%

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

      1. unpow2 65.4%

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

      2. unpow2 65.4%

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

      3. difference-of-squares 65.4%

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

   3. Applied egg-rr 65.4%

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

   if 4.9999999999999999e-224 < (pow.f64 b 2)

   1. Initial program 46.1%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in angle around 0 44.7%

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

      1. unpow2 16.4%

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

      2. unpow2 16.4%

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

      3. difference-of-squares 18.6%

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

   6. Applied egg-rr 52.5%

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

      1. expm1-log1p-u 32.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-udef 24.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

      3. *-commutative 24.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

      4. associate-*r* 24.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

      5. +-commutative 24.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

   8. Applied egg-rr 24.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 32.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

      2. expm1-log1p 52.5%

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

      3. *-commutative 52.5%

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

      4. associate-*r* 66.9%

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

      5. +-commutative 66.9%

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

   10. Simplified 66.9%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-224}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(a - b\right)\right)\\ \end{array} \]

Alternative 5: 63.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;\frac{angle}{180} \leq -4 \cdot 10^{+43}:\\ \;\;\;\;2 \cdot \left(\sin t_0 \cdot \left({b}^{2} - {a}^{2}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+114}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(a - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<= (/ angle 180.0) -4e+43)
     (* 2.0 (* (sin t_0) (- (pow b 2.0) (pow a 2.0))))
     (if (<= (/ angle 180.0) 2e+114)
       (* -0.011111111111111112 (* (* (* PI angle) (+ b a)) (- a b)))
       (*
        (cos t_0)
        (* 0.011111111111111112 (* angle (* PI (* (- b a) (+ b a))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((angle / 180.0) <= -4e+43) {
		tmp = 2.0 * (sin(t_0) * (pow(b, 2.0) - pow(a, 2.0)));
	} else if ((angle / 180.0) <= 2e+114) {
		tmp = -0.011111111111111112 * (((((double) M_PI) * angle) * (b + a)) * (a - b));
	} else {
		tmp = cos(t_0) * (0.011111111111111112 * (angle * (((double) M_PI) * ((b - a) * (b + a)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double tmp;
	if ((angle / 180.0) <= -4e+43) {
		tmp = 2.0 * (Math.sin(t_0) * (Math.pow(b, 2.0) - Math.pow(a, 2.0)));
	} else if ((angle / 180.0) <= 2e+114) {
		tmp = -0.011111111111111112 * (((Math.PI * angle) * (b + a)) * (a - b));
	} else {
		tmp = Math.cos(t_0) * (0.011111111111111112 * (angle * (Math.PI * ((b - a) * (b + a)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	tmp = 0
	if (angle / 180.0) <= -4e+43:
		tmp = 2.0 * (math.sin(t_0) * (math.pow(b, 2.0) - math.pow(a, 2.0)))
	elif (angle / 180.0) <= 2e+114:
		tmp = -0.011111111111111112 * (((math.pi * angle) * (b + a)) * (a - b))
	else:
		tmp = math.cos(t_0) * (0.011111111111111112 * (angle * (math.pi * ((b - a) * (b + a)))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -4e+43)
		tmp = Float64(2.0 * Float64(sin(t_0) * Float64((b ^ 2.0) - (a ^ 2.0))));
	elseif (Float64(angle / 180.0) <= 2e+114)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(Float64(pi * angle) * Float64(b + a)) * Float64(a - b)));
	else
		tmp = Float64(cos(t_0) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(b - a) * Float64(b + a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = 0.0;
	if ((angle / 180.0) <= -4e+43)
		tmp = 2.0 * (sin(t_0) * ((b ^ 2.0) - (a ^ 2.0)));
	elseif ((angle / 180.0) <= 2e+114)
		tmp = -0.011111111111111112 * (((pi * angle) * (b + a)) * (a - b));
	else
		tmp = cos(t_0) * (0.011111111111111112 * (angle * (pi * ((b - a) * (b + a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -4e+43], N[(2.0 * N[(N[Sin[t$95$0], $MachinePrecision] * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+114], N[(-0.011111111111111112 * N[(N[(N[(Pi * angle), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[t$95$0], $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -4.00000000000000006e43

   1. Initial program 38.1%

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

      1. associate-*l* 38.1%

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

      2. associate-*l* 38.1%

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

      3. cos-neg 38.1%

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

      4. distribute-rgt-neg-out 38.1%

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

      5. distribute-frac-neg 38.1%

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

      6. neg-mul-1 38.1%

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

      7. associate-/l* 32.7%

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

      8. associate-*r/ 38.4%

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

      9. associate-/r/ 28.7%

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

      10. associate-/l* 28.7%

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

      11. metadata-eval 28.7%

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

   3. Simplified 28.7%

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

   4. Taylor expanded in angle around 0 40.9%

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

   if -4.00000000000000006e43 < (/.f64 angle 180) < 2e114

   1. Initial program 59.6%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in angle around 0 58.8%

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

      1. unpow2 16.8%

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

      2. unpow2 16.8%

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

      3. difference-of-squares 18.0%

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

   6. Applied egg-rr 64.8%

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

      1. expm1-log1p-u 46.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-udef 31.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

      3. *-commutative 31.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

      4. associate-*r* 31.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

      5. +-commutative 31.0%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

   8. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 46.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

      2. expm1-log1p 64.8%

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

      3. *-commutative 64.8%

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

      4. associate-*r* 82.0%

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

      5. +-commutative 82.0%

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

   10. Simplified 82.0%

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

   if 2e114 < (/.f64 angle 180)

   1. Initial program 35.8%

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

      1. unpow2 35.8%

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

      2. unpow2 35.8%

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

      3. difference-of-squares 41.1%

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

   3. Applied egg-rr 41.1%

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

   4. Taylor expanded in angle around 0 40.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -4 \cdot 10^{+43}:\\ \;\;\;\;2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+114}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(a - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 6: 49.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.2 \cdot 10^{-195}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{+119}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 5.2e-195)
   (* -0.011111111111111112 (* (- a b) (* angle (* b PI))))
   (if (<= a 6.4e+119)
     (* -0.011111111111111112 (* angle (* PI (* (+ b a) (- a b)))))
     (* -0.011111111111111112 (* (- a b) (* a (* PI angle)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.2e-195) {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * ((double) M_PI))));
	} else if (a <= 6.4e+119) {
		tmp = -0.011111111111111112 * (angle * (((double) M_PI) * ((b + a) * (a - b))));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (a * (((double) M_PI) * angle)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.2e-195) {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * Math.PI)));
	} else if (a <= 6.4e+119) {
		tmp = -0.011111111111111112 * (angle * (Math.PI * ((b + a) * (a - b))));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (a * (Math.PI * angle)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 5.2e-195:
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * math.pi)))
	elif a <= 6.4e+119:
		tmp = -0.011111111111111112 * (angle * (math.pi * ((b + a) * (a - b))))
	else:
		tmp = -0.011111111111111112 * ((a - b) * (a * (math.pi * angle)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 5.2e-195)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(angle * Float64(b * pi))));
	elseif (a <= 6.4e+119)
		tmp = Float64(-0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(b + a) * Float64(a - b)))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(a * Float64(pi * angle))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 5.2e-195)
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * pi)));
	elseif (a <= 6.4e+119)
		tmp = -0.011111111111111112 * (angle * (pi * ((b + a) * (a - b))));
	else
		tmp = -0.011111111111111112 * ((a - b) * (a * (pi * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 5.2e-195], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6.4e+119], N[(-0.011111111111111112 * N[(angle * N[(Pi * N[(N[(b + a), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 5.2000000000000003e-195

   1. Initial program 52.2%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in angle around 0 48.8%

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

      1. unpow2 22.9%

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

      2. unpow2 22.9%

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

      3. difference-of-squares 24.2%

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

   6. Applied egg-rr 53.4%

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

      1. expm1-log1p-u 36.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-udef 28.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

      3. *-commutative 28.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

      4. associate-*r* 28.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

      5. +-commutative 28.1%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

   8. Applied egg-rr 28.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 36.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

      2. expm1-log1p 53.4%

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

      3. *-commutative 53.4%

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

      4. associate-*r* 63.4%

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

      5. +-commutative 63.4%

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

   10. Simplified 63.4%

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

   11. Taylor expanded in a around 0 43.8%

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

       1. *-commutative 43.8%

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

   13. Simplified 43.8%

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

   if 5.2000000000000003e-195 < a < 6.39999999999999979e119

   1. Initial program 55.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in angle around 0 49.5%

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

      1. unpow2 20.3%

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

      2. unpow2 20.3%

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

      3. difference-of-squares 20.3%

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

   6. Applied egg-rr 49.5%

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

   if 6.39999999999999979e119 < a

   1. Initial program 45.5%

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

   3. Simplified 38.7%

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

   4. Taylor expanded in angle around 0 44.4%

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

      1. unpow2 22.9%

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

      2. unpow2 22.9%

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

      3. difference-of-squares 27.4%

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

   6. Applied egg-rr 60.5%

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

      1. expm1-log1p-u 29.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-udef 29.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

      3. *-commutative 29.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

      4. associate-*r* 29.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

      5. +-commutative 29.1%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

   8. Applied egg-rr 29.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 29.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

      2. expm1-log1p 60.5%

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

      3. *-commutative 60.5%

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

      4. associate-*r* 79.3%

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

      5. +-commutative 79.3%

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

   10. Simplified 79.3%

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

   11. Taylor expanded in a around inf 68.0%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.2 \cdot 10^{-195}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{+119}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 7: 46.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+50}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.5e+50)
   (* -0.011111111111111112 (* (- a b) (* angle (* a PI))))
   (* -0.011111111111111112 (* (- a b) (* angle (* b PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.5e+50) {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (a * ((double) M_PI))));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.5e+50) {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (a * Math.PI)));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.5e+50:
		tmp = -0.011111111111111112 * ((a - b) * (angle * (a * math.pi)))
	else:
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.5e+50)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(angle * Float64(a * pi))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(angle * Float64(b * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.5e+50)
		tmp = -0.011111111111111112 * ((a - b) * (angle * (a * pi)));
	else
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.5e+50], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4999999999999999e50

   1. Initial program 55.7%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in angle around 0 51.4%

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

      1. unpow2 24.8%

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

      2. unpow2 24.8%

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

      3. difference-of-squares 24.8%

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

   6. Applied egg-rr 53.9%

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

      1. expm1-log1p-u 37.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-udef 25.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

      3. *-commutative 25.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

      4. associate-*r* 25.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

      5. +-commutative 25.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

   8. Applied egg-rr 25.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 37.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

      2. expm1-log1p 53.9%

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

      3. *-commutative 53.9%

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

      4. associate-*r* 62.9%

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

      5. +-commutative 62.9%

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

   10. Simplified 62.9%

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

   11. Taylor expanded in a around inf 47.9%

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

       1. *-commutative 47.9%

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

       2. associate-*l* 47.9%

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

   13. Simplified 47.9%

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

   if 1.4999999999999999e50 < b

   1. Initial program 37.7%

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

   3. Simplified 36.0%

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

   4. Taylor expanded in angle around 0 36.4%

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

      1. unpow2 13.2%

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

      2. unpow2 13.2%

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

      3. difference-of-squares 20.5%

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

   6. Applied egg-rr 52.9%

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

      1. expm1-log1p-u 30.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-udef 28.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

      3. *-commutative 28.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

      4. associate-*r* 28.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

      5. +-commutative 28.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

   8. Applied egg-rr 28.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 30.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

      2. expm1-log1p 52.9%

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

      3. *-commutative 52.9%

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

      4. associate-*r* 66.3%

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

      5. +-commutative 66.3%

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

   10. Simplified 66.3%

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

   11. Taylor expanded in a around 0 57.7%

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

       1. *-commutative 57.7%

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

   13. Simplified 57.7%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+50}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 8: 46.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{+50}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 6.2e+50)
   (* -0.011111111111111112 (* (- a b) (* angle (* a PI))))
   (* -0.011111111111111112 (* (- a b) (* PI (* b angle))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.2e+50) {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (a * ((double) M_PI))));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (((double) M_PI) * (b * angle)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.2e+50) {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (a * Math.PI)));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (Math.PI * (b * angle)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 6.2e+50:
		tmp = -0.011111111111111112 * ((a - b) * (angle * (a * math.pi)))
	else:
		tmp = -0.011111111111111112 * ((a - b) * (math.pi * (b * angle)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 6.2e+50)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(angle * Float64(a * pi))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(pi * Float64(b * angle))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 6.2e+50)
		tmp = -0.011111111111111112 * ((a - b) * (angle * (a * pi)));
	else
		tmp = -0.011111111111111112 * ((a - b) * (pi * (b * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 6.2e+50], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(Pi * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.20000000000000006e50

   1. Initial program 55.7%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in angle around 0 51.4%

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

      1. unpow2 24.8%

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

      2. unpow2 24.8%

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

      3. difference-of-squares 24.8%

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

   6. Applied egg-rr 53.9%

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

      1. expm1-log1p-u 37.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-udef 25.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

      3. *-commutative 25.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

      4. associate-*r* 25.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

      5. +-commutative 25.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

   8. Applied egg-rr 25.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 37.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

      2. expm1-log1p 53.9%

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

      3. *-commutative 53.9%

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

      4. associate-*r* 62.9%

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

      5. +-commutative 62.9%

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

   10. Simplified 62.9%

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

   11. Taylor expanded in a around inf 47.9%

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

       1. *-commutative 47.9%

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

       2. associate-*l* 47.9%

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

   13. Simplified 47.9%

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

   if 6.20000000000000006e50 < b

   1. Initial program 37.7%

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

   3. Simplified 36.0%

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

   4. Taylor expanded in angle around 0 36.4%

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

      1. unpow2 13.2%

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

      2. unpow2 13.2%

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

      3. difference-of-squares 20.5%

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

   6. Applied egg-rr 52.9%

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

      1. expm1-log1p-u 30.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-udef 28.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

      3. *-commutative 28.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

      4. associate-*r* 28.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

      5. +-commutative 28.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

   8. Applied egg-rr 28.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 30.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

      2. expm1-log1p 52.9%

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

      3. *-commutative 52.9%

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

      4. associate-*r* 66.3%

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

      5. +-commutative 66.3%

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

   10. Simplified 66.3%

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

   11. Taylor expanded in a around 0 57.7%

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

       1. associate-*r* 57.8%

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

   13. Simplified 57.8%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{+50}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 9: 62.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.8%

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

3. Simplified 48.7%

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

4. Taylor expanded in angle around 0 48.2%

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

   1. unpow2 22.3%

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

   2. unpow2 22.3%

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

   3. difference-of-squares 23.9%

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

6. Applied egg-rr 53.7%

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

   1. expm1-log1p-u 36.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

   2. expm1-udef 26.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

   3. *-commutative 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

   4. associate-*r* 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

   5. +-commutative 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

8. Applied egg-rr 26.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 36.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

   2. expm1-log1p 53.7%

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

   3. *-commutative 53.7%

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

   4. associate-*r* 63.7%

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

   5. +-commutative 63.7%

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

10. Simplified 63.7%

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

11. Taylor expanded in angle around 0 63.6%

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

12. Final simplification 63.6%

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

Alternative 10: 62.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ -0.011111111111111112 \cdot \left(\left(\left(\pi \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(a - b\right)\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.8%

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

3. Simplified 48.7%

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

4. Taylor expanded in angle around 0 48.2%

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

   1. unpow2 22.3%

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

   2. unpow2 22.3%

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

   3. difference-of-squares 23.9%

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

6. Applied egg-rr 53.7%

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

   1. expm1-log1p-u 36.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

   2. expm1-udef 26.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

   3. *-commutative 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

   4. associate-*r* 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

   5. +-commutative 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

8. Applied egg-rr 26.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 36.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

   2. expm1-log1p 53.7%

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

   3. *-commutative 53.7%

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

   4. associate-*r* 63.7%

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

   5. +-commutative 63.7%

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

10. Simplified 63.7%

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

11. Final simplification 63.7%

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

Alternative 11: 40.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.8%

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

3. Simplified 48.7%

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

4. Taylor expanded in angle around 0 48.2%

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

   1. unpow2 22.3%

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

   2. unpow2 22.3%

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

   3. difference-of-squares 23.9%

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

6. Applied egg-rr 53.7%

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

   1. expm1-log1p-u 36.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

   2. expm1-udef 26.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

   3. *-commutative 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

   4. associate-*r* 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

   5. +-commutative 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

8. Applied egg-rr 26.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 36.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

   2. expm1-log1p 53.7%

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

   3. *-commutative 53.7%

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

   4. associate-*r* 63.7%

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

   5. +-commutative 63.7%

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

10. Simplified 63.7%

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

11. Taylor expanded in a around inf 44.6%

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

12. Final simplification 44.6%

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

Alternative 12: 40.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.8%

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

3. Simplified 48.7%

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

4. Taylor expanded in angle around 0 48.2%

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

   1. unpow2 22.3%

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

   2. unpow2 22.3%

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

   3. difference-of-squares 23.9%

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

6. Applied egg-rr 53.7%

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

   1. expm1-log1p-u 36.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

   2. expm1-udef 26.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)} - 1} \]

   3. *-commutative 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112}\right)} - 1 \]

   4. associate-*r* 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)} - 1 \]

   5. +-commutative 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1 \]

8. Applied egg-rr 26.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 36.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)\right)} \]

   2. expm1-log1p 53.7%

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

   3. *-commutative 53.7%

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

   4. associate-*r* 63.7%

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

   5. +-commutative 63.7%

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

10. Simplified 63.7%

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

11. Taylor expanded in a around inf 44.6%

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

    1. *-commutative 44.6%

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

    2. associate-*l* 44.6%

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

13. Simplified 44.6%

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

14. Final simplification 44.6%

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

Alternative 13: 13.8% accurate, 617.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 0.0)

C

double code(double a, double b, double angle) {
	return 0.0;
}

Fortran

real(8) function code(a, b, angle)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = 0.0d0
end function

Java

public static double code(double a, double b, double angle) {
	return 0.0;
}

Python

def code(a, b, angle):
	return 0.0

Julia

function code(a, b, angle)
	return 0.0
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.0;
end

Wolfram

code[a_, b_, angle_] := 0.0

Derivation

1. Initial program 51.8%

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

   1. associate-*l* 51.8%

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

   2. associate-*l* 51.8%

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

   3. cos-neg 51.8%

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

   4. distribute-rgt-neg-out 51.8%

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

   5. distribute-frac-neg 51.8%

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

   6. neg-mul-1 51.8%

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

   7. associate-/l* 50.5%

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

   8. associate-*r/ 50.9%

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

   9. associate-/r/ 47.5%

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

   10. associate-/l* 47.5%

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

   11. metadata-eval 47.5%

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

3. Simplified 47.5%

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

4. Taylor expanded in b around 0 47.5%

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

   1. neg-mul-1 47.5%

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

   2. +-commutative 47.5%

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

   3. sub-neg 47.5%

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

   4. rem-square-sqrt 23.8%

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

   5. fabs-sqr 23.8%

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

   6. rem-square-sqrt 31.9%

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

   7. fabs-sub 31.9%

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

   8. rem-square-sqrt 12.9%

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

   9. fabs-sqr 12.9%

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

   10. rem-square-sqrt 22.3%

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

6. Simplified 22.3%

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

8. Applied egg-rr 2.8%

   \[\leadsto 2 \cdot \color{blue}{\frac{\left({a}^{6} - {b}^{6}\right) \cdot \left(\sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right) - angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(\pi, angle \cdot 0.005555555555555556, angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}{\left({a}^{4} + \left({b}^{4} + {\left(a \cdot b\right)}^{2}\right)\right) \cdot 2}} \]
9. Step-by-step derivation

   1. times-frac 2.8%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{a}^{6} - {b}^{6}}{{a}^{4} + \left({b}^{4} + {\left(a \cdot b\right)}^{2}\right)} \cdot \frac{\sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right) - angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(\pi, angle \cdot 0.005555555555555556, angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)}{2}\right)} \]

10. Simplified 11.2%

    \[\leadsto 2 \cdot \color{blue}{0} \]

11. Final simplification 11.2%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023314 
(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)))))