ab-angle->ABCF B

Percentage Accurate: 54.5% → 67.4%

Time: 40.1s

Alternatives: 20

Speedup: 5.5×

• 32.3% 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: 54.5% 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: 67.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \left(\left(\sin \left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b + a\right)\right) \cdot \cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi \cdot \left(\pi \cdot \pi\right)}\right)\right) \cdot \left(2 \cdot \left(b - a\right)\right) \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := N[(N[(N[(N[Sin[N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[Power[N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

   1. associate-*l* 56.8%

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

   2. unpow2 56.8%

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

   3. unpow2 56.8%

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

   4. difference-of-squares 58.8%

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

3. Simplified 58.8%

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

4. Taylor expanded in angle around inf 69.4%

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

   1. associate-*r* 69.4%

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

   2. *-commutative 69.4%

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

   3. *-commutative 69.4%

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

   4. *-commutative 69.4%

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

   5. *-commutative 69.4%

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

   6. associate-*r* 70.2%

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

   7. +-commutative 70.2%

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

   8. *-commutative 70.2%

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

   9. *-commutative 70.2%

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

   10. associate-*r* 70.3%

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

6. Simplified 70.3%

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

   1. add-cbrt-cube 71.0%

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

8. Applied egg-rr 71.0%

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

   1. add-sqr-sqrt 72.5%

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

10. Applied egg-rr 72.5%

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

11. Final simplification 72.5%

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

Alternative 2: 67.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} t_0 := 2 \cdot \left(b - a\right)\\ t_1 := \left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ t_2 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t_2\right) \cdot \cos t_2 \leq -\infty:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (- b a)))
        (t_1 (* (+ b a) (sin (* PI (* angle 0.005555555555555556)))))
        (t_2 (* PI (/ angle 180.0))))
   (if (<=
        (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_2)) (cos t_2))
        (- INFINITY))
     (*
      t_0
      (*
       t_1
       (+ 1.0 (* -1.54320987654321e-5 (* (* angle angle) (pow PI 2.0))))))
     (* t_0 (* t_1 (cos (* 0.005555555555555556 (* PI angle))))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = (b + a) * sin((((double) M_PI) * (angle * 0.005555555555555556)));
	double t_2 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_2)) * cos(t_2)) <= -((double) INFINITY)) {
		tmp = t_0 * (t_1 * (1.0 + (-1.54320987654321e-5 * ((angle * angle) * pow(((double) M_PI), 2.0)))));
	} else {
		tmp = t_0 * (t_1 * cos((0.005555555555555556 * (((double) M_PI) * angle))));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = (b + a) * Math.sin((Math.PI * (angle * 0.005555555555555556)));
	double t_2 = Math.PI * (angle / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_2)) * Math.cos(t_2)) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0 * (t_1 * (1.0 + (-1.54320987654321e-5 * ((angle * angle) * Math.pow(Math.PI, 2.0)))));
	} else {
		tmp = t_0 * (t_1 * Math.cos((0.005555555555555556 * (Math.PI * angle))));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	t_0 = 2.0 * (b - a)
	t_1 = (b + a) * math.sin((math.pi * (angle * 0.005555555555555556)))
	t_2 = math.pi * (angle / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_2)) * math.cos(t_2)) <= -math.inf:
		tmp = t_0 * (t_1 * (1.0 + (-1.54320987654321e-5 * ((angle * angle) * math.pow(math.pi, 2.0)))))
	else:
		tmp = t_0 * (t_1 * math.cos((0.005555555555555556 * (math.pi * angle))))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	t_0 = Float64(2.0 * Float64(b - a))
	t_1 = Float64(Float64(b + a) * sin(Float64(pi * Float64(angle * 0.005555555555555556))))
	t_2 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_2)) * cos(t_2)) <= Float64(-Inf))
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 + Float64(-1.54320987654321e-5 * Float64(Float64(angle * angle) * (pi ^ 2.0))))));
	else
		tmp = Float64(t_0 * Float64(t_1 * cos(Float64(0.005555555555555556 * Float64(pi * angle)))));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	t_0 = 2.0 * (b - a);
	t_1 = (b + a) * sin((pi * (angle * 0.005555555555555556)));
	t_2 = pi * (angle / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_2)) * cos(t_2)) <= -Inf)
		tmp = t_0 * (t_1 * (1.0 + (-1.54320987654321e-5 * ((angle * angle) * (pi ^ 2.0)))));
	else
		tmp = t_0 * (t_1 * cos((0.005555555555555556 * (pi * angle))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + a), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(t$95$0 * N[(t$95$1 * N[(1.0 + N[(-1.54320987654321e-5 * N[(N[(angle * angle), $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$1 * N[Cos[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) < -inf.0

   1. Initial program 48.2%

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

      1. associate-*l* 48.2%

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

      2. unpow2 48.2%

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

      3. unpow2 48.2%

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

      4. difference-of-squares 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in angle around inf 73.9%

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

      1. associate-*r* 73.9%

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

      2. *-commutative 73.9%

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

      3. *-commutative 73.9%

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

      4. *-commutative 73.9%

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

      5. *-commutative 73.9%

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

      6. associate-*r* 73.7%

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

      7. +-commutative 73.7%

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

      8. *-commutative 73.7%

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

      9. *-commutative 73.7%

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

      10. associate-*r* 75.6%

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

   6. Simplified 75.6%

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

   7. Taylor expanded in angle around 0 83.0%

      \[\leadsto \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)}\right) \cdot \left(2 \cdot \left(b - a\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 83.0%

         \[\leadsto \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b + a\right)\right) \cdot \left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right)\right)\right) \cdot \left(2 \cdot \left(b - a\right)\right) \]

   9. Simplified 83.0%

      \[\leadsto \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)}\right) \cdot \left(2 \cdot \left(b - a\right)\right) \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))

   1. Initial program 59.2%

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

      1. associate-*l* 59.2%

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

      2. unpow2 59.2%

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

      3. unpow2 59.2%

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

      4. difference-of-squares 61.7%

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

   3. Simplified 61.7%

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

   4. Taylor expanded in angle around inf 68.2%

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

      1. associate-*r* 68.2%

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

      2. *-commutative 68.2%

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

      3. *-commutative 68.2%

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

      4. *-commutative 68.2%

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

      5. *-commutative 68.2%

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

      6. associate-*r* 69.2%

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

      7. +-commutative 69.2%

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

      8. *-commutative 69.2%

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

      9. *-commutative 69.2%

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

      10. associate-*r* 68.9%

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

   6. Simplified 68.9%

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

   7. Taylor expanded in angle around inf 69.2%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq -\infty:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 3: 67.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \sin \left({\left(\sqrt[3]{t_0}\right)}^{3}\right)\right) \cdot \cos t_0\right) \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle 0.005555555555555556))))
   (* (* 2.0 (- b a)) (* (* (+ b a) (sin (pow (cbrt t_0) 3.0))) (cos t_0)))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle * 0.005555555555555556);
	return (2.0 * (b - a)) * (((b + a) * sin(pow(cbrt(t_0), 3.0))) * cos(t_0));
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle * 0.005555555555555556);
	return (2.0 * (b - a)) * (((b + a) * Math.sin(Math.pow(Math.cbrt(t_0), 3.0))) * Math.cos(t_0));
}

Julia

a = abs(a)
function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	return Float64(Float64(2.0 * Float64(b - a)) * Float64(Float64(Float64(b + a) * sin((cbrt(t_0) ^ 3.0))) * cos(t_0)))
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b + a), $MachinePrecision] * N[Sin[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 56.8%

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

   1. associate-*l* 56.8%

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

   2. unpow2 56.8%

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

   3. unpow2 56.8%

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

   4. difference-of-squares 58.8%

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

3. Simplified 58.8%

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

4. Taylor expanded in angle around inf 69.4%

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

   1. associate-*r* 69.4%

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

   2. *-commutative 69.4%

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

   3. *-commutative 69.4%

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

   4. *-commutative 69.4%

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

   5. *-commutative 69.4%

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

   6. associate-*r* 70.2%

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

   7. +-commutative 70.2%

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

   8. *-commutative 70.2%

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

   9. *-commutative 70.2%

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

   10. associate-*r* 70.3%

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

6. Simplified 70.3%

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

   1. expm1-log1p-u 62.4%

      \[\leadsto \left(\left(\sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \left(b + a\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(2 \cdot \left(b - a\right)\right) \]

8. Applied egg-rr 62.4%

   \[\leadsto \left(\left(\sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \left(b + a\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(2 \cdot \left(b - a\right)\right) \]
9. Step-by-step derivation

   1. expm1-log1p-u 70.3%

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

   2. *-commutative 70.3%

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

   3. *-commutative 70.3%

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

   4. associate-*r* 69.7%

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

   5. add-cube-cbrt 71.0%

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

   6. pow3 71.6%

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

   7. associate-*r* 71.6%

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

   8. *-commutative 71.6%

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

   9. *-commutative 71.6%

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

10. Applied egg-rr 71.6%

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

11. Final simplification 71.6%

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

Alternative 4: 67.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ t_1 := \sin t_0\\ t_2 := 2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+225}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \cos t_0\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0)))
        (t_1 (sin t_0))
        (t_2 (* 2.0 (* (+ b a) (- b a)))))
   (if (<= (/ angle 180.0) -5e+225)
     (* t_2 (* t_1 (cos t_0)))
     (if (<= (/ angle 180.0) 2e-35)
       (*
        (* 2.0 (- b a))
        (* (+ b a) (sin (* PI (* angle 0.005555555555555556)))))
       (* t_2 (* t_1 (cos (* 0.005555555555555556 (* PI angle)))))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double t_1 = sin(t_0);
	double t_2 = 2.0 * ((b + a) * (b - a));
	double tmp;
	if ((angle / 180.0) <= -5e+225) {
		tmp = t_2 * (t_1 * cos(t_0));
	} else if ((angle / 180.0) <= 2e-35) {
		tmp = (2.0 * (b - a)) * ((b + a) * sin((((double) M_PI) * (angle * 0.005555555555555556))));
	} else {
		tmp = t_2 * (t_1 * cos((0.005555555555555556 * (((double) M_PI) * angle))));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double t_1 = Math.sin(t_0);
	double t_2 = 2.0 * ((b + a) * (b - a));
	double tmp;
	if ((angle / 180.0) <= -5e+225) {
		tmp = t_2 * (t_1 * Math.cos(t_0));
	} else if ((angle / 180.0) <= 2e-35) {
		tmp = (2.0 * (b - a)) * ((b + a) * Math.sin((Math.PI * (angle * 0.005555555555555556))));
	} else {
		tmp = t_2 * (t_1 * Math.cos((0.005555555555555556 * (Math.PI * angle))));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	t_1 = math.sin(t_0)
	t_2 = 2.0 * ((b + a) * (b - a))
	tmp = 0
	if (angle / 180.0) <= -5e+225:
		tmp = t_2 * (t_1 * math.cos(t_0))
	elif (angle / 180.0) <= 2e-35:
		tmp = (2.0 * (b - a)) * ((b + a) * math.sin((math.pi * (angle * 0.005555555555555556))))
	else:
		tmp = t_2 * (t_1 * math.cos((0.005555555555555556 * (math.pi * angle))))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	t_1 = sin(t_0)
	t_2 = Float64(2.0 * Float64(Float64(b + a) * Float64(b - a)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+225)
		tmp = Float64(t_2 * Float64(t_1 * cos(t_0)));
	elseif (Float64(angle / 180.0) <= 2e-35)
		tmp = Float64(Float64(2.0 * Float64(b - a)) * Float64(Float64(b + a) * sin(Float64(pi * Float64(angle * 0.005555555555555556)))));
	else
		tmp = Float64(t_2 * Float64(t_1 * cos(Float64(0.005555555555555556 * Float64(pi * angle)))));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	t_1 = sin(t_0);
	t_2 = 2.0 * ((b + a) * (b - a));
	tmp = 0.0;
	if ((angle / 180.0) <= -5e+225)
		tmp = t_2 * (t_1 * cos(t_0));
	elseif ((angle / 180.0) <= 2e-35)
		tmp = (2.0 * (b - a)) * ((b + a) * sin((pi * (angle * 0.005555555555555556))));
	else
		tmp = t_2 * (t_1 * cos((0.005555555555555556 * (pi * angle))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+225], N[(t$95$2 * N[(t$95$1 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e-35], N[(N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$1 * N[Cos[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -4.99999999999999981e225

   1. Initial program 52.1%

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

      1. associate-*l* 52.1%

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

      2. unpow2 52.1%

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

      3. unpow2 52.1%

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

      4. difference-of-squares 66.4%

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

   3. Simplified 66.4%

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

   if -4.99999999999999981e225 < (/.f64 angle 180) < 2.00000000000000002e-35

   1. Initial program 63.2%

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

      1. associate-*l* 63.2%

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

      2. unpow2 63.2%

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

      3. unpow2 63.2%

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

      4. difference-of-squares 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in angle around inf 81.1%

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

      1. associate-*r* 81.1%

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

      2. *-commutative 81.1%

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

      3. *-commutative 81.1%

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

      4. *-commutative 81.1%

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

      5. *-commutative 81.1%

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

      6. associate-*r* 81.9%

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

      7. +-commutative 81.9%

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

      8. *-commutative 81.9%

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

      9. *-commutative 81.9%

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

      10. associate-*r* 81.4%

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

   6. Simplified 81.4%

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

   7. Taylor expanded in angle around 0 83.7%

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

   if 2.00000000000000002e-35 < (/.f64 angle 180)

   1. Initial program 42.4%

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

      1. associate-*l* 42.4%

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

      2. unpow2 42.4%

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

      3. unpow2 42.4%

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

      4. difference-of-squares 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in angle around inf 44.8%

      \[\leadsto \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+225}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 5: 67.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_1 := \cos t_0\\ t_2 := 2 \cdot \left(b - a\right)\\ t_3 := \left(b + a\right) \cdot \sin t_0\\ \mathbf{if}\;a \leq 1.6 \cdot 10^{-51}:\\ \;\;\;\;t_2 \cdot \left(\left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot t_1\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+185}:\\ \;\;\;\;t_2 \cdot t_3\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(t_1 \cdot t_3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* PI angle)))
        (t_1 (cos t_0))
        (t_2 (* 2.0 (- b a)))
        (t_3 (* (+ b a) (sin t_0))))
   (if (<= a 1.6e-51)
     (* t_2 (* (* (+ b a) (sin (* PI (* angle 0.005555555555555556)))) t_1))
     (if (<= a 2e+185) (* t_2 t_3) (* 2.0 (* (- b a) (* t_1 t_3)))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * angle);
	double t_1 = cos(t_0);
	double t_2 = 2.0 * (b - a);
	double t_3 = (b + a) * sin(t_0);
	double tmp;
	if (a <= 1.6e-51) {
		tmp = t_2 * (((b + a) * sin((((double) M_PI) * (angle * 0.005555555555555556)))) * t_1);
	} else if (a <= 2e+185) {
		tmp = t_2 * t_3;
	} else {
		tmp = 2.0 * ((b - a) * (t_1 * t_3));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (Math.PI * angle);
	double t_1 = Math.cos(t_0);
	double t_2 = 2.0 * (b - a);
	double t_3 = (b + a) * Math.sin(t_0);
	double tmp;
	if (a <= 1.6e-51) {
		tmp = t_2 * (((b + a) * Math.sin((Math.PI * (angle * 0.005555555555555556)))) * t_1);
	} else if (a <= 2e+185) {
		tmp = t_2 * t_3;
	} else {
		tmp = 2.0 * ((b - a) * (t_1 * t_3));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	t_0 = 0.005555555555555556 * (math.pi * angle)
	t_1 = math.cos(t_0)
	t_2 = 2.0 * (b - a)
	t_3 = (b + a) * math.sin(t_0)
	tmp = 0
	if a <= 1.6e-51:
		tmp = t_2 * (((b + a) * math.sin((math.pi * (angle * 0.005555555555555556)))) * t_1)
	elif a <= 2e+185:
		tmp = t_2 * t_3
	else:
		tmp = 2.0 * ((b - a) * (t_1 * t_3))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 * Float64(pi * angle))
	t_1 = cos(t_0)
	t_2 = Float64(2.0 * Float64(b - a))
	t_3 = Float64(Float64(b + a) * sin(t_0))
	tmp = 0.0
	if (a <= 1.6e-51)
		tmp = Float64(t_2 * Float64(Float64(Float64(b + a) * sin(Float64(pi * Float64(angle * 0.005555555555555556)))) * t_1));
	elseif (a <= 2e+185)
		tmp = Float64(t_2 * t_3);
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(t_1 * t_3)));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	t_0 = 0.005555555555555556 * (pi * angle);
	t_1 = cos(t_0);
	t_2 = 2.0 * (b - a);
	t_3 = (b + a) * sin(t_0);
	tmp = 0.0;
	if (a <= 1.6e-51)
		tmp = t_2 * (((b + a) * sin((pi * (angle * 0.005555555555555556)))) * t_1);
	elseif (a <= 2e+185)
		tmp = t_2 * t_3;
	else
		tmp = 2.0 * ((b - a) * (t_1 * t_3));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.6e-51], N[(t$95$2 * N[(N[(N[(b + a), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2e+185], N[(t$95$2 * t$95$3), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.6e-51

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

      2. unpow2 57.6%

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

      3. unpow2 57.6%

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

      4. difference-of-squares 58.8%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in angle around inf 68.1%

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

      1. associate-*r* 68.1%

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

      2. *-commutative 68.1%

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

      3. *-commutative 68.1%

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

      4. *-commutative 68.1%

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

      5. *-commutative 68.1%

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

      6. associate-*r* 68.9%

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

      7. +-commutative 68.9%

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

      8. *-commutative 68.9%

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

      9. *-commutative 68.9%

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

      10. associate-*r* 69.2%

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

   6. Simplified 69.2%

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

   7. Taylor expanded in angle around inf 68.9%

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

   if 1.6e-51 < a < 2e185

   1. Initial program 55.1%

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

      1. associate-*l* 55.1%

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

      2. unpow2 55.1%

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

      3. unpow2 55.1%

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

      4. difference-of-squares 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in angle around inf 64.3%

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

      1. associate-*r* 64.3%

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

      2. *-commutative 64.3%

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

      3. *-commutative 64.3%

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

      4. *-commutative 64.3%

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

      5. *-commutative 64.3%

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

      6. associate-*r* 65.3%

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

      7. +-commutative 65.3%

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

      8. *-commutative 65.3%

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

      9. *-commutative 65.3%

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

      10. associate-*r* 65.0%

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

   6. Simplified 65.0%

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

      1. add-cbrt-cube 69.6%

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

   8. Applied egg-rr 69.6%

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

   9. Taylor expanded in angle around inf 69.2%

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

   10. Taylor expanded in angle around 0 72.8%

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

   if 2e185 < a

   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. associate-*l* 55.7%

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

      2. unpow2 55.7%

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

      3. unpow2 55.7%

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

      4. difference-of-squares 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in angle around inf 91.3%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-51}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+185}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 6: 67.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} t_0 := 2 \cdot \left(b - a\right)\\ t_1 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_2 := \left(b + a\right) \cdot \sin t_1\\ t_3 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;a \leq 4 \cdot 10^{-49}:\\ \;\;\;\;t_0 \cdot \left(\cos t_3 \cdot \left(\left(b + a\right) \cdot \sin t_3\right)\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+185}:\\ \;\;\;\;t_0 \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\cos t_1 \cdot t_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (- b a)))
        (t_1 (* 0.005555555555555556 (* PI angle)))
        (t_2 (* (+ b a) (sin t_1)))
        (t_3 (* PI (* angle 0.005555555555555556))))
   (if (<= a 4e-49)
     (* t_0 (* (cos t_3) (* (+ b a) (sin t_3))))
     (if (<= a 2e+185) (* t_0 t_2) (* 2.0 (* (- b a) (* (cos t_1) t_2)))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = 0.005555555555555556 * (((double) M_PI) * angle);
	double t_2 = (b + a) * sin(t_1);
	double t_3 = ((double) M_PI) * (angle * 0.005555555555555556);
	double tmp;
	if (a <= 4e-49) {
		tmp = t_0 * (cos(t_3) * ((b + a) * sin(t_3)));
	} else if (a <= 2e+185) {
		tmp = t_0 * t_2;
	} else {
		tmp = 2.0 * ((b - a) * (cos(t_1) * t_2));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = 0.005555555555555556 * (Math.PI * angle);
	double t_2 = (b + a) * Math.sin(t_1);
	double t_3 = Math.PI * (angle * 0.005555555555555556);
	double tmp;
	if (a <= 4e-49) {
		tmp = t_0 * (Math.cos(t_3) * ((b + a) * Math.sin(t_3)));
	} else if (a <= 2e+185) {
		tmp = t_0 * t_2;
	} else {
		tmp = 2.0 * ((b - a) * (Math.cos(t_1) * t_2));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	t_0 = 2.0 * (b - a)
	t_1 = 0.005555555555555556 * (math.pi * angle)
	t_2 = (b + a) * math.sin(t_1)
	t_3 = math.pi * (angle * 0.005555555555555556)
	tmp = 0
	if a <= 4e-49:
		tmp = t_0 * (math.cos(t_3) * ((b + a) * math.sin(t_3)))
	elif a <= 2e+185:
		tmp = t_0 * t_2
	else:
		tmp = 2.0 * ((b - a) * (math.cos(t_1) * t_2))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	t_0 = Float64(2.0 * Float64(b - a))
	t_1 = Float64(0.005555555555555556 * Float64(pi * angle))
	t_2 = Float64(Float64(b + a) * sin(t_1))
	t_3 = Float64(pi * Float64(angle * 0.005555555555555556))
	tmp = 0.0
	if (a <= 4e-49)
		tmp = Float64(t_0 * Float64(cos(t_3) * Float64(Float64(b + a) * sin(t_3))));
	elseif (a <= 2e+185)
		tmp = Float64(t_0 * t_2);
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(cos(t_1) * t_2)));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	t_0 = 2.0 * (b - a);
	t_1 = 0.005555555555555556 * (pi * angle);
	t_2 = (b + a) * sin(t_1);
	t_3 = pi * (angle * 0.005555555555555556);
	tmp = 0.0;
	if (a <= 4e-49)
		tmp = t_0 * (cos(t_3) * ((b + a) * sin(t_3)));
	elseif (a <= 2e+185)
		tmp = t_0 * t_2;
	else
		tmp = 2.0 * ((b - a) * (cos(t_1) * t_2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 4e-49], N[(t$95$0 * N[(N[Cos[t$95$3], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2e+185], N[(t$95$0 * t$95$2), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[Cos[t$95$1], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 3.99999999999999975e-49

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

      2. unpow2 57.6%

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

      3. unpow2 57.6%

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

      4. difference-of-squares 58.8%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in angle around inf 68.1%

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

      1. associate-*r* 68.1%

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

      2. *-commutative 68.1%

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

      3. *-commutative 68.1%

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

      4. *-commutative 68.1%

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

      5. *-commutative 68.1%

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

      6. associate-*r* 68.9%

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

      7. +-commutative 68.9%

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

      8. *-commutative 68.9%

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

      9. *-commutative 68.9%

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

      10. associate-*r* 69.2%

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

   6. Simplified 69.2%

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

   if 3.99999999999999975e-49 < a < 2e185

   1. Initial program 55.1%

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

      1. associate-*l* 55.1%

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

      2. unpow2 55.1%

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

      3. unpow2 55.1%

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

      4. difference-of-squares 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in angle around inf 64.3%

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

      1. associate-*r* 64.3%

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

      2. *-commutative 64.3%

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

      3. *-commutative 64.3%

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

      4. *-commutative 64.3%

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

      5. *-commutative 64.3%

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

      6. associate-*r* 65.3%

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

      7. +-commutative 65.3%

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

      8. *-commutative 65.3%

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

      9. *-commutative 65.3%

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

      10. associate-*r* 65.0%

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

   6. Simplified 65.0%

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

      1. add-cbrt-cube 69.6%

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

   8. Applied egg-rr 69.6%

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

   9. Taylor expanded in angle around inf 69.2%

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

   10. Taylor expanded in angle around 0 72.8%

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

   if 2e185 < a

   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. associate-*l* 55.7%

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

      2. unpow2 55.7%

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

      3. unpow2 55.7%

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

      4. difference-of-squares 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in angle around inf 91.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{-49}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+185}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 67.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ 2 \cdot \left(\left(b - a\right) \cdot \left(\cos t_0 \cdot \left(\left(b + a\right) \cdot \sin t_0\right)\right)\right) \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* PI angle))))
   (* 2.0 (* (- b a) (* (cos t_0) (* (+ b a) (sin t_0)))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * angle);
	return 2.0 * ((b - a) * (cos(t_0) * ((b + a) * sin(t_0))));
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (Math.PI * angle);
	return 2.0 * ((b - a) * (Math.cos(t_0) * ((b + a) * Math.sin(t_0))));
}

Python

a = abs(a)
def code(a, b, angle):
	t_0 = 0.005555555555555556 * (math.pi * angle)
	return 2.0 * ((b - a) * (math.cos(t_0) * ((b + a) * math.sin(t_0))))

Julia

a = abs(a)
function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 * Float64(pi * angle))
	return Float64(2.0 * Float64(Float64(b - a) * Float64(cos(t_0) * Float64(Float64(b + a) * sin(t_0)))))
end

MATLAB

a = abs(a)
function tmp = code(a, b, angle)
	t_0 = 0.005555555555555556 * (pi * angle);
	tmp = 2.0 * ((b - a) * (cos(t_0) * ((b + a) * sin(t_0))));
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[Cos[t$95$0], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 56.8%

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

   1. associate-*l* 56.8%

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

   2. unpow2 56.8%

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

   3. unpow2 56.8%

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

   4. difference-of-squares 58.8%

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

3. Simplified 58.8%

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

4. Taylor expanded in angle around inf 69.4%

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

5. Final simplification 69.4%

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

Alternative 8: 66.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-79}:\\ \;\;\;\;\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -2e-79)
   (* (sin (* PI (/ angle 180.0))) (* 2.0 (* (+ b a) (- b a))))
   (*
    (* 2.0 (- b a))
    (* (+ b a) (sin (expm1 (log1p (* PI (* angle 0.005555555555555556)))))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -2e-79) {
		tmp = sin((((double) M_PI) * (angle / 180.0))) * (2.0 * ((b + a) * (b - a)));
	} else {
		tmp = (2.0 * (b - a)) * ((b + a) * sin(expm1(log1p((((double) M_PI) * (angle * 0.005555555555555556))))));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -2e-79) {
		tmp = Math.sin((Math.PI * (angle / 180.0))) * (2.0 * ((b + a) * (b - a)));
	} else {
		tmp = (2.0 * (b - a)) * ((b + a) * Math.sin(Math.expm1(Math.log1p((Math.PI * (angle * 0.005555555555555556))))));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= -2e-79:
		tmp = math.sin((math.pi * (angle / 180.0))) * (2.0 * ((b + a) * (b - a)))
	else:
		tmp = (2.0 * (b - a)) * ((b + a) * math.sin(math.expm1(math.log1p((math.pi * (angle * 0.005555555555555556))))))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e-79)
		tmp = Float64(sin(Float64(pi * Float64(angle / 180.0))) * Float64(2.0 * Float64(Float64(b + a) * Float64(b - a))));
	else
		tmp = Float64(Float64(2.0 * Float64(b - a)) * Float64(Float64(b + a) * sin(expm1(log1p(Float64(pi * Float64(angle * 0.005555555555555556)))))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e-79], N[(N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(Exp[N[Log[1 + N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -2e-79

   1. Initial program 48.7%

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

      1. associate-*l* 48.7%

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

      2. unpow2 48.7%

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

      3. unpow2 48.7%

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

      4. difference-of-squares 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in angle around 0 52.7%

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

   if -2e-79 < (/.f64 angle 180)

   1. Initial program 60.1%

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

      1. associate-*l* 60.1%

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

      2. unpow2 60.1%

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

      3. unpow2 60.1%

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

      4. difference-of-squares 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in angle around inf 78.0%

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

      1. associate-*r* 78.0%

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

      2. *-commutative 78.0%

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

      3. *-commutative 78.0%

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

      4. *-commutative 78.0%

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

      5. *-commutative 78.0%

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

      6. associate-*r* 78.2%

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

      7. +-commutative 78.2%

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

      8. *-commutative 78.2%

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

      9. *-commutative 78.2%

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

      10. associate-*r* 78.4%

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

   6. Simplified 78.4%

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

      1. expm1-log1p-u 77.9%

         \[\leadsto \left(\left(\sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \left(b + a\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(2 \cdot \left(b - a\right)\right) \]

   8. Applied egg-rr 77.9%

      \[\leadsto \left(\left(\sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \left(b + a\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(2 \cdot \left(b - a\right)\right) \]

   9. Taylor expanded in angle around 0 74.8%

      \[\leadsto \left(\left(\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{1}\right) \cdot \left(2 \cdot \left(b - a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-79}:\\ \;\;\;\;\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 9: 66.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{-124}:\\ \;\;\;\;{\left(\sqrt[3]{b \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 4e-124)
   (pow (cbrt (* b (* b (sin (* angle (* PI 0.011111111111111112)))))) 3.0)
   (*
    (* 2.0 (- b a))
    (* (+ b a) (sin (* PI (* angle 0.005555555555555556)))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4e-124) {
		tmp = pow(cbrt((b * (b * sin((angle * (((double) M_PI) * 0.011111111111111112)))))), 3.0);
	} else {
		tmp = (2.0 * (b - a)) * ((b + a) * sin((((double) M_PI) * (angle * 0.005555555555555556))));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4e-124) {
		tmp = Math.pow(Math.cbrt((b * (b * Math.sin((angle * (Math.PI * 0.011111111111111112)))))), 3.0);
	} else {
		tmp = (2.0 * (b - a)) * ((b + a) * Math.sin((Math.PI * (angle * 0.005555555555555556))));
	}
	return tmp;
}

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 4e-124)
		tmp = cbrt(Float64(b * Float64(b * sin(Float64(angle * Float64(pi * 0.011111111111111112)))))) ^ 3.0;
	else
		tmp = Float64(Float64(2.0 * Float64(b - a)) * Float64(Float64(b + a) * sin(Float64(pi * Float64(angle * 0.005555555555555556)))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 4e-124], N[Power[N[Power[N[(b * N[(b * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], N[(N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.99999999999999973e-124

   1. Initial program 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-*l* 59.4%

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

      3. unpow2 59.4%

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

      4. fma-neg 60.0%

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

      5. unpow2 60.0%

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

   3. Simplified 60.0%

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

   5. Applied egg-rr 46.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(2 \cdot {\left(b + a\right)}^{2}\right) \cdot \left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right)}\right)}^{3}} \]

   6. Taylor expanded in b around inf 43.5%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot {b}^{2}}}\right)}^{3} \]
   7. Step-by-step derivation

      1. *-commutative 43.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{{b}^{2} \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}}\right)}^{3} \]

      2. unpow2 43.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(b \cdot b\right)} \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{3} \]

      3. associate-*l* 47.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{b \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)}^{3} \]

      4. *-commutative 47.5%

         \[\leadsto {\left(\sqrt[3]{b \cdot \left(b \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}\right)}\right)}^{3} \]

      5. associate-*l* 47.8%

         \[\leadsto {\left(\sqrt[3]{b \cdot \left(b \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)}\right)}^{3} \]

   8. Simplified 47.8%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{b \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)}}\right)}^{3} \]

   if 3.99999999999999973e-124 < a

   1. Initial program 52.6%

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

      1. associate-*l* 52.6%

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

      2. unpow2 52.6%

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

      3. unpow2 52.6%

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

      4. difference-of-squares 55.7%

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

   3. Simplified 55.7%

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

   4. Taylor expanded in angle around inf 70.1%

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

      1. associate-*r* 70.1%

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

      2. *-commutative 70.1%

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

      3. *-commutative 70.1%

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

      4. *-commutative 70.1%

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

      5. *-commutative 70.1%

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

      6. associate-*r* 70.7%

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

      7. +-commutative 70.7%

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

      8. *-commutative 70.7%

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

      9. *-commutative 70.7%

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

      10. associate-*r* 70.5%

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

   6. Simplified 70.5%

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

   7. Taylor expanded in angle around 0 70.9%

      \[\leadsto \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{1}\right) \cdot \left(2 \cdot \left(b - a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{-124}:\\ \;\;\;\;{\left(\sqrt[3]{b \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]

Alternative 10: 65.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;a \leq 7.2 \cdot 10^{-208}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \left(\cos t_0 \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \sin t_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* PI angle))))
   (if (<= a 7.2e-208)
     (* 2.0 (* t_0 (* (cos t_0) (* b b))))
     (* (* 2.0 (- b a)) (* (+ b a) (sin t_0))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * angle);
	double tmp;
	if (a <= 7.2e-208) {
		tmp = 2.0 * (t_0 * (cos(t_0) * (b * b)));
	} else {
		tmp = (2.0 * (b - a)) * ((b + a) * sin(t_0));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (Math.PI * angle);
	double tmp;
	if (a <= 7.2e-208) {
		tmp = 2.0 * (t_0 * (Math.cos(t_0) * (b * b)));
	} else {
		tmp = (2.0 * (b - a)) * ((b + a) * Math.sin(t_0));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	t_0 = 0.005555555555555556 * (math.pi * angle)
	tmp = 0
	if a <= 7.2e-208:
		tmp = 2.0 * (t_0 * (math.cos(t_0) * (b * b)))
	else:
		tmp = (2.0 * (b - a)) * ((b + a) * math.sin(t_0))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 * Float64(pi * angle))
	tmp = 0.0
	if (a <= 7.2e-208)
		tmp = Float64(2.0 * Float64(t_0 * Float64(cos(t_0) * Float64(b * b))));
	else
		tmp = Float64(Float64(2.0 * Float64(b - a)) * Float64(Float64(b + a) * sin(t_0)));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	t_0 = 0.005555555555555556 * (pi * angle);
	tmp = 0.0;
	if (a <= 7.2e-208)
		tmp = 2.0 * (t_0 * (cos(t_0) * (b * b)));
	else
		tmp = (2.0 * (b - a)) * ((b + a) * sin(t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 7.2e-208], N[(2.0 * N[(t$95$0 * N[(N[Cos[t$95$0], $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 7.1999999999999997e-208

   1. Initial program 59.2%

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

      1. associate-*l* 59.2%

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

      2. unpow2 59.2%

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

      3. unpow2 59.2%

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

      4. difference-of-squares 60.7%

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

   3. Simplified 60.7%

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

   4. Taylor expanded in angle around inf 58.6%

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

   5. Taylor expanded in b around inf 40.6%

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

      1. associate-*r* 40.6%

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

      2. unpow2 40.6%

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

   7. Simplified 40.6%

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

   8. Taylor expanded in angle around 0 40.8%

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

   if 7.1999999999999997e-208 < a

   1. Initial program 53.9%

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

      1. associate-*l* 53.9%

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

      2. unpow2 53.9%

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

      3. unpow2 53.9%

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

      4. difference-of-squares 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in angle around inf 70.4%

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

      1. associate-*r* 70.4%

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

      2. *-commutative 70.4%

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

      3. *-commutative 70.4%

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

      4. *-commutative 70.4%

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

      5. *-commutative 70.4%

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

      6. associate-*r* 70.9%

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

      7. +-commutative 70.9%

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

      8. *-commutative 70.9%

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

      9. *-commutative 70.9%

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

      10. associate-*r* 69.7%

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

   6. Simplified 69.7%

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

      1. add-cbrt-cube 71.9%

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

   8. Applied egg-rr 71.9%

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

   9. Taylor expanded in angle around inf 71.7%

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

   10. Taylor expanded in angle around 0 69.5%

       \[\leadsto \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{1}\right) \cdot \left(2 \cdot \left(b - a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.2 \cdot 10^{-208}:\\ \;\;\;\;2 \cdot \left(\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 11: 66.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (+ b a) (- b a)))))
   (if (<= (/ angle 180.0) -2e-15)
     (* (sin (* PI (/ angle 180.0))) t_0)
     (if (<= (/ angle 180.0) 5e-9)
       (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))
       (* (sin (* 0.005555555555555556 (* PI angle))) t_0)))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double t_0 = 2.0 * ((b + a) * (b - a));
	double tmp;
	if ((angle / 180.0) <= -2e-15) {
		tmp = sin((((double) M_PI) * (angle / 180.0))) * t_0;
	} else if ((angle / 180.0) <= 5e-9) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	} else {
		tmp = sin((0.005555555555555556 * (((double) M_PI) * angle))) * t_0;
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double t_0 = 2.0 * ((b + a) * (b - a));
	double tmp;
	if ((angle / 180.0) <= -2e-15) {
		tmp = Math.sin((Math.PI * (angle / 180.0))) * t_0;
	} else if ((angle / 180.0) <= 5e-9) {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	} else {
		tmp = Math.sin((0.005555555555555556 * (Math.PI * angle))) * t_0;
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	t_0 = 2.0 * ((b + a) * (b - a))
	tmp = 0
	if (angle / 180.0) <= -2e-15:
		tmp = math.sin((math.pi * (angle / 180.0))) * t_0
	elif (angle / 180.0) <= 5e-9:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	else:
		tmp = math.sin((0.005555555555555556 * (math.pi * angle))) * t_0
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	t_0 = Float64(2.0 * Float64(Float64(b + a) * Float64(b - a)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e-15)
		tmp = Float64(sin(Float64(pi * Float64(angle / 180.0))) * t_0);
	elseif (Float64(angle / 180.0) <= 5e-9)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	else
		tmp = Float64(sin(Float64(0.005555555555555556 * Float64(pi * angle))) * t_0);
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	t_0 = 2.0 * ((b + a) * (b - a));
	tmp = 0.0;
	if ((angle / 180.0) <= -2e-15)
		tmp = sin((pi * (angle / 180.0))) * t_0;
	elseif ((angle / 180.0) <= 5e-9)
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	else
		tmp = sin((0.005555555555555556 * (pi * angle))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e-15], N[(N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e-9], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -2.0000000000000002e-15

   1. Initial program 37.8%

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

      1. associate-*l* 37.8%

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

      2. unpow2 37.8%

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

      3. unpow2 37.8%

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

      4. difference-of-squares 41.1%

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

   3. Simplified 41.1%

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

   4. Taylor expanded in angle around 0 42.6%

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

   if -2.0000000000000002e-15 < (/.f64 angle 180) < 5.0000000000000001e-9

   1. Initial program 76.2%

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

      1. associate-*l* 76.2%

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

      2. unpow2 76.2%

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

      3. unpow2 76.2%

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

      4. difference-of-squares 77.7%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in angle around 0 77.9%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   if 5.0000000000000001e-9 < (/.f64 angle 180)

   1. Initial program 34.1%

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

      1. associate-*l* 34.1%

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

      2. unpow2 34.1%

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

      3. unpow2 34.1%

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

      4. difference-of-squares 35.7%

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

   3. Simplified 35.7%

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

   4. Taylor expanded in angle around inf 35.5%

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

   5. Taylor expanded in angle around 0 22.1%

      \[\leadsto \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{1}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \]

Alternative 12: 66.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;angle \leq -1 \cdot 10^{-13} \lor \neg \left(angle \leq 0.000116\right):\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (or (<= angle -1e-13) (not (<= angle 0.000116)))
   (* (sin (* 0.005555555555555556 (* PI angle))) (* 2.0 (* (+ b a) (- b a))))
   (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if ((angle <= -1e-13) || !(angle <= 0.000116)) {
		tmp = sin((0.005555555555555556 * (((double) M_PI) * angle))) * (2.0 * ((b + a) * (b - a)));
	} else {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle <= -1e-13) || !(angle <= 0.000116)) {
		tmp = Math.sin((0.005555555555555556 * (Math.PI * angle))) * (2.0 * ((b + a) * (b - a)));
	} else {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	tmp = 0
	if (angle <= -1e-13) or not (angle <= 0.000116):
		tmp = math.sin((0.005555555555555556 * (math.pi * angle))) * (2.0 * ((b + a) * (b - a)))
	else:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if ((angle <= -1e-13) || !(angle <= 0.000116))
		tmp = Float64(sin(Float64(0.005555555555555556 * Float64(pi * angle))) * Float64(2.0 * Float64(Float64(b + a) * Float64(b - a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle <= -1e-13) || ~((angle <= 0.000116)))
		tmp = sin((0.005555555555555556 * (pi * angle))) * (2.0 * ((b + a) * (b - a)));
	else
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[Or[LessEqual[angle, -1e-13], N[Not[LessEqual[angle, 0.000116]], $MachinePrecision]], N[(N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < -1e-13 or 1.16e-4 < angle

   1. Initial program 35.9%

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

      1. associate-*l* 35.9%

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

      2. unpow2 35.9%

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

      3. unpow2 35.9%

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

      4. difference-of-squares 38.4%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in angle around inf 35.3%

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

   5. Taylor expanded in angle around 0 30.7%

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

   if -1e-13 < angle < 1.16e-4

   1. Initial program 76.2%

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

      1. associate-*l* 76.2%

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

      2. unpow2 76.2%

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

      3. unpow2 76.2%

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

      4. difference-of-squares 77.7%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in angle around 0 77.9%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -1 \cdot 10^{-13} \lor \neg \left(angle \leq 0.000116\right):\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \]

Alternative 13: 66.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

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

   1. associate-*l* 56.8%

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

   2. unpow2 56.8%

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

   3. unpow2 56.8%

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

   4. difference-of-squares 58.8%

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

3. Simplified 58.8%

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

4. Taylor expanded in angle around inf 69.4%

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

   1. associate-*r* 69.4%

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

   2. *-commutative 69.4%

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

   3. *-commutative 69.4%

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

   4. *-commutative 69.4%

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

   5. *-commutative 69.4%

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

   6. associate-*r* 70.2%

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

   7. +-commutative 70.2%

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

   8. *-commutative 70.2%

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

   9. *-commutative 70.2%

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

   10. associate-*r* 70.3%

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

6. Simplified 70.3%

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

   1. add-cbrt-cube 71.0%

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

8. Applied egg-rr 71.0%

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

9. Taylor expanded in angle around inf 71.2%

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

10. Taylor expanded in angle around 0 66.4%

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

11. Final simplification 66.4%

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

Alternative 14: 57.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 2.15 \cdot 10^{+141}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(a \cdot \left(angle \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.15e+141)
   (* 0.011111111111111112 (* angle (* (- b a) (* PI (+ b a)))))
   (* PI (* a (* angle (* a -0.011111111111111112))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.15e+141) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (((double) M_PI) * (b + a))));
	} else {
		tmp = ((double) M_PI) * (a * (angle * (a * -0.011111111111111112)));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.15e+141) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (Math.PI * (b + a))));
	} else {
		tmp = Math.PI * (a * (angle * (a * -0.011111111111111112)));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	tmp = 0
	if a <= 2.15e+141:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (math.pi * (b + a))))
	else:
		tmp = math.pi * (a * (angle * (a * -0.011111111111111112)))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.15e+141)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(pi * Float64(b + a)))));
	else
		tmp = Float64(pi * Float64(a * Float64(angle * Float64(a * -0.011111111111111112))));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.15e+141)
		tmp = 0.011111111111111112 * (angle * ((b - a) * (pi * (b + a))));
	else
		tmp = pi * (a * (angle * (a * -0.011111111111111112)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 2.15e+141], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(a * N[(angle * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.1499999999999999e141

   1. Initial program 58.0%

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

      1. associate-*l* 58.0%

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

      2. unpow2 58.0%

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

      3. unpow2 58.0%

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

      4. difference-of-squares 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in angle around 0 54.5%

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

   if 2.1499999999999999e141 < a

   1. Initial program 49.0%

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

      1. associate-*l* 49.0%

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

      2. unpow2 49.0%

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

      3. unpow2 49.0%

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

      4. difference-of-squares 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in angle around 0 39.6%

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

   5. Taylor expanded in b around 0 39.6%

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

      1. *-commutative 39.6%

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

      2. associate-*l* 39.6%

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

      3. *-commutative 39.6%

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

      4. unpow2 39.6%

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

   7. Simplified 39.6%

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

   8. Taylor expanded in angle around 0 39.6%

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

      1. associate-*r* 39.6%

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

      2. unpow2 39.6%

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

   10. Simplified 39.6%

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

   11. Taylor expanded in angle around 0 39.6%

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

       1. associate-*r* 39.6%

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

       2. unpow2 39.6%

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

       3. associate-*r* 39.6%

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

       4. *-commutative 39.6%

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

       5. associate-*r* 61.8%

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

       6. *-commutative 61.8%

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

       7. *-commutative 61.8%

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

       8. associate-*l* 61.8%

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

   13. Simplified 61.8%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.15 \cdot 10^{+141}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(a \cdot \left(angle \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 15: 50.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 2.05 \cdot 10^{+63}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(a \cdot \left(angle \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.05e+63)
   (* 0.011111111111111112 (* angle (* (- b a) (* PI b))))
   (* PI (* a (* angle (* a -0.011111111111111112))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.05e+63) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (((double) M_PI) * b)));
	} else {
		tmp = ((double) M_PI) * (a * (angle * (a * -0.011111111111111112)));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.05e+63) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (Math.PI * b)));
	} else {
		tmp = Math.PI * (a * (angle * (a * -0.011111111111111112)));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	tmp = 0
	if a <= 2.05e+63:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (math.pi * b)))
	else:
		tmp = math.pi * (a * (angle * (a * -0.011111111111111112)))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.05e+63)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(pi * b))));
	else
		tmp = Float64(pi * Float64(a * Float64(angle * Float64(a * -0.011111111111111112))));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.05e+63)
		tmp = 0.011111111111111112 * (angle * ((b - a) * (pi * b)));
	else
		tmp = pi * (a * (angle * (a * -0.011111111111111112)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 2.05e+63], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(a * N[(angle * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.04999999999999996e63

   1. Initial program 57.9%

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

      1. associate-*l* 57.9%

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

      2. unpow2 57.9%

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

      3. unpow2 57.9%

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

      4. difference-of-squares 58.9%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in angle around 0 53.6%

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

   5. Taylor expanded in a around 0 43.3%

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

      1. *-commutative 43.3%

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

   7. Simplified 43.3%

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

   if 2.04999999999999996e63 < a

   1. Initial program 53.3%

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

      1. associate-*l* 53.3%

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

      2. unpow2 53.3%

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

      3. unpow2 53.3%

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

      4. difference-of-squares 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in angle around 0 48.7%

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

   5. Taylor expanded in b around 0 41.5%

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

      1. *-commutative 41.5%

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

      2. associate-*l* 41.4%

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

      3. *-commutative 41.4%

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

      4. unpow2 41.4%

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

   7. Simplified 41.4%

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

   8. Taylor expanded in angle around 0 41.5%

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

      1. associate-*r* 41.4%

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

      2. unpow2 41.4%

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

   10. Simplified 41.4%

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

   11. Taylor expanded in angle around 0 41.5%

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

       1. associate-*r* 41.4%

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

       2. unpow2 41.4%

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

       3. associate-*r* 41.4%

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

       4. *-commutative 41.4%

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

       5. associate-*r* 54.6%

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

       6. *-commutative 54.6%

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

       7. *-commutative 54.6%

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

       8. associate-*l* 54.6%

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

   13. Simplified 54.6%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.05 \cdot 10^{+63}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(a \cdot \left(angle \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 16: 62.2% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

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

   1. associate-*l* 56.8%

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

   2. unpow2 56.8%

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

   3. unpow2 56.8%

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

   4. difference-of-squares 58.8%

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

3. Simplified 58.8%

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

4. Taylor expanded in angle around 0 52.5%

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

   1. associate-*r* 61.6%

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

   2. *-commutative 61.6%

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

   3. +-commutative 61.6%

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

6. Simplified 61.6%

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

7. Final simplification 61.6%

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

Alternative 17: 46.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{+58}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(a \cdot \left(\pi \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.5e+58)
   (* angle (* 0.011111111111111112 (* PI (* b b))))
   (* angle (* a (* PI (* a -0.011111111111111112))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.5e+58) {
		tmp = angle * (0.011111111111111112 * (((double) M_PI) * (b * b)));
	} else {
		tmp = angle * (a * (((double) M_PI) * (a * -0.011111111111111112)));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.5e+58) {
		tmp = angle * (0.011111111111111112 * (Math.PI * (b * b)));
	} else {
		tmp = angle * (a * (Math.PI * (a * -0.011111111111111112)));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	tmp = 0
	if a <= 3.5e+58:
		tmp = angle * (0.011111111111111112 * (math.pi * (b * b)))
	else:
		tmp = angle * (a * (math.pi * (a * -0.011111111111111112)))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.5e+58)
		tmp = Float64(angle * Float64(0.011111111111111112 * Float64(pi * Float64(b * b))));
	else
		tmp = Float64(angle * Float64(a * Float64(pi * Float64(a * -0.011111111111111112))));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 3.5e+58)
		tmp = angle * (0.011111111111111112 * (pi * (b * b)));
	else
		tmp = angle * (a * (pi * (a * -0.011111111111111112)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 3.5e+58], N[(angle * N[(0.011111111111111112 * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[(a * N[(Pi * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.4999999999999997e58

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. unpow2 58.1%

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

      3. unpow2 58.1%

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

      4. difference-of-squares 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in angle around 0 53.8%

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

   5. Taylor expanded in b around inf 39.7%

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

      1. *-commutative 39.7%

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

      2. associate-*l* 39.7%

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

      3. *-commutative 39.7%

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

      4. unpow2 39.7%

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

   7. Simplified 39.7%

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

   if 3.4999999999999997e58 < a

   1. Initial program 52.7%

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

      1. associate-*l* 52.7%

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

      2. unpow2 52.7%

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

      3. unpow2 52.7%

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

      4. difference-of-squares 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in angle around 0 47.9%

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

   5. Taylor expanded in b around 0 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-*l* 40.8%

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

      3. *-commutative 40.8%

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

      4. unpow2 40.8%

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

   7. Simplified 40.8%

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

   8. Taylor expanded in angle around 0 40.8%

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

      1. associate-*r* 40.7%

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

      2. *-commutative 40.7%

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

      3. *-commutative 40.7%

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

      4. unpow2 40.7%

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

      5. associate-*r* 40.8%

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

      6. unpow2 40.8%

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

      7. *-commutative 40.8%

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

      8. associate-*r* 40.7%

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

      9. *-commutative 40.7%

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

      10. unpow2 40.7%

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

      11. associate-*l* 42.4%

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

      12. associate-*l* 42.4%

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

      13. *-commutative 42.4%

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

   10. Simplified 42.4%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{+58}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(a \cdot \left(\pi \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 18: 50.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 6.2 \cdot 10^{+53}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(a \cdot \left(angle \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 6.2e+53)
   (* angle (* 0.011111111111111112 (* PI (* b b))))
   (* PI (* a (* angle (* a -0.011111111111111112))))))

C

a = abs(a);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6.2e+53) {
		tmp = angle * (0.011111111111111112 * (((double) M_PI) * (b * b)));
	} else {
		tmp = ((double) M_PI) * (a * (angle * (a * -0.011111111111111112)));
	}
	return tmp;
}

Java

a = Math.abs(a);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6.2e+53) {
		tmp = angle * (0.011111111111111112 * (Math.PI * (b * b)));
	} else {
		tmp = Math.PI * (a * (angle * (a * -0.011111111111111112)));
	}
	return tmp;
}

Python

a = abs(a)
def code(a, b, angle):
	tmp = 0
	if a <= 6.2e+53:
		tmp = angle * (0.011111111111111112 * (math.pi * (b * b)))
	else:
		tmp = math.pi * (a * (angle * (a * -0.011111111111111112)))
	return tmp

Julia

a = abs(a)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 6.2e+53)
		tmp = Float64(angle * Float64(0.011111111111111112 * Float64(pi * Float64(b * b))));
	else
		tmp = Float64(pi * Float64(a * Float64(angle * Float64(a * -0.011111111111111112))));
	end
	return tmp
end

MATLAB

a = abs(a)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 6.2e+53)
		tmp = angle * (0.011111111111111112 * (pi * (b * b)));
	else
		tmp = pi * (a * (angle * (a * -0.011111111111111112)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 6.2e+53], N[(angle * N[(0.011111111111111112 * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(a * N[(angle * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.20000000000000038e53

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. unpow2 58.1%

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

      3. unpow2 58.1%

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

      4. difference-of-squares 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in angle around 0 53.8%

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

   5. Taylor expanded in b around inf 39.7%

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

      1. *-commutative 39.7%

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

      2. associate-*l* 39.7%

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

      3. *-commutative 39.7%

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

      4. unpow2 39.7%

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

   7. Simplified 39.7%

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

   if 6.20000000000000038e53 < a

   1. Initial program 52.7%

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

      1. associate-*l* 52.7%

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

      2. unpow2 52.7%

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

      3. unpow2 52.7%

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

      4. difference-of-squares 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in angle around 0 47.9%

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

   5. Taylor expanded in b around 0 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-*l* 40.8%

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

      3. *-commutative 40.8%

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

      4. unpow2 40.8%

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

   7. Simplified 40.8%

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

   8. Taylor expanded in angle around 0 40.8%

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

      1. associate-*r* 40.8%

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

      2. unpow2 40.8%

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

   10. Simplified 40.8%

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

   11. Taylor expanded in angle around 0 40.8%

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

       1. associate-*r* 40.7%

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

       2. unpow2 40.7%

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

       3. associate-*r* 40.7%

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

       4. *-commutative 40.7%

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

       5. associate-*r* 53.7%

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

       6. *-commutative 53.7%

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

       7. *-commutative 53.7%

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

       8. associate-*l* 53.8%

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

   13. Simplified 53.8%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.2 \cdot 10^{+53}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(a \cdot \left(angle \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 19: 34.9% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

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

   1. associate-*l* 56.8%

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

   2. unpow2 56.8%

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

   3. unpow2 56.8%

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

   4. difference-of-squares 58.8%

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

3. Simplified 58.8%

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

4. Taylor expanded in angle around 0 52.5%

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

5. Taylor expanded in b around 0 32.9%

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

   1. *-commutative 32.9%

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

   2. associate-*l* 32.8%

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

   3. *-commutative 32.8%

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

   4. unpow2 32.8%

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

7. Simplified 32.8%

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

8. Taylor expanded in angle around 0 32.9%

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

   1. associate-*r* 32.8%

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

   2. unpow2 32.8%

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

10. Simplified 32.8%

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

11. Taylor expanded in angle around 0 32.9%

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

    1. unpow2 32.9%

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

    2. *-commutative 32.9%

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

    3. associate-*r* 32.9%

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

13. Simplified 32.9%

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

14. Final simplification 32.9%

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

Alternative 20: 34.9% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

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

   1. associate-*l* 56.8%

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

   2. unpow2 56.8%

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

   3. unpow2 56.8%

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

   4. difference-of-squares 58.8%

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

3. Simplified 58.8%

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

4. Taylor expanded in angle around 0 52.5%

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

5. Taylor expanded in b around 0 32.9%

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

   1. *-commutative 32.9%

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

   2. associate-*l* 32.8%

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

   3. *-commutative 32.8%

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

   4. unpow2 32.8%

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

7. Simplified 32.8%

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

8. Taylor expanded in angle around 0 32.9%

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

   1. associate-*r* 32.8%

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

   2. *-commutative 32.8%

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

   3. *-commutative 32.8%

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

   4. unpow2 32.8%

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

   5. associate-*r* 32.8%

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

   6. unpow2 32.8%

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

   7. *-commutative 32.8%

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

   8. associate-*r* 32.8%

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

   9. *-commutative 32.8%

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

   10. unpow2 32.8%

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

   11. associate-*l* 33.2%

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

   12. associate-*l* 33.2%

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

   13. *-commutative 33.2%

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

10. Simplified 33.2%

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

11. Final simplification 33.2%

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

Reproduce

herbie shell --seed 2023258 
(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)))))