ab-angle->ABCF B

Percentage Accurate: 54.0% → 67.0%

Time: 38.6s

Alternatives: 15

Speedup: 5.5×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 3.7 \cdot 10^{+247}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \sin \left(2 \cdot \frac{0.005555555555555556}{\frac{1}{\pi} \cdot \frac{1}{angle}}\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt{\pi} \cdot \sqrt{\pi}}{180}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.7e+247)
   (*
    (*
     (- b a)
     (sin (* 2.0 (/ 0.005555555555555556 (* (/ 1.0 PI) (/ 1.0 angle))))))
    (+ a b))
   (*
    (- b a)
    (* (+ a b) (* 2.0 (sin (* angle (/ (* (sqrt PI) (sqrt PI)) 180.0))))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.7e+247) {
		tmp = ((b - a) * sin((2.0 * (0.005555555555555556 / ((1.0 / ((double) M_PI)) * (1.0 / angle)))))) * (a + b);
	} else {
		tmp = (b - a) * ((a + b) * (2.0 * sin((angle * ((sqrt(((double) M_PI)) * sqrt(((double) M_PI))) / 180.0)))));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.7e+247) {
		tmp = ((b - a) * Math.sin((2.0 * (0.005555555555555556 / ((1.0 / Math.PI) * (1.0 / angle)))))) * (a + b);
	} else {
		tmp = (b - a) * ((a + b) * (2.0 * Math.sin((angle * ((Math.sqrt(Math.PI) * Math.sqrt(Math.PI)) / 180.0)))));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	tmp = 0
	if a <= 3.7e+247:
		tmp = ((b - a) * math.sin((2.0 * (0.005555555555555556 / ((1.0 / math.pi) * (1.0 / angle)))))) * (a + b)
	else:
		tmp = (b - a) * ((a + b) * (2.0 * math.sin((angle * ((math.sqrt(math.pi) * math.sqrt(math.pi)) / 180.0)))))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.7e+247)
		tmp = Float64(Float64(Float64(b - a) * sin(Float64(2.0 * Float64(0.005555555555555556 / Float64(Float64(1.0 / pi) * Float64(1.0 / angle)))))) * Float64(a + b));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(a + b) * Float64(2.0 * sin(Float64(angle * Float64(Float64(sqrt(pi) * sqrt(pi)) / 180.0))))));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 3.7e+247)
		tmp = ((b - a) * sin((2.0 * (0.005555555555555556 / ((1.0 / pi) * (1.0 / angle)))))) * (a + b);
	else
		tmp = (b - a) * ((a + b) * (2.0 * sin((angle * ((sqrt(pi) * sqrt(pi)) / 180.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 3.7e+247], N[(N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(0.005555555555555556 / N[(N[(1.0 / Pi), $MachinePrecision] * N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(2.0 * N[Sin[N[(angle * N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.6999999999999998e247

   1. Initial program 53.7%

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

      1. *-commutative 53.7%

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

      2. associate-*l* 53.7%

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

      3. associate-*l* 53.7%

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

      4. unpow2 53.7%

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

      5. unpow2 53.7%

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

      6. sqr-neg 53.7%

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

      7. difference-of-squares 58.3%

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

      8. sub-neg 58.3%

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

      9. sub-neg 58.3%

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

      10. remove-double-neg 58.3%

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

   3. Simplified 58.3%

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

      1. *-commutative 58.3%

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

      2. associate-*l* 69.0%

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

      3. flip-+ 53.7%

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

      4. *-commutative 53.7%

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

      5. flip-- 28.4%

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

      6. div-inv 28.4%

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

      7. associate-/r* 28.4%

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

      8. *-inverses 69.0%

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

      9. associate-*l/ 69.0%

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

   5. Applied egg-rr 68.5%

      \[\leadsto \color{blue}{\frac{\left(b - a\right) \cdot \sin \left(\left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \pi\right)}{\frac{1}{b + a}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 68.5%

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

      2. /-rgt-identity 68.5%

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

      3. associate-*l* 68.5%

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

      4. *-commutative 68.5%

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

      5. *-commutative 68.5%

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

   7. Simplified 68.5%

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

      1. *-commutative 68.5%

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

      2. associate-*l* 67.8%

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

      3. metadata-eval 67.8%

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

      4. associate-/r/ 65.6%

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

      5. div-inv 66.4%

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

      6. associate-/r* 69.0%

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

      7. metadata-eval 69.0%

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

      8. *-commutative 69.0%

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

      9. associate-/r* 69.5%

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

   9. Applied egg-rr 69.5%

      \[\leadsto \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\frac{0.005555555555555556}{\frac{\frac{1}{\pi}}{angle}}}\right)\right) \cdot \left(b + a\right) \]
   10. Step-by-step derivation

       1. div-inv 69.8%

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

   11. Applied egg-rr 69.8%

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

   if 3.6999999999999998e247 < a

   1. Initial program 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-*l* 40.8%

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

      3. associate-*l* 40.8%

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

      4. unpow2 40.8%

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

      5. unpow2 40.8%

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

      6. sqr-neg 40.8%

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

      7. difference-of-squares 54.1%

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

      8. sub-neg 54.1%

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

      9. sub-neg 54.1%

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

      10. remove-double-neg 54.1%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in angle around 0 47.4%

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

      1. log1p-expm1-u_binary64 47.4%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot 1\right)\right)\right)} \]

   6. Applied rewrite-once 47.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot 1\right)\right)\right)} \]
   7. Step-by-step derivation

      1. log1p-expm1 47.4%

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

      2. *-rgt-identity 47.4%

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

      3. associate-*l* 59.9%

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

      4. +-commutative 59.9%

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

      5. associate-*r/ 66.6%

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

      6. associate-*l/ 53.3%

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

      7. *-commutative 53.3%

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

   8. Simplified 53.3%

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

      1. add-sqr-sqrt_binary64 93.2%

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

   10. Applied rewrite-once 93.2%

       \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}{180}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.7 \cdot 10^{+247}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \sin \left(2 \cdot \frac{0.005555555555555556}{\frac{1}{\pi} \cdot \frac{1}{angle}}\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt{\pi} \cdot \sqrt{\pi}}{180}\right)\right)\right)\\ \end{array} \]

Alternative 2: 67.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \left(\left(b - a\right) \cdot \sin \left(2 \cdot \frac{0.005555555555555556}{\frac{1}{\pi} \cdot \frac{1}{angle}}\right)\right) \cdot \left(a + b\right) \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (*
  (*
   (- b a)
   (sin (* 2.0 (/ 0.005555555555555556 (* (/ 1.0 PI) (/ 1.0 angle))))))
  (+ a b)))

C

a = abs(a);
double code(double a, double b, double angle) {
	return ((b - a) * sin((2.0 * (0.005555555555555556 / ((1.0 / ((double) M_PI)) * (1.0 / angle)))))) * (a + b);
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	return ((b - a) * Math.sin((2.0 * (0.005555555555555556 / ((1.0 / Math.PI) * (1.0 / angle)))))) * (a + b);
}

Python

a = abs(a)
def code(a, b, angle):
	return ((b - a) * math.sin((2.0 * (0.005555555555555556 / ((1.0 / math.pi) * (1.0 / angle)))))) * (a + b)

Julia

a = abs(a)
function code(a, b, angle)
	return Float64(Float64(Float64(b - a) * sin(Float64(2.0 * Float64(0.005555555555555556 / Float64(Float64(1.0 / pi) * Float64(1.0 / angle)))))) * Float64(a + b))
end

MATLAB

a = abs(a)
function tmp = code(a, b, angle)
	tmp = ((b - a) * sin((2.0 * (0.005555555555555556 / ((1.0 / pi) * (1.0 / angle)))))) * (a + b);
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := N[(N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(0.005555555555555556 / N[(N[(1.0 / Pi), $MachinePrecision] * N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

   1. *-commutative 53.0%

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

   2. associate-*l* 53.0%

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

   3. associate-*l* 53.0%

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

   4. unpow2 53.0%

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

   5. unpow2 53.0%

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

   6. sqr-neg 53.0%

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

   7. difference-of-squares 58.1%

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

   8. sub-neg 58.1%

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

   9. sub-neg 58.1%

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

   10. remove-double-neg 58.1%

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

3. Simplified 58.1%

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

   1. *-commutative 58.1%

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

   2. associate-*l* 68.9%

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

   3. flip-+ 52.9%

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

   4. *-commutative 52.9%

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

   5. flip-- 26.7%

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

   6. div-inv 26.7%

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

   7. associate-/r* 26.7%

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

   8. *-inverses 68.9%

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

   9. associate-*l/ 68.9%

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

5. Applied egg-rr 68.4%

   \[\leadsto \color{blue}{\frac{\left(b - a\right) \cdot \sin \left(\left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \pi\right)}{\frac{1}{b + a}}} \]
6. Step-by-step derivation

   1. associate-/r/ 68.4%

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

   2. /-rgt-identity 68.4%

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

   3. associate-*l* 68.4%

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

   4. *-commutative 68.4%

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

   5. *-commutative 68.4%

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

7. Simplified 68.4%

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

   1. *-commutative 68.4%

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

   2. associate-*l* 68.1%

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

   3. metadata-eval 68.1%

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

   4. associate-/r/ 65.6%

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

   5. div-inv 66.8%

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

   6. associate-/r* 69.2%

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

   7. metadata-eval 69.2%

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

   8. *-commutative 69.2%

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

   9. associate-/r* 69.7%

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

9. Applied egg-rr 69.7%

   \[\leadsto \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\frac{0.005555555555555556}{\frac{\frac{1}{\pi}}{angle}}}\right)\right) \cdot \left(b + a\right) \]
10. Step-by-step derivation

    1. div-inv 70.0%

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

11. Applied egg-rr 70.0%

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

12. Final simplification 70.0%

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

Alternative 3: 67.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \frac{0.005555555555555556}{\frac{1}{\pi \cdot angle}}\right)\right) \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (*
  (+ a b)
  (* (- b a) (sin (* 2.0 (/ 0.005555555555555556 (/ 1.0 (* PI angle))))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	return (a + b) * ((b - a) * sin((2.0 * (0.005555555555555556 / (1.0 / (((double) M_PI) * angle))))));
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	return (a + b) * ((b - a) * Math.sin((2.0 * (0.005555555555555556 / (1.0 / (Math.PI * angle))))));
}

Python

a = abs(a)
def code(a, b, angle):
	return (a + b) * ((b - a) * math.sin((2.0 * (0.005555555555555556 / (1.0 / (math.pi * angle))))))

Julia

a = abs(a)
function code(a, b, angle)
	return Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(0.005555555555555556 / Float64(1.0 / Float64(pi * angle)))))))
end

MATLAB

a = abs(a)
function tmp = code(a, b, angle)
	tmp = (a + b) * ((b - a) * sin((2.0 * (0.005555555555555556 / (1.0 / (pi * angle))))));
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(0.005555555555555556 / N[(1.0 / N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

   1. *-commutative 53.0%

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

   2. associate-*l* 53.0%

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

   3. associate-*l* 53.0%

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

   4. unpow2 53.0%

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

   5. unpow2 53.0%

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

   6. sqr-neg 53.0%

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

   7. difference-of-squares 58.1%

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

   8. sub-neg 58.1%

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

   9. sub-neg 58.1%

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

   10. remove-double-neg 58.1%

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

3. Simplified 58.1%

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

   1. *-commutative 58.1%

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

   2. associate-*l* 68.9%

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

   3. flip-+ 52.9%

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

   4. *-commutative 52.9%

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

   5. flip-- 26.7%

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

   6. div-inv 26.7%

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

   7. associate-/r* 26.7%

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

   8. *-inverses 68.9%

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

   9. associate-*l/ 68.9%

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

5. Applied egg-rr 68.4%

   \[\leadsto \color{blue}{\frac{\left(b - a\right) \cdot \sin \left(\left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \pi\right)}{\frac{1}{b + a}}} \]
6. Step-by-step derivation

   1. associate-/r/ 68.4%

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

   2. /-rgt-identity 68.4%

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

   3. associate-*l* 68.4%

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

   4. *-commutative 68.4%

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

   5. *-commutative 68.4%

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

7. Simplified 68.4%

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

   1. *-commutative 68.4%

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

   2. associate-*l* 68.1%

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

   3. metadata-eval 68.1%

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

   4. associate-/r/ 65.6%

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

   5. div-inv 66.8%

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

   6. associate-/r* 69.2%

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

   7. metadata-eval 69.2%

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

   8. *-commutative 69.2%

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

   9. associate-/r* 69.7%

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

9. Applied egg-rr 69.7%

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

10. Taylor expanded in angle around 0 69.2%

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

11. Final simplification 69.2%

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

Alternative 4: 67.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \frac{0.005555555555555556}{\frac{\frac{1}{\pi}}{angle}}\right)\right) \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (*
  (+ a b)
  (* (- b a) (sin (* 2.0 (/ 0.005555555555555556 (/ (/ 1.0 PI) angle)))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	return (a + b) * ((b - a) * sin((2.0 * (0.005555555555555556 / ((1.0 / ((double) M_PI)) / angle)))));
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	return (a + b) * ((b - a) * Math.sin((2.0 * (0.005555555555555556 / ((1.0 / Math.PI) / angle)))));
}

Python

a = abs(a)
def code(a, b, angle):
	return (a + b) * ((b - a) * math.sin((2.0 * (0.005555555555555556 / ((1.0 / math.pi) / angle)))))

Julia

a = abs(a)
function code(a, b, angle)
	return Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(0.005555555555555556 / Float64(Float64(1.0 / pi) / angle))))))
end

MATLAB

a = abs(a)
function tmp = code(a, b, angle)
	tmp = (a + b) * ((b - a) * sin((2.0 * (0.005555555555555556 / ((1.0 / pi) / angle)))));
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(0.005555555555555556 / N[(N[(1.0 / Pi), $MachinePrecision] / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

   1. *-commutative 53.0%

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

   2. associate-*l* 53.0%

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

   3. associate-*l* 53.0%

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

   4. unpow2 53.0%

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

   5. unpow2 53.0%

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

   6. sqr-neg 53.0%

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

   7. difference-of-squares 58.1%

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

   8. sub-neg 58.1%

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

   9. sub-neg 58.1%

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

   10. remove-double-neg 58.1%

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

3. Simplified 58.1%

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

   1. *-commutative 58.1%

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

   2. associate-*l* 68.9%

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

   3. flip-+ 52.9%

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

   4. *-commutative 52.9%

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

   5. flip-- 26.7%

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

   6. div-inv 26.7%

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

   7. associate-/r* 26.7%

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

   8. *-inverses 68.9%

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

   9. associate-*l/ 68.9%

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

5. Applied egg-rr 68.4%

   \[\leadsto \color{blue}{\frac{\left(b - a\right) \cdot \sin \left(\left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \pi\right)}{\frac{1}{b + a}}} \]
6. Step-by-step derivation

   1. associate-/r/ 68.4%

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

   2. /-rgt-identity 68.4%

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

   3. associate-*l* 68.4%

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

   4. *-commutative 68.4%

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

   5. *-commutative 68.4%

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

7. Simplified 68.4%

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

   1. *-commutative 68.4%

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

   2. associate-*l* 68.1%

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

   3. metadata-eval 68.1%

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

   4. associate-/r/ 65.6%

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

   5. div-inv 66.8%

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

   6. associate-/r* 69.2%

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

   7. metadata-eval 69.2%

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

   8. *-commutative 69.2%

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

   9. associate-/r* 69.7%

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

9. Applied egg-rr 69.7%

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

10. Final simplification 69.7%

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

Alternative 5: 67.4% accurate, 2.9× speedup

Math

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

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* (+ a b) (* (- b a) (sin (* PI (* angle 0.011111111111111112))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

   1. *-commutative 53.0%

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

   2. associate-*l* 53.0%

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

   3. associate-*l* 53.0%

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

   4. unpow2 53.0%

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

   5. unpow2 53.0%

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

   6. sqr-neg 53.0%

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

   7. difference-of-squares 58.1%

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

   8. sub-neg 58.1%

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

   9. sub-neg 58.1%

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

   10. remove-double-neg 58.1%

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

3. Simplified 58.1%

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

   1. *-commutative 58.1%

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

   2. associate-*l* 68.9%

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

   3. flip-+ 52.9%

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

   4. *-commutative 52.9%

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

   5. flip-- 26.7%

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

   6. div-inv 26.7%

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

   7. associate-/r* 26.7%

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

   8. *-inverses 68.9%

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

   9. associate-*l/ 68.9%

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

5. Applied egg-rr 68.4%

   \[\leadsto \color{blue}{\frac{\left(b - a\right) \cdot \sin \left(\left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \pi\right)}{\frac{1}{b + a}}} \]
6. Step-by-step derivation

   1. associate-/r/ 68.4%

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

   2. /-rgt-identity 68.4%

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

   3. associate-*l* 68.4%

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

   4. *-commutative 68.4%

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

   5. *-commutative 68.4%

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

7. Simplified 68.4%

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

   1. add-log-exp_binary64 27.9%

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

9. Applied rewrite-once 27.9%

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

    1. rem-log-exp 68.4%

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

    2. *-commutative 68.4%

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

    3. associate-*l* 68.1%

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

    4. metadata-eval 68.1%

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

    5. associate-/r/ 69.2%

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

    6. associate-/l/ 69.7%

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

    7. associate-*r/ 69.7%

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

    8. metadata-eval 69.7%

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

    9. associate-/l/ 69.2%

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

    10. associate-/r/ 68.1%

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

    11. metadata-eval 68.1%

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

    12. associate-*r* 68.4%

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

    13. *-commutative 68.4%

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

    14. *-commutative 68.4%

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

11. Applied egg-rr 68.4%

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

12. Final simplification 68.4%

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

Alternative 6: 51.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.06 \cdot 10^{-176}:\\ \;\;\;\;a \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.06e-176)
   (* a (* (sin (* (* PI angle) 0.011111111111111112)) (- a)))
   (* (+ a b) (* (- b a) (* PI (* angle 0.011111111111111112))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.06e-176) {
		tmp = a * (sin(((((double) M_PI) * angle) * 0.011111111111111112)) * -a);
	} else {
		tmp = (a + b) * ((b - a) * (((double) M_PI) * (angle * 0.011111111111111112)));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.06e-176) {
		tmp = a * (Math.sin(((Math.PI * angle) * 0.011111111111111112)) * -a);
	} else {
		tmp = (a + b) * ((b - a) * (Math.PI * (angle * 0.011111111111111112)));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	tmp = 0
	if b <= 1.06e-176:
		tmp = a * (math.sin(((math.pi * angle) * 0.011111111111111112)) * -a)
	else:
		tmp = (a + b) * ((b - a) * (math.pi * (angle * 0.011111111111111112)))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.06e-176)
		tmp = Float64(a * Float64(sin(Float64(Float64(pi * angle) * 0.011111111111111112)) * Float64(-a)));
	else
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(pi * Float64(angle * 0.011111111111111112))));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.06e-176)
		tmp = a * (sin(((pi * angle) * 0.011111111111111112)) * -a);
	else
		tmp = (a + b) * ((b - a) * (pi * (angle * 0.011111111111111112)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 1.06e-176], N[(a * N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.06000000000000006e-176

   1. Initial program 54.2%

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

      1. *-commutative 54.2%

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

      2. associate-*l* 54.2%

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

      3. associate-*l* 54.3%

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

      4. unpow2 54.3%

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

      5. unpow2 54.3%

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

      6. sqr-neg 54.3%

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

      7. difference-of-squares 59.4%

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

      8. sub-neg 59.4%

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

      9. sub-neg 59.4%

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

      10. remove-double-neg 59.4%

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

   3. Simplified 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-*l* 69.9%

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

      3. flip-+ 54.2%

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

      4. *-commutative 54.2%

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

      5. flip-- 29.9%

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

      6. div-inv 29.9%

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

      7. associate-/r* 29.9%

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

      8. *-inverses 69.9%

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

      9. associate-*l/ 69.9%

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

   5. Applied egg-rr 71.1%

      \[\leadsto \color{blue}{\frac{\left(b - a\right) \cdot \sin \left(\left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \pi\right)}{\frac{1}{b + a}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 71.1%

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

      2. /-rgt-identity 71.1%

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

      3. associate-*l* 71.1%

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

      4. *-commutative 71.1%

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

      5. *-commutative 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in b around 0 39.5%

      \[\leadsto \color{blue}{-1 \cdot \left({a}^{2} \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 39.5%

         \[\leadsto \color{blue}{\left(-1 \cdot {a}^{2}\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

      2. mul-1-neg 39.5%

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

      3. unpow2 39.5%

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

      4. distribute-rgt-neg-out 39.5%

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

      5. *-commutative 39.5%

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

      6. *-commutative 39.5%

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

      7. associate-*r* 39.7%

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

      8. associate-*l* 46.7%

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

      9. *-commutative 46.7%

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

      10. associate-*r* 46.6%

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

      11. *-commutative 46.6%

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

      12. *-commutative 46.6%

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

   10. Simplified 46.6%

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

   if 1.06000000000000006e-176 < b

   1. Initial program 51.0%

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

      1. *-commutative 51.0%

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

      2. associate-*l* 51.0%

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

      3. associate-*l* 51.0%

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

      4. unpow2 51.0%

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

      5. unpow2 51.0%

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

      6. sqr-neg 51.0%

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

      7. difference-of-squares 56.0%

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

      8. sub-neg 56.0%

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

      9. sub-neg 56.0%

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

      10. remove-double-neg 56.0%

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

   3. Simplified 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-*l* 67.2%

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

      3. flip-+ 51.0%

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

      4. *-commutative 51.0%

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

      5. flip-- 21.7%

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

      6. div-inv 21.6%

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

      7. associate-/r* 21.6%

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

      8. *-inverses 67.2%

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

      9. associate-*l/ 67.2%

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

   5. Applied egg-rr 64.1%

      \[\leadsto \color{blue}{\frac{\left(b - a\right) \cdot \sin \left(\left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \pi\right)}{\frac{1}{b + a}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 64.1%

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

      2. /-rgt-identity 64.1%

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

      3. associate-*l* 64.1%

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

      4. *-commutative 64.1%

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

      5. *-commutative 64.1%

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

   7. Simplified 64.1%

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

      1. *-commutative 64.1%

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

      2. associate-*l* 64.3%

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

      3. metadata-eval 64.3%

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

      4. associate-/r/ 61.2%

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

      5. div-inv 64.1%

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

      6. associate-/r* 65.3%

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

      7. metadata-eval 65.3%

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

      8. *-commutative 65.3%

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

      9. associate-/r* 66.4%

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

   9. Applied egg-rr 66.4%

      \[\leadsto \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\frac{0.005555555555555556}{\frac{\frac{1}{\pi}}{angle}}}\right)\right) \cdot \left(b + a\right) \]
   10. Step-by-step derivation

       1. div-inv 66.7%

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

   11. Applied egg-rr 66.7%

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

   12. Taylor expanded in angle around 0 62.4%

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

   14. Simplified 62.3%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.06 \cdot 10^{-176}:\\ \;\;\;\;a \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 7: 65.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 2.25 \cdot 10^{-142}:\\ \;\;\;\;b \cdot \left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.25e-142)
   (* b (* b (sin (* (* PI angle) 0.011111111111111112))))
   (* (- b a) (* 0.011111111111111112 (* angle (* PI (+ a b)))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.25e-142) {
		tmp = b * (b * sin(((((double) M_PI) * angle) * 0.011111111111111112)));
	} else {
		tmp = (b - a) * (0.011111111111111112 * (angle * (((double) M_PI) * (a + b))));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.25e-142) {
		tmp = b * (b * Math.sin(((Math.PI * angle) * 0.011111111111111112)));
	} else {
		tmp = (b - a) * (0.011111111111111112 * (angle * (Math.PI * (a + b))));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	tmp = 0
	if a <= 2.25e-142:
		tmp = b * (b * math.sin(((math.pi * angle) * 0.011111111111111112)))
	else:
		tmp = (b - a) * (0.011111111111111112 * (angle * (math.pi * (a + b))))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.25e-142)
		tmp = Float64(b * Float64(b * sin(Float64(Float64(pi * angle) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(b - a) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(a + b)))));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.25e-142)
		tmp = b * (b * sin(((pi * angle) * 0.011111111111111112)));
	else
		tmp = (b - a) * (0.011111111111111112 * (angle * (pi * (a + b))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 2.25e-142], N[(b * N[(b * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.25000000000000009e-142

   1. Initial program 60.0%

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

      1. *-commutative 60.0%

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

      2. associate-*l* 60.0%

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

      3. associate-*l* 60.0%

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

      4. unpow2 60.0%

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

      5. unpow2 60.0%

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

      6. sqr-neg 60.0%

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

      7. difference-of-squares 65.2%

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

      8. sub-neg 65.2%

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

      9. sub-neg 65.2%

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

      10. remove-double-neg 65.2%

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

   3. Simplified 65.2%

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

      1. *-commutative 65.2%

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

      2. associate-*l* 71.2%

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

      3. flip-+ 59.9%

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

      4. *-commutative 59.9%

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

      5. flip-- 27.9%

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

      6. div-inv 27.9%

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

      7. associate-/r* 27.9%

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

      8. *-inverses 71.2%

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

      9. associate-*l/ 71.2%

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

   5. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\frac{\left(b - a\right) \cdot \sin \left(\left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \pi\right)}{\frac{1}{b + a}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 69.3%

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

      2. /-rgt-identity 69.3%

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

      3. associate-*l* 69.3%

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

      4. *-commutative 69.3%

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

      5. *-commutative 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in b around inf 43.2%

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

      1. unpow2 43.2%

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

      2. *-commutative 43.2%

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

      3. *-commutative 43.2%

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

      4. associate-*r* 43.0%

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

      5. associate-*l* 46.0%

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

      6. associate-*r* 46.2%

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

      7. *-commutative 46.2%

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

      8. *-commutative 46.2%

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

   10. Simplified 46.2%

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

   if 2.25000000000000009e-142 < a

   1. Initial program 42.2%

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

      1. *-commutative 42.2%

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

      2. associate-*l* 42.2%

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

      3. associate-*l* 42.2%

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

      4. unpow2 42.2%

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

      5. unpow2 42.2%

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

      6. sqr-neg 42.2%

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

      7. difference-of-squares 47.1%

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

      8. sub-neg 47.1%

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

      9. sub-neg 47.1%

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

      10. remove-double-neg 47.1%

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

   3. Simplified 47.1%

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

   4. Taylor expanded in angle around 0 43.1%

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

      1. log1p-expm1-u_binary64 36.2%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot 1\right)\right)\right)} \]

   6. Applied rewrite-once 36.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot 1\right)\right)\right)} \]
   7. Step-by-step derivation

      1. log1p-expm1 43.1%

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

      2. *-rgt-identity 43.1%

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

      3. associate-*l* 61.3%

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

      4. +-commutative 61.3%

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

      5. associate-*r/ 63.7%

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

      6. associate-*l/ 61.4%

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

      7. *-commutative 61.4%

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

   8. Simplified 61.4%

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

   9. Taylor expanded in angle around 0 62.8%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.25 \cdot 10^{-142}:\\ \;\;\;\;b \cdot \left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 57.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 6.9 \cdot 10^{+176}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot \left(-angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 6.9e+176)
   (* 0.011111111111111112 (* angle (* PI (* (- b a) (+ a b)))))
   (* 0.011111111111111112 (* a (* a (* PI (- angle)))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6.9e+176) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * ((b - a) * (a + b))));
	} else {
		tmp = 0.011111111111111112 * (a * (a * (((double) M_PI) * -angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6.9e+176) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * ((b - a) * (a + b))));
	} else {
		tmp = 0.011111111111111112 * (a * (a * (Math.PI * -angle)));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	tmp = 0
	if a <= 6.9e+176:
		tmp = 0.011111111111111112 * (angle * (math.pi * ((b - a) * (a + b))))
	else:
		tmp = 0.011111111111111112 * (a * (a * (math.pi * -angle)))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 6.9e+176)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(b - a) * Float64(a + b)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(a * Float64(a * Float64(pi * Float64(-angle)))));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 6.9e+176)
		tmp = 0.011111111111111112 * (angle * (pi * ((b - a) * (a + b))));
	else
		tmp = 0.011111111111111112 * (a * (a * (pi * -angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 6.9e+176], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(a * N[(a * N[(Pi * (-angle)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.9000000000000002e176

   1. Initial program 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*l* 55.7%

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

      3. associate-*l* 55.7%

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

      4. unpow2 55.7%

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

      5. unpow2 55.7%

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

      6. sqr-neg 55.7%

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

      7. difference-of-squares 60.3%

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

      8. sub-neg 60.3%

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

      9. sub-neg 60.3%

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

      10. remove-double-neg 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in angle around 0 53.5%

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

   if 6.9000000000000002e176 < a

   1. Initial program 33.6%

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

      1. *-commutative 33.6%

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

      2. associate-*l* 33.6%

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

      3. associate-*l* 33.6%

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

      4. unpow2 33.6%

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

      5. unpow2 33.6%

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

      6. sqr-neg 33.6%

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

      7. difference-of-squares 43.0%

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

      8. sub-neg 43.0%

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

      9. sub-neg 43.0%

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

      10. remove-double-neg 43.0%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in angle around 0 49.3%

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

   5. Taylor expanded in a around inf 33.6%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(-1 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 33.6%

         \[\leadsto 0.011111111111111112 \cdot \left(-1 \cdot \left({a}^{2} \cdot \color{blue}{\left(\pi \cdot angle\right)}\right)\right) \]

      2. associate-*r* 33.6%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(-1 \cdot {a}^{2}\right) \cdot \left(\pi \cdot angle\right)\right)} \]

      3. mul-1-neg 33.6%

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

      4. unpow2 33.6%

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

      5. distribute-rgt-neg-out 33.6%

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

      6. associate-*l* 65.2%

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

      7. *-commutative 65.2%

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

   7. Simplified 65.2%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.9 \cdot 10^{+176}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot \left(-angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 9: 51.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 5.1 \cdot 10^{+29}:\\ \;\;\;\;\pi \cdot \frac{\left(b \cdot b\right) \cdot -0.011111111111111112}{\frac{-1}{angle}}\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot \left(-angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 5.1e+29)
   (* PI (/ (* (* b b) -0.011111111111111112) (/ -1.0 angle)))
   (* 0.011111111111111112 (* a (* a (* PI (- angle)))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.1e+29) {
		tmp = ((double) M_PI) * (((b * b) * -0.011111111111111112) / (-1.0 / angle));
	} else {
		tmp = 0.011111111111111112 * (a * (a * (((double) M_PI) * -angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.1e+29) {
		tmp = Math.PI * (((b * b) * -0.011111111111111112) / (-1.0 / angle));
	} else {
		tmp = 0.011111111111111112 * (a * (a * (Math.PI * -angle)));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	tmp = 0
	if a <= 5.1e+29:
		tmp = math.pi * (((b * b) * -0.011111111111111112) / (-1.0 / angle))
	else:
		tmp = 0.011111111111111112 * (a * (a * (math.pi * -angle)))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 5.1e+29)
		tmp = Float64(pi * Float64(Float64(Float64(b * b) * -0.011111111111111112) / Float64(-1.0 / angle)));
	else
		tmp = Float64(0.011111111111111112 * Float64(a * Float64(a * Float64(pi * Float64(-angle)))));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 5.1e+29)
		tmp = pi * (((b * b) * -0.011111111111111112) / (-1.0 / angle));
	else
		tmp = 0.011111111111111112 * (a * (a * (pi * -angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 5.1e+29], N[(Pi * N[(N[(N[(b * b), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] / N[(-1.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(a * N[(a * N[(Pi * (-angle)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.1000000000000001e29

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*l* 60.3%

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

      3. associate-*l* 60.3%

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

      4. unpow2 60.3%

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

      5. unpow2 60.3%

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

      6. sqr-neg 60.3%

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

      7. difference-of-squares 64.6%

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

      8. sub-neg 64.6%

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

      9. sub-neg 64.6%

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

      10. remove-double-neg 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in angle around 0 58.0%

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

   5. Taylor expanded in a around 0 41.1%

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

      1. *-commutative 41.1%

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

      2. unpow2 41.1%

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

   7. Simplified 41.1%

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

      1. associate-*r* 41.1%

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

      2. associate-*r* 41.1%

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

      3. metadata-eval 41.1%

         \[\leadsto \left(\color{blue}{\frac{0.011111111111111112}{1}} \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b\right) \]

      4. associate-/r/ 41.1%

         \[\leadsto \color{blue}{\frac{0.011111111111111112}{\frac{1}{angle \cdot \pi}}} \cdot \left(b \cdot b\right) \]

      5. metadata-eval 41.1%

         \[\leadsto \frac{\color{blue}{2 \cdot 0.005555555555555556}}{\frac{1}{angle \cdot \pi}} \cdot \left(b \cdot b\right) \]

      6. associate-/l/ 41.1%

         \[\leadsto \frac{2 \cdot 0.005555555555555556}{\color{blue}{\frac{\frac{1}{\pi}}{angle}}} \cdot \left(b \cdot b\right) \]

      7. associate-*r/ 41.1%

         \[\leadsto \color{blue}{\left(2 \cdot \frac{0.005555555555555556}{\frac{\frac{1}{\pi}}{angle}}\right)} \cdot \left(b \cdot b\right) \]

      8. frac-2neg 41.1%

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

      9. associate-*r/ 41.1%

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

      10. associate-*l/ 41.1%

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

      11. metadata-eval 41.1%

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

      12. metadata-eval 41.1%

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

      13. associate-/l/ 41.1%

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

      14. distribute-neg-frac 41.1%

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

      15. metadata-eval 41.1%

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

      16. associate-/r* 41.1%

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

   9. Applied egg-rr 41.1%

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

       1. associate-/r/ 41.1%

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

       2. *-commutative 41.1%

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

   11. Simplified 41.1%

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

   if 5.1000000000000001e29 < a

   1. Initial program 33.6%

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

      1. *-commutative 33.6%

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

      2. associate-*l* 33.6%

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

      3. associate-*l* 33.6%

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

      4. unpow2 33.6%

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

      5. unpow2 33.6%

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

      6. sqr-neg 33.6%

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

      7. difference-of-squares 40.8%

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

      8. sub-neg 40.8%

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

      9. sub-neg 40.8%

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

      10. remove-double-neg 40.8%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in angle around 0 39.8%

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

   5. Taylor expanded in a around inf 30.9%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(-1 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 30.9%

         \[\leadsto 0.011111111111111112 \cdot \left(-1 \cdot \left({a}^{2} \cdot \color{blue}{\left(\pi \cdot angle\right)}\right)\right) \]

      2. associate-*r* 30.9%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(-1 \cdot {a}^{2}\right) \cdot \left(\pi \cdot angle\right)\right)} \]

      3. mul-1-neg 30.9%

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

      4. unpow2 30.9%

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

      5. distribute-rgt-neg-out 30.9%

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

      6. associate-*l* 48.1%

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

      7. *-commutative 48.1%

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

   7. Simplified 48.1%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.1 \cdot 10^{+29}:\\ \;\;\;\;\pi \cdot \frac{\left(b \cdot b\right) \cdot -0.011111111111111112}{\frac{-1}{angle}}\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot \left(-angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 10: 51.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{+34}:\\ \;\;\;\;\frac{2 \cdot \left(b \cdot b\right)}{\frac{\frac{180}{angle}}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot \left(-angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.15e+34)
   (/ (* 2.0 (* b b)) (/ (/ 180.0 angle) PI))
   (* 0.011111111111111112 (* a (* a (* PI (- angle)))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.15e+34) {
		tmp = (2.0 * (b * b)) / ((180.0 / angle) / ((double) M_PI));
	} else {
		tmp = 0.011111111111111112 * (a * (a * (((double) M_PI) * -angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.15e+34) {
		tmp = (2.0 * (b * b)) / ((180.0 / angle) / Math.PI);
	} else {
		tmp = 0.011111111111111112 * (a * (a * (Math.PI * -angle)));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	tmp = 0
	if a <= 1.15e+34:
		tmp = (2.0 * (b * b)) / ((180.0 / angle) / math.pi)
	else:
		tmp = 0.011111111111111112 * (a * (a * (math.pi * -angle)))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.15e+34)
		tmp = Float64(Float64(2.0 * Float64(b * b)) / Float64(Float64(180.0 / angle) / pi));
	else
		tmp = Float64(0.011111111111111112 * Float64(a * Float64(a * Float64(pi * Float64(-angle)))));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.15e+34)
		tmp = (2.0 * (b * b)) / ((180.0 / angle) / pi);
	else
		tmp = 0.011111111111111112 * (a * (a * (pi * -angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 1.15e+34], N[(N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(180.0 / angle), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(a * N[(a * N[(Pi * (-angle)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.1499999999999999e34

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*l* 60.3%

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

      3. associate-*l* 60.3%

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

      4. unpow2 60.3%

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

      5. unpow2 60.3%

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

      6. sqr-neg 60.3%

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

      7. difference-of-squares 64.6%

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

      8. sub-neg 64.6%

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

      9. sub-neg 64.6%

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

      10. remove-double-neg 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in angle around 0 58.0%

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

   5. Taylor expanded in a around 0 41.1%

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

      1. *-commutative 41.1%

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

      2. unpow2 41.1%

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

   7. Simplified 41.1%

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

      1. associate-*r* 41.1%

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

      2. associate-*r* 41.1%

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

      3. metadata-eval 41.1%

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

      4. associate-*r* 41.1%

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

      5. metadata-eval 41.1%

         \[\leadsto \left(2 \cdot \left(\color{blue}{\frac{0.005555555555555556}{1}} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(b \cdot b\right) \]

      6. associate-/r/ 41.1%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}}\right) \cdot \left(b \cdot b\right) \]

      7. associate-/l/ 41.1%

         \[\leadsto \left(2 \cdot \frac{0.005555555555555556}{\color{blue}{\frac{\frac{1}{\pi}}{angle}}}\right) \cdot \left(b \cdot b\right) \]

      8. clear-num 41.1%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{1}{\frac{\frac{\frac{1}{\pi}}{angle}}{0.005555555555555556}}}\right) \cdot \left(b \cdot b\right) \]

      9. un-div-inv 41.1%

         \[\leadsto \color{blue}{\frac{2}{\frac{\frac{\frac{1}{\pi}}{angle}}{0.005555555555555556}}} \cdot \left(b \cdot b\right) \]

      10. associate-*l/ 41.1%

          \[\leadsto \color{blue}{\frac{2 \cdot \left(b \cdot b\right)}{\frac{\frac{\frac{1}{\pi}}{angle}}{0.005555555555555556}}} \]

      11. associate-/l/ 41.1%

          \[\leadsto \frac{2 \cdot \left(b \cdot b\right)}{\frac{\color{blue}{\frac{1}{angle \cdot \pi}}}{0.005555555555555556}} \]

      12. associate-/l/ 41.1%

          \[\leadsto \frac{2 \cdot \left(b \cdot b\right)}{\color{blue}{\frac{1}{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}}} \]

      13. associate-/r* 41.1%

          \[\leadsto \frac{2 \cdot \left(b \cdot b\right)}{\color{blue}{\frac{\frac{1}{0.005555555555555556}}{angle \cdot \pi}}} \]

      14. metadata-eval 41.1%

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

      15. associate-/r* 41.2%

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

   9. Applied egg-rr 41.2%

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

   if 1.1499999999999999e34 < a

   1. Initial program 33.6%

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

      1. *-commutative 33.6%

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

      2. associate-*l* 33.6%

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

      3. associate-*l* 33.6%

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

      4. unpow2 33.6%

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

      5. unpow2 33.6%

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

      6. sqr-neg 33.6%

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

      7. difference-of-squares 40.8%

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

      8. sub-neg 40.8%

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

      9. sub-neg 40.8%

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

      10. remove-double-neg 40.8%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in angle around 0 39.8%

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

   5. Taylor expanded in a around inf 30.9%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(-1 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 30.9%

         \[\leadsto 0.011111111111111112 \cdot \left(-1 \cdot \left({a}^{2} \cdot \color{blue}{\left(\pi \cdot angle\right)}\right)\right) \]

      2. associate-*r* 30.9%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(-1 \cdot {a}^{2}\right) \cdot \left(\pi \cdot angle\right)\right)} \]

      3. mul-1-neg 30.9%

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

      4. unpow2 30.9%

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

      5. distribute-rgt-neg-out 30.9%

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

      6. associate-*l* 48.1%

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

      7. *-commutative 48.1%

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

   7. Simplified 48.1%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{+34}:\\ \;\;\;\;\frac{2 \cdot \left(b \cdot b\right)}{\frac{\frac{180}{angle}}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot \left(-angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 11: 62.3% accurate, 5.5× speedup

Math

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

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* (+ a b) (* (- b a) (* (* PI angle) 0.011111111111111112))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

   1. *-commutative 53.0%

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

   2. associate-*l* 53.0%

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

   3. associate-*l* 53.0%

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

   4. unpow2 53.0%

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

   5. unpow2 53.0%

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

   6. sqr-neg 53.0%

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

   7. difference-of-squares 58.1%

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

   8. sub-neg 58.1%

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

   9. sub-neg 58.1%

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

   10. remove-double-neg 58.1%

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

3. Simplified 58.1%

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

   1. *-commutative 58.1%

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

   2. associate-*l* 68.9%

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

   3. flip-+ 52.9%

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

   4. *-commutative 52.9%

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

   5. flip-- 26.7%

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

   6. div-inv 26.7%

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

   7. associate-/r* 26.7%

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

   8. *-inverses 68.9%

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

   9. associate-*l/ 68.9%

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

5. Applied egg-rr 68.4%

   \[\leadsto \color{blue}{\frac{\left(b - a\right) \cdot \sin \left(\left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \pi\right)}{\frac{1}{b + a}}} \]
6. Step-by-step derivation

   1. associate-/r/ 68.4%

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

   2. /-rgt-identity 68.4%

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

   3. associate-*l* 68.4%

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

   4. *-commutative 68.4%

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

   5. *-commutative 68.4%

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

7. Simplified 68.4%

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

8. Taylor expanded in angle around 0 61.9%

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

9. Final simplification 61.9%

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

Alternative 12: 62.3% accurate, 5.5× speedup

Math

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

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* (- b a) (* 0.011111111111111112 (* angle (* PI (+ a b))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := N[(N[(b - a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

   1. *-commutative 53.0%

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

   2. associate-*l* 53.0%

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

   3. associate-*l* 53.0%

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

   4. unpow2 53.0%

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

   5. unpow2 53.0%

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

   6. sqr-neg 53.0%

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

   7. difference-of-squares 58.1%

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

   8. sub-neg 58.1%

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

   9. sub-neg 58.1%

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

   10. remove-double-neg 58.1%

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

3. Simplified 58.1%

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

4. Taylor expanded in angle around 0 53.1%

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

   1. log1p-expm1-u_binary64 45.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot 1\right)\right)\right)} \]

6. Applied rewrite-once 45.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot 1\right)\right)\right)} \]
7. Step-by-step derivation

   1. log1p-expm1 53.1%

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

   2. *-rgt-identity 53.1%

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

   3. associate-*l* 63.9%

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

   4. +-commutative 63.9%

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

   5. associate-*r/ 65.8%

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

   6. associate-*l/ 64.0%

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

   7. *-commutative 64.0%

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

8. Simplified 64.0%

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

9. Taylor expanded in angle around 0 61.9%

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

10. Final simplification 61.9%

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

Alternative 13: 51.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 7.8 \cdot 10^{+31}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot \left(-angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 7.8e+31)
   (* 0.011111111111111112 (* angle (* PI (* b b))))
   (* 0.011111111111111112 (* a (* a (* PI (- angle)))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 7.8e+31) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	} else {
		tmp = 0.011111111111111112 * (a * (a * (((double) M_PI) * -angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 7.8e+31) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	} else {
		tmp = 0.011111111111111112 * (a * (a * (Math.PI * -angle)));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	tmp = 0
	if a <= 7.8e+31:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	else:
		tmp = 0.011111111111111112 * (a * (a * (math.pi * -angle)))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 7.8e+31)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	else
		tmp = Float64(0.011111111111111112 * Float64(a * Float64(a * Float64(pi * Float64(-angle)))));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 7.8e+31)
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	else
		tmp = 0.011111111111111112 * (a * (a * (pi * -angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 7.8e+31], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(a * N[(a * N[(Pi * (-angle)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.79999999999999999e31

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*l* 60.3%

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

      3. associate-*l* 60.3%

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

      4. unpow2 60.3%

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

      5. unpow2 60.3%

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

      6. sqr-neg 60.3%

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

      7. difference-of-squares 64.6%

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

      8. sub-neg 64.6%

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

      9. sub-neg 64.6%

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

      10. remove-double-neg 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in angle around 0 58.0%

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

   5. Taylor expanded in a around 0 41.1%

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

      1. *-commutative 41.1%

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

      2. unpow2 41.1%

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

   7. Simplified 41.1%

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

   if 7.79999999999999999e31 < a

   1. Initial program 33.6%

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

      1. *-commutative 33.6%

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

      2. associate-*l* 33.6%

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

      3. associate-*l* 33.6%

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

      4. unpow2 33.6%

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

      5. unpow2 33.6%

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

      6. sqr-neg 33.6%

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

      7. difference-of-squares 40.8%

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

      8. sub-neg 40.8%

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

      9. sub-neg 40.8%

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

      10. remove-double-neg 40.8%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in angle around 0 39.8%

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

   5. Taylor expanded in a around inf 30.9%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(-1 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 30.9%

         \[\leadsto 0.011111111111111112 \cdot \left(-1 \cdot \left({a}^{2} \cdot \color{blue}{\left(\pi \cdot angle\right)}\right)\right) \]

      2. associate-*r* 30.9%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(-1 \cdot {a}^{2}\right) \cdot \left(\pi \cdot angle\right)\right)} \]

      3. mul-1-neg 30.9%

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

      4. unpow2 30.9%

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

      5. distribute-rgt-neg-out 30.9%

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

      6. associate-*l* 48.1%

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

      7. *-commutative 48.1%

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

   7. Simplified 48.1%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.8 \cdot 10^{+31}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot \left(-angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 14: 46.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{+30}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.5e+30)
   (* 0.011111111111111112 (* angle (* PI (* b b))))
   (* (* PI angle) (* -0.011111111111111112 (* a a)))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.5e+30) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	} else {
		tmp = (((double) M_PI) * angle) * (-0.011111111111111112 * (a * a));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.5e+30) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	} else {
		tmp = (Math.PI * angle) * (-0.011111111111111112 * (a * a));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	tmp = 0
	if a <= 4.5e+30:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	else:
		tmp = (math.pi * angle) * (-0.011111111111111112 * (a * a))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 4.5e+30)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	else
		tmp = Float64(Float64(pi * angle) * Float64(-0.011111111111111112 * Float64(a * a)));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 4.5e+30)
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	else
		tmp = (pi * angle) * (-0.011111111111111112 * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 4.5e+30], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * angle), $MachinePrecision] * N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.49999999999999995e30

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*l* 60.3%

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

      3. associate-*l* 60.3%

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

      4. unpow2 60.3%

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

      5. unpow2 60.3%

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

      6. sqr-neg 60.3%

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

      7. difference-of-squares 64.6%

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

      8. sub-neg 64.6%

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

      9. sub-neg 64.6%

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

      10. remove-double-neg 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in angle around 0 58.0%

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

   5. Taylor expanded in a around 0 41.1%

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

      1. *-commutative 41.1%

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

      2. unpow2 41.1%

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

   7. Simplified 41.1%

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

   if 4.49999999999999995e30 < a

   1. Initial program 33.6%

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

      1. *-commutative 33.6%

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

      2. associate-*l* 33.6%

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

      3. associate-*l* 33.6%

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

      4. unpow2 33.6%

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

      5. unpow2 33.6%

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

      6. sqr-neg 33.6%

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

      7. difference-of-squares 40.8%

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

      8. sub-neg 40.8%

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

      9. sub-neg 40.8%

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

      10. remove-double-neg 40.8%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in angle around 0 39.8%

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

   5. Taylor expanded in a around inf 30.9%

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

      1. *-commutative 30.9%

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

      2. associate-*r* 31.0%

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

      3. unpow2 31.0%

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

      4. *-commutative 31.0%

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

   7. Simplified 31.0%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{+30}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 15: 35.9% accurate, 5.7× speedup

Math

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

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* angle (* PI (* b b)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

   1. *-commutative 53.0%

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

   2. associate-*l* 53.0%

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

   3. associate-*l* 53.0%

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

   4. unpow2 53.0%

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

   5. unpow2 53.0%

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

   6. sqr-neg 53.0%

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

   7. difference-of-squares 58.1%

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

   8. sub-neg 58.1%

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

   9. sub-neg 58.1%

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

   10. remove-double-neg 58.1%

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

3. Simplified 58.1%

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

4. Taylor expanded in angle around 0 53.0%

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

5. Taylor expanded in a around 0 34.3%

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

   1. *-commutative 34.3%

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

   2. unpow2 34.3%

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

7. Simplified 34.3%

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

8. Final simplification 34.3%

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

Reproduce

herbie shell --seed 2023297 
(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)))))