ab-angle->ABCF B

Percentage Accurate: 53.9% → 67.1%

Time: 16.6s

Alternatives: 20

Speedup: 16.8×

• 33.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% 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: 67.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ t_1 := \left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \left(2 \cdot t\_0\right)\right)\\ t_2 := \frac{1}{b\_m - a}\\ \mathbf{if}\;b\_m \leq 10^{+70}:\\ \;\;\;\;t\_1 \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathbf{elif}\;b\_m \leq 5 \cdot 10^{+170}:\\ \;\;\;\;t\_1 \cdot \cos \left(\left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m + a\right) \cdot \left(t\_2 \cdot \frac{t\_0 \cdot \left(\left(b\_m - a\right) \cdot 2\right)}{t\_2}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (sin (* PI (* angle 0.005555555555555556))))
        (t_1 (* (+ b_m a) (* (- b_m a) (* 2.0 t_0))))
        (t_2 (/ 1.0 (- b_m a))))
   (if (<= b_m 1e+70)
     (* t_1 (cos (* angle (* PI 0.005555555555555556))))
     (if (<= b_m 5e+170)
       (*
        t_1
        (cos (* (* (sqrt (* PI (sqrt PI))) (sqrt (sqrt PI))) (/ angle 180.0))))
       (* (* (+ b_m a) (* t_2 (/ (* t_0 (* (- b_m a) 2.0)) t_2))) 1.0)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = sin((((double) M_PI) * (angle * 0.005555555555555556)));
	double t_1 = (b_m + a) * ((b_m - a) * (2.0 * t_0));
	double t_2 = 1.0 / (b_m - a);
	double tmp;
	if (b_m <= 1e+70) {
		tmp = t_1 * cos((angle * (((double) M_PI) * 0.005555555555555556)));
	} else if (b_m <= 5e+170) {
		tmp = t_1 * cos(((sqrt((((double) M_PI) * sqrt(((double) M_PI)))) * sqrt(sqrt(((double) M_PI)))) * (angle / 180.0)));
	} else {
		tmp = ((b_m + a) * (t_2 * ((t_0 * ((b_m - a) * 2.0)) / t_2))) * 1.0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = Math.sin((Math.PI * (angle * 0.005555555555555556)));
	double t_1 = (b_m + a) * ((b_m - a) * (2.0 * t_0));
	double t_2 = 1.0 / (b_m - a);
	double tmp;
	if (b_m <= 1e+70) {
		tmp = t_1 * Math.cos((angle * (Math.PI * 0.005555555555555556)));
	} else if (b_m <= 5e+170) {
		tmp = t_1 * Math.cos(((Math.sqrt((Math.PI * Math.sqrt(Math.PI))) * Math.sqrt(Math.sqrt(Math.PI))) * (angle / 180.0)));
	} else {
		tmp = ((b_m + a) * (t_2 * ((t_0 * ((b_m - a) * 2.0)) / t_2))) * 1.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = math.sin((math.pi * (angle * 0.005555555555555556)))
	t_1 = (b_m + a) * ((b_m - a) * (2.0 * t_0))
	t_2 = 1.0 / (b_m - a)
	tmp = 0
	if b_m <= 1e+70:
		tmp = t_1 * math.cos((angle * (math.pi * 0.005555555555555556)))
	elif b_m <= 5e+170:
		tmp = t_1 * math.cos(((math.sqrt((math.pi * math.sqrt(math.pi))) * math.sqrt(math.sqrt(math.pi))) * (angle / 180.0)))
	else:
		tmp = ((b_m + a) * (t_2 * ((t_0 * ((b_m - a) * 2.0)) / t_2))) * 1.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = sin(Float64(pi * Float64(angle * 0.005555555555555556)))
	t_1 = Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * Float64(2.0 * t_0)))
	t_2 = Float64(1.0 / Float64(b_m - a))
	tmp = 0.0
	if (b_m <= 1e+70)
		tmp = Float64(t_1 * cos(Float64(angle * Float64(pi * 0.005555555555555556))));
	elseif (b_m <= 5e+170)
		tmp = Float64(t_1 * cos(Float64(Float64(sqrt(Float64(pi * sqrt(pi))) * sqrt(sqrt(pi))) * Float64(angle / 180.0))));
	else
		tmp = Float64(Float64(Float64(b_m + a) * Float64(t_2 * Float64(Float64(t_0 * Float64(Float64(b_m - a) * 2.0)) / t_2))) * 1.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = sin((pi * (angle * 0.005555555555555556)));
	t_1 = (b_m + a) * ((b_m - a) * (2.0 * t_0));
	t_2 = 1.0 / (b_m - a);
	tmp = 0.0;
	if (b_m <= 1e+70)
		tmp = t_1 * cos((angle * (pi * 0.005555555555555556)));
	elseif (b_m <= 5e+170)
		tmp = t_1 * cos(((sqrt((pi * sqrt(pi))) * sqrt(sqrt(pi))) * (angle / 180.0)));
	else
		tmp = ((b_m + a) * (t_2 * ((t_0 * ((b_m - a) * 2.0)) / t_2))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1e+70], N[(t$95$1 * N[Cos[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5e+170], N[(t$95$1 * N[Cos[N[(N[(N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(t$95$2 * N[(N[(t$95$0 * N[(N[(b$95$m - a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.00000000000000007e70

   1. Initial program 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 65.5

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 65.1

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

   4. Applied rewrites 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 67.7

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

   6. Applied rewrites 67.7%

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

   if 1.00000000000000007e70 < b < 4.99999999999999977e170

   1. Initial program 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 65.7

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 68.7

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

   4. Applied rewrites 68.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lift-PI.f64 N/A

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

      4. add-sqr-sqrt N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-sqrt.f64 77.9

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

   6. Applied rewrites 77.9%

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

   if 4.99999999999999977e170 < b

   1. Initial program 44.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 79.9

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 77.0

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 88.4%

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

   9. Applied rewrites 88.5%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{+70}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+170}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \cos \left(\left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\frac{1}{b - a} \cdot \frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot 2\right)}{\frac{1}{b - a}}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<=
        (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
        5e+274)
     (* (+ b_m a) (* (- b_m a) (sin (* 0.011111111111111112 (* PI angle)))))
     (*
      1.0
      (*
       (+ b_m a)
       (*
        (- b_m a)
        (*
         2.0
         (*
          angle
          (fma
           -2.8577960676726107e-8
           (* (* angle angle) (* PI (* PI PI)))
           (* PI 0.005555555555555556))))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= 5e+274) {
		tmp = (b_m + a) * ((b_m - a) * sin((0.011111111111111112 * (((double) M_PI) * angle))));
	} else {
		tmp = 1.0 * ((b_m + a) * ((b_m - a) * (2.0 * (angle * fma(-2.8577960676726107e-8, ((angle * angle) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.005555555555555556))))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 5e+274)
		tmp = Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * sin(Float64(0.011111111111111112 * Float64(pi * angle)))));
	else
		tmp = Float64(1.0 * Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * Float64(2.0 * Float64(angle * fma(-2.8577960676726107e-8, Float64(Float64(angle * angle) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.005555555555555556)))))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b$95$m, 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], 5e+274], N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(2.0 * N[(angle * N[(-2.8577960676726107e-8 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 4.9999999999999998e274

   1. Initial program 60.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 63.6%

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

   if 4.9999999999999998e274 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 74.8

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 73.4

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

   4. Applied rewrites 73.4%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 82.7%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \cdot 1 \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \cdot 1 \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{34992000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right) \cdot 1 \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \color{blue}{{angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \color{blue}{\left(angle \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \color{blue}{\left(angle \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      6. cube-mult N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      7. unpow2 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      10. unpow2 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      12. lower-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{180} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)\right)\right) \cdot 1 \]

      15. lower-PI.f64 70.9

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.005555555555555556 \cdot \color{blue}{\pi}\right)\right)\right)\right)\right) \cdot 1 \]

   10. Applied rewrites 70.9%

       \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.005555555555555556 \cdot \pi\right)\right)}\right)\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;{b\_m}^{2} - {a}^{2} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;a \cdot \left(a \cdot \left(\mathsf{fma}\left(b\_m, \frac{b\_m}{a \cdot a}, -1\right) \cdot \sin \left(2 \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \left(2 \cdot \sin t\_0\right)\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle 0.005555555555555556))))
   (if (<= (- (pow b_m 2.0) (pow a 2.0)) -1e-129)
     (* a (* a (* (fma b_m (/ b_m (* a a)) -1.0) (sin (* 2.0 t_0)))))
     (* (* (+ b_m a) (* (- b_m a) (* 2.0 (sin t_0)))) 1.0))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = ((double) M_PI) * (angle * 0.005555555555555556);
	double tmp;
	if ((pow(b_m, 2.0) - pow(a, 2.0)) <= -1e-129) {
		tmp = a * (a * (fma(b_m, (b_m / (a * a)), -1.0) * sin((2.0 * t_0))));
	} else {
		tmp = ((b_m + a) * ((b_m - a) * (2.0 * sin(t_0)))) * 1.0;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	tmp = 0.0
	if (Float64((b_m ^ 2.0) - (a ^ 2.0)) <= -1e-129)
		tmp = Float64(a * Float64(a * Float64(fma(b_m, Float64(b_m / Float64(a * a)), -1.0) * sin(Float64(2.0 * t_0)))));
	else
		tmp = Float64(Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * Float64(2.0 * sin(t_0)))) * 1.0);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -1e-129], N[(a * N[(a * N[(N[(b$95$m * N[(b$95$m / N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[Sin[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(2.0 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -9.9999999999999993e-130

   1. Initial program 61.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow1 N/A

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

      8. sqr-pow N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. metadata-eval N/A

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

      14. unpow1/2 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 39.1

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 39.1

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

   4. Applied rewrites 39.1%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 56.1%

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

   9. Applied rewrites 71.0%

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

   if -9.9999999999999993e-130 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 48.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 64.9

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 64.6

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

   4. Applied rewrites 64.6%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 71.8%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;a \cdot \left(a \cdot \left(\mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ t_1 := \frac{1}{b\_m - a}\\ \mathbf{if}\;{b\_m}^{2} \leq 10^{+197}:\\ \;\;\;\;\left(\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \left(2 \cdot t\_0\right)\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m + a\right) \cdot \left(t\_1 \cdot \frac{t\_0 \cdot \left(\left(b\_m - a\right) \cdot 2\right)}{t\_1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (sin (* PI (* angle 0.005555555555555556))))
        (t_1 (/ 1.0 (- b_m a))))
   (if (<= (pow b_m 2.0) 1e+197)
     (*
      (* (+ b_m a) (* (- b_m a) (* 2.0 t_0)))
      (cos (* angle (* PI 0.005555555555555556))))
     (* (* (+ b_m a) (* t_1 (/ (* t_0 (* (- b_m a) 2.0)) t_1))) 1.0))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = sin((((double) M_PI) * (angle * 0.005555555555555556)));
	double t_1 = 1.0 / (b_m - a);
	double tmp;
	if (pow(b_m, 2.0) <= 1e+197) {
		tmp = ((b_m + a) * ((b_m - a) * (2.0 * t_0))) * cos((angle * (((double) M_PI) * 0.005555555555555556)));
	} else {
		tmp = ((b_m + a) * (t_1 * ((t_0 * ((b_m - a) * 2.0)) / t_1))) * 1.0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = Math.sin((Math.PI * (angle * 0.005555555555555556)));
	double t_1 = 1.0 / (b_m - a);
	double tmp;
	if (Math.pow(b_m, 2.0) <= 1e+197) {
		tmp = ((b_m + a) * ((b_m - a) * (2.0 * t_0))) * Math.cos((angle * (Math.PI * 0.005555555555555556)));
	} else {
		tmp = ((b_m + a) * (t_1 * ((t_0 * ((b_m - a) * 2.0)) / t_1))) * 1.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = math.sin((math.pi * (angle * 0.005555555555555556)))
	t_1 = 1.0 / (b_m - a)
	tmp = 0
	if math.pow(b_m, 2.0) <= 1e+197:
		tmp = ((b_m + a) * ((b_m - a) * (2.0 * t_0))) * math.cos((angle * (math.pi * 0.005555555555555556)))
	else:
		tmp = ((b_m + a) * (t_1 * ((t_0 * ((b_m - a) * 2.0)) / t_1))) * 1.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = sin(Float64(pi * Float64(angle * 0.005555555555555556)))
	t_1 = Float64(1.0 / Float64(b_m - a))
	tmp = 0.0
	if ((b_m ^ 2.0) <= 1e+197)
		tmp = Float64(Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * Float64(2.0 * t_0))) * cos(Float64(angle * Float64(pi * 0.005555555555555556))));
	else
		tmp = Float64(Float64(Float64(b_m + a) * Float64(t_1 * Float64(Float64(t_0 * Float64(Float64(b_m - a) * 2.0)) / t_1))) * 1.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = sin((pi * (angle * 0.005555555555555556)));
	t_1 = 1.0 / (b_m - a);
	tmp = 0.0;
	if ((b_m ^ 2.0) <= 1e+197)
		tmp = ((b_m + a) * ((b_m - a) * (2.0 * t_0))) * cos((angle * (pi * 0.005555555555555556)));
	else
		tmp = ((b_m + a) * (t_1 * ((t_0 * ((b_m - a) * 2.0)) / t_1))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[b$95$m, 2.0], $MachinePrecision], 1e+197], N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(t$95$1 * N[(N[(t$95$0 * N[(N[(b$95$m - a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 9.9999999999999995e196

   1. Initial program 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 65.5

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 65.4

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

   4. Applied rewrites 65.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 67.4

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

   6. Applied rewrites 67.4%

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

   if 9.9999999999999995e196 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 41.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 71.1

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 70.0

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

   4. Applied rewrites 70.0%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 83.4%

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

   9. Applied rewrites 83.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 10^{+197}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\frac{1}{b - a} \cdot \frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot 2\right)}{\frac{1}{b - a}}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \sin t\_0 \cdot \left(\left(b\_m - a\right) \cdot 2\right)\\ t_2 := \frac{1}{b\_m - a}\\ \mathbf{if}\;{b\_m}^{2} \leq 2 \cdot 10^{-115}:\\ \;\;\;\;t\_1 \cdot \left(\left(b\_m + a\right) \cdot \cos t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m + a\right) \cdot \left(t\_2 \cdot \frac{t\_1}{t\_2}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle 0.005555555555555556)))
        (t_1 (* (sin t_0) (* (- b_m a) 2.0)))
        (t_2 (/ 1.0 (- b_m a))))
   (if (<= (pow b_m 2.0) 2e-115)
     (* t_1 (* (+ b_m a) (cos t_0)))
     (* (* (+ b_m a) (* t_2 (/ t_1 t_2))) 1.0))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_1 = sin(t_0) * ((b_m - a) * 2.0);
	double t_2 = 1.0 / (b_m - a);
	double tmp;
	if (pow(b_m, 2.0) <= 2e-115) {
		tmp = t_1 * ((b_m + a) * cos(t_0));
	} else {
		tmp = ((b_m + a) * (t_2 * (t_1 / t_2))) * 1.0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = Math.PI * (angle * 0.005555555555555556);
	double t_1 = Math.sin(t_0) * ((b_m - a) * 2.0);
	double t_2 = 1.0 / (b_m - a);
	double tmp;
	if (Math.pow(b_m, 2.0) <= 2e-115) {
		tmp = t_1 * ((b_m + a) * Math.cos(t_0));
	} else {
		tmp = ((b_m + a) * (t_2 * (t_1 / t_2))) * 1.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = math.pi * (angle * 0.005555555555555556)
	t_1 = math.sin(t_0) * ((b_m - a) * 2.0)
	t_2 = 1.0 / (b_m - a)
	tmp = 0
	if math.pow(b_m, 2.0) <= 2e-115:
		tmp = t_1 * ((b_m + a) * math.cos(t_0))
	else:
		tmp = ((b_m + a) * (t_2 * (t_1 / t_2))) * 1.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	t_1 = Float64(sin(t_0) * Float64(Float64(b_m - a) * 2.0))
	t_2 = Float64(1.0 / Float64(b_m - a))
	tmp = 0.0
	if ((b_m ^ 2.0) <= 2e-115)
		tmp = Float64(t_1 * Float64(Float64(b_m + a) * cos(t_0)));
	else
		tmp = Float64(Float64(Float64(b_m + a) * Float64(t_2 * Float64(t_1 / t_2))) * 1.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = pi * (angle * 0.005555555555555556);
	t_1 = sin(t_0) * ((b_m - a) * 2.0);
	t_2 = 1.0 / (b_m - a);
	tmp = 0.0;
	if ((b_m ^ 2.0) <= 2e-115)
		tmp = t_1 * ((b_m + a) * cos(t_0));
	else
		tmp = ((b_m + a) * (t_2 * (t_1 / t_2))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[t$95$0], $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[b$95$m, 2.0], $MachinePrecision], 2e-115], N[(t$95$1 * N[(N[(b$95$m + a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(t$95$2 * N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 2.0000000000000001e-115

   1. Initial program 61.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 68.2

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 68.0

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

   4. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 67.9%

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

   if 2.0000000000000001e-115 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 48.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 67.0

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 75.3%

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

   9. Applied rewrites 75.3%

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

Alternative 6: 66.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{1}{b\_m - a}\\ t_1 := \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ \mathbf{if}\;b\_m \leq 4.7 \cdot 10^{+67}:\\ \;\;\;\;\left(\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \left(2 \cdot t\_1\right)\right)\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m + a\right) \cdot \left(t\_0 \cdot \frac{t\_1 \cdot \left(\left(b\_m - a\right) \cdot 2\right)}{t\_0}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- b_m a)))
        (t_1 (sin (* PI (* angle 0.005555555555555556)))))
   (if (<= b_m 4.7e+67)
     (* (* (+ b_m a) (* (- b_m a) (* 2.0 t_1))) (cos (/ PI (/ 180.0 angle))))
     (* (* (+ b_m a) (* t_0 (/ (* t_1 (* (- b_m a) 2.0)) t_0))) 1.0))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = 1.0 / (b_m - a);
	double t_1 = sin((((double) M_PI) * (angle * 0.005555555555555556)));
	double tmp;
	if (b_m <= 4.7e+67) {
		tmp = ((b_m + a) * ((b_m - a) * (2.0 * t_1))) * cos((((double) M_PI) / (180.0 / angle)));
	} else {
		tmp = ((b_m + a) * (t_0 * ((t_1 * ((b_m - a) * 2.0)) / t_0))) * 1.0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = 1.0 / (b_m - a);
	double t_1 = Math.sin((Math.PI * (angle * 0.005555555555555556)));
	double tmp;
	if (b_m <= 4.7e+67) {
		tmp = ((b_m + a) * ((b_m - a) * (2.0 * t_1))) * Math.cos((Math.PI / (180.0 / angle)));
	} else {
		tmp = ((b_m + a) * (t_0 * ((t_1 * ((b_m - a) * 2.0)) / t_0))) * 1.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = 1.0 / (b_m - a)
	t_1 = math.sin((math.pi * (angle * 0.005555555555555556)))
	tmp = 0
	if b_m <= 4.7e+67:
		tmp = ((b_m + a) * ((b_m - a) * (2.0 * t_1))) * math.cos((math.pi / (180.0 / angle)))
	else:
		tmp = ((b_m + a) * (t_0 * ((t_1 * ((b_m - a) * 2.0)) / t_0))) * 1.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(1.0 / Float64(b_m - a))
	t_1 = sin(Float64(pi * Float64(angle * 0.005555555555555556)))
	tmp = 0.0
	if (b_m <= 4.7e+67)
		tmp = Float64(Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * Float64(2.0 * t_1))) * cos(Float64(pi / Float64(180.0 / angle))));
	else
		tmp = Float64(Float64(Float64(b_m + a) * Float64(t_0 * Float64(Float64(t_1 * Float64(Float64(b_m - a) * 2.0)) / t_0))) * 1.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = 1.0 / (b_m - a);
	t_1 = sin((pi * (angle * 0.005555555555555556)));
	tmp = 0.0;
	if (b_m <= 4.7e+67)
		tmp = ((b_m + a) * ((b_m - a) * (2.0 * t_1))) * cos((pi / (180.0 / angle)));
	else
		tmp = ((b_m + a) * (t_0 * ((t_1 * ((b_m - a) * 2.0)) / t_0))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(1.0 / N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 4.7e+67], N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(t$95$0 * N[(N[(t$95$1 * N[(N[(b$95$m - a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.70000000000000017e67

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 65.3

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 64.9

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

   4. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 66.6

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

   6. Applied rewrites 66.6%

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

   if 4.70000000000000017e67 < b

   1. Initial program 47.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 74.4

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 74.0

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 79.7%

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

   9. Applied rewrites 79.7%

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

Alternative 7: 66.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;{b\_m}^{2} \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (pow b_m 2.0) 2e-150)
   (* (+ b_m a) (* (- b_m a) (sin (* 0.011111111111111112 (* PI angle)))))
   (*
    (*
     (+ b_m a)
     (* (- b_m a) (* 2.0 (sin (* PI (* angle 0.005555555555555556))))))
    1.0)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (pow(b_m, 2.0) <= 2e-150) {
		tmp = (b_m + a) * ((b_m - a) * sin((0.011111111111111112 * (((double) M_PI) * angle))));
	} else {
		tmp = ((b_m + a) * ((b_m - a) * (2.0 * sin((((double) M_PI) * (angle * 0.005555555555555556)))))) * 1.0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if (Math.pow(b_m, 2.0) <= 2e-150) {
		tmp = (b_m + a) * ((b_m - a) * Math.sin((0.011111111111111112 * (Math.PI * angle))));
	} else {
		tmp = ((b_m + a) * ((b_m - a) * (2.0 * Math.sin((Math.PI * (angle * 0.005555555555555556)))))) * 1.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if math.pow(b_m, 2.0) <= 2e-150:
		tmp = (b_m + a) * ((b_m - a) * math.sin((0.011111111111111112 * (math.pi * angle))))
	else:
		tmp = ((b_m + a) * ((b_m - a) * (2.0 * math.sin((math.pi * (angle * 0.005555555555555556)))))) * 1.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if ((b_m ^ 2.0) <= 2e-150)
		tmp = Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * sin(Float64(0.011111111111111112 * Float64(pi * angle)))));
	else
		tmp = Float64(Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * Float64(2.0 * sin(Float64(pi * Float64(angle * 0.005555555555555556)))))) * 1.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if ((b_m ^ 2.0) <= 2e-150)
		tmp = (b_m + a) * ((b_m - a) * sin((0.011111111111111112 * (pi * angle))));
	else
		tmp = ((b_m + a) * ((b_m - a) * (2.0 * sin((pi * (angle * 0.005555555555555556)))))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[Power[b$95$m, 2.0], $MachinePrecision], 2e-150], N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(2.0 * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 2.00000000000000001e-150

   1. Initial program 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 66.0%

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

   if 2.00000000000000001e-150 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 49.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 66.8

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 74.5%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;{b\_m}^{2} - {a}^{2} \leq -5 \cdot 10^{-254}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (- (pow b_m 2.0) (pow a 2.0)) -5e-254)
   (* -0.011111111111111112 (* a (* PI (* a angle))))
   (* 0.011111111111111112 (* angle (* PI (* b_m b_m))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if ((pow(b_m, 2.0) - pow(a, 2.0)) <= -5e-254) {
		tmp = -0.011111111111111112 * (a * (((double) M_PI) * (a * angle)));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b_m * b_m)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if ((Math.pow(b_m, 2.0) - Math.pow(a, 2.0)) <= -5e-254) {
		tmp = -0.011111111111111112 * (a * (Math.PI * (a * angle)));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b_m * b_m)));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if (math.pow(b_m, 2.0) - math.pow(a, 2.0)) <= -5e-254:
		tmp = -0.011111111111111112 * (a * (math.pi * (a * angle)))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b_m * b_m)))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (Float64((b_m ^ 2.0) - (a ^ 2.0)) <= -5e-254)
		tmp = Float64(-0.011111111111111112 * Float64(a * Float64(pi * Float64(a * angle))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b_m * b_m))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if (((b_m ^ 2.0) - (a ^ 2.0)) <= -5e-254)
		tmp = -0.011111111111111112 * (a * (pi * (a * angle)));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b_m * b_m)));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -5e-254], N[(-0.011111111111111112 * N[(a * N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -5.0000000000000003e-254

   1. Initial program 61.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 57.5

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

   5. Applied rewrites 57.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 57.4%

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

   10. Applied rewrites 67.0%

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

   if -5.0000000000000003e-254 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 48.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 23.2%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 50.2%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{-254}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{1}{b\_m - a}\\ t_1 := \left(b\_m - a\right) \cdot 2\\ \mathbf{if}\;b\_m \leq 5.5 \cdot 10^{+204}:\\ \;\;\;\;\left(b\_m + a\right) \cdot \left(t\_1 \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m + a\right) \cdot \left(t\_0 \cdot \frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot t\_1}{t\_0}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- b_m a))) (t_1 (* (- b_m a) 2.0)))
   (if (<= b_m 5.5e+204)
     (* (+ b_m a) (* t_1 (* (sin (* angle (* PI 0.011111111111111112))) 0.5)))
     (*
      (*
       (+ b_m a)
       (* t_0 (/ (* (sin (* PI (* angle 0.005555555555555556))) t_1) t_0)))
      1.0))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = 1.0 / (b_m - a);
	double t_1 = (b_m - a) * 2.0;
	double tmp;
	if (b_m <= 5.5e+204) {
		tmp = (b_m + a) * (t_1 * (sin((angle * (((double) M_PI) * 0.011111111111111112))) * 0.5));
	} else {
		tmp = ((b_m + a) * (t_0 * ((sin((((double) M_PI) * (angle * 0.005555555555555556))) * t_1) / t_0))) * 1.0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = 1.0 / (b_m - a);
	double t_1 = (b_m - a) * 2.0;
	double tmp;
	if (b_m <= 5.5e+204) {
		tmp = (b_m + a) * (t_1 * (Math.sin((angle * (Math.PI * 0.011111111111111112))) * 0.5));
	} else {
		tmp = ((b_m + a) * (t_0 * ((Math.sin((Math.PI * (angle * 0.005555555555555556))) * t_1) / t_0))) * 1.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = 1.0 / (b_m - a)
	t_1 = (b_m - a) * 2.0
	tmp = 0
	if b_m <= 5.5e+204:
		tmp = (b_m + a) * (t_1 * (math.sin((angle * (math.pi * 0.011111111111111112))) * 0.5))
	else:
		tmp = ((b_m + a) * (t_0 * ((math.sin((math.pi * (angle * 0.005555555555555556))) * t_1) / t_0))) * 1.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(1.0 / Float64(b_m - a))
	t_1 = Float64(Float64(b_m - a) * 2.0)
	tmp = 0.0
	if (b_m <= 5.5e+204)
		tmp = Float64(Float64(b_m + a) * Float64(t_1 * Float64(sin(Float64(angle * Float64(pi * 0.011111111111111112))) * 0.5)));
	else
		tmp = Float64(Float64(Float64(b_m + a) * Float64(t_0 * Float64(Float64(sin(Float64(pi * Float64(angle * 0.005555555555555556))) * t_1) / t_0))) * 1.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = 1.0 / (b_m - a);
	t_1 = (b_m - a) * 2.0;
	tmp = 0.0;
	if (b_m <= 5.5e+204)
		tmp = (b_m + a) * (t_1 * (sin((angle * (pi * 0.011111111111111112))) * 0.5));
	else
		tmp = ((b_m + a) * (t_0 * ((sin((pi * (angle * 0.005555555555555556))) * t_1) / t_0))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(1.0 / N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m - a), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[b$95$m, 5.5e+204], N[(N[(b$95$m + a), $MachinePrecision] * N[(t$95$1 * N[(N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(t$95$0 * N[(N[(N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.4999999999999996e204

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 66.3

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 66.2

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

   4. Applied rewrites 66.2%

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

   6. Applied rewrites 67.6%

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

   if 5.4999999999999996e204 < b

   1. Initial program 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 77.7

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 74.0

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 88.8%

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

   9. Applied rewrites 88.8%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{+204}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\frac{1}{b - a} \cdot \frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot 2\right)}{\frac{1}{b - a}}\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;{b\_m}^{2} \leq 2 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(a, -0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot angle\right)\right), \mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(angle \cdot \left(b\_m \cdot b\_m\right)\right), 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (pow b_m 2.0) 2e+69)
   (fma
    a
    (* -0.011111111111111112 (* a (* PI angle)))
    (fma 0.011111111111111112 (* PI (* angle (* b_m b_m))) 0.0))
   (*
    1.0
    (*
     (+ b_m a)
     (*
      (- b_m a)
      (*
       2.0
       (*
        angle
        (fma
         -2.8577960676726107e-8
         (* (* angle angle) (* PI (* PI PI)))
         (* PI 0.005555555555555556)))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (pow(b_m, 2.0) <= 2e+69) {
		tmp = fma(a, (-0.011111111111111112 * (a * (((double) M_PI) * angle))), fma(0.011111111111111112, (((double) M_PI) * (angle * (b_m * b_m))), 0.0));
	} else {
		tmp = 1.0 * ((b_m + a) * ((b_m - a) * (2.0 * (angle * fma(-2.8577960676726107e-8, ((angle * angle) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.005555555555555556))))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if ((b_m ^ 2.0) <= 2e+69)
		tmp = fma(a, Float64(-0.011111111111111112 * Float64(a * Float64(pi * angle))), fma(0.011111111111111112, Float64(pi * Float64(angle * Float64(b_m * b_m))), 0.0));
	else
		tmp = Float64(1.0 * Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * Float64(2.0 * Float64(angle * fma(-2.8577960676726107e-8, Float64(Float64(angle * angle) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.005555555555555556)))))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[Power[b$95$m, 2.0], $MachinePrecision], 2e+69], N[(a * N[(-0.011111111111111112 * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(2.0 * N[(angle * N[(-2.8577960676726107e-8 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 2.0000000000000001e69

   1. Initial program 60.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 61.2%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)}, \mathsf{fma}\left(0.011111111111111112, \left(angle \cdot \left(b \cdot b\right)\right) \cdot \pi, 0\right)\right) \]

   if 2.0000000000000001e69 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 45.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.0

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 68.0

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

   4. Applied rewrites 68.0%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \cdot 1 \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \cdot 1 \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{34992000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right) \cdot 1 \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \color{blue}{{angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \color{blue}{\left(angle \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \color{blue}{\left(angle \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      6. cube-mult N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      7. unpow2 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      10. unpow2 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      12. lower-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot 1 \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{180} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)\right)\right) \cdot 1 \]

      15. lower-PI.f64 72.9

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.005555555555555556 \cdot \color{blue}{\pi}\right)\right)\right)\right)\right) \cdot 1 \]

   10. Applied rewrites 72.9%

       \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.005555555555555556 \cdot \pi\right)\right)}\right)\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(a, -0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot angle\right)\right), \mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(angle \cdot \left(b \cdot b\right)\right), 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 4.3 \cdot 10^{+197}:\\ \;\;\;\;1 \cdot \left(\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot \sqrt{\pi}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(\mathsf{fma}\left(b\_m, \frac{b\_m}{a \cdot a}, -1\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= a 4.3e+197)
   (*
    1.0
    (*
     (+ b_m a)
     (*
      (- b_m a)
      (*
       2.0
       (sin (* 0.005555555555555556 (* (sqrt PI) (* angle (sqrt PI)))))))))
   (*
    a
    (*
     a
     (*
      (fma b_m (/ b_m (* a a)) -1.0)
      (sin (* 2.0 (* PI (* angle 0.005555555555555556)))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (a <= 4.3e+197) {
		tmp = 1.0 * ((b_m + a) * ((b_m - a) * (2.0 * sin((0.005555555555555556 * (sqrt(((double) M_PI)) * (angle * sqrt(((double) M_PI)))))))));
	} else {
		tmp = a * (a * (fma(b_m, (b_m / (a * a)), -1.0) * sin((2.0 * (((double) M_PI) * (angle * 0.005555555555555556))))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (a <= 4.3e+197)
		tmp = Float64(1.0 * Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * Float64(2.0 * sin(Float64(0.005555555555555556 * Float64(sqrt(pi) * Float64(angle * sqrt(pi)))))))));
	else
		tmp = Float64(a * Float64(a * Float64(fma(b_m, Float64(b_m / Float64(a * a)), -1.0) * sin(Float64(2.0 * Float64(pi * Float64(angle * 0.005555555555555556)))))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[a, 4.3e+197], N[(1.0 * N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(2.0 * N[Sin[N[(0.005555555555555556 * N[(N[Sqrt[Pi], $MachinePrecision] * N[(angle * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(N[(b$95$m * N[(b$95$m / N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.29999999999999996e197

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 65.2

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 64.7

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

   4. Applied rewrites 64.7%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 69.5

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

   9. Applied rewrites 69.5%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r/ N/A

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

       4. /-rgt-identity N/A

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

       5. *-commutative N/A

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

       6. lift-PI.f64 N/A

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

       7. add-sqr-sqrt N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-sqrt.f64 N/A

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

       13. lift-PI.f64 N/A

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

       14. lower-sqrt.f64 70.4

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

   11. Applied rewrites 70.4%

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

   if 4.29999999999999996e197 < a

   1. Initial program 65.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow1 N/A

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

      8. sqr-pow N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. metadata-eval N/A

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

      14. unpow1/2 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 65.6

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 65.6

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

   4. Applied rewrites 65.6%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 80.6%

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

   9. Applied rewrites 95.0%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.3 \cdot 10^{+197}:\\ \;\;\;\;1 \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot \sqrt{\pi}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(\mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+24}:\\ \;\;\;\;\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+92}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \sqrt{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 4e+24)
   (* (+ b_m a) (* (- b_m a) (sin (* 0.011111111111111112 (* PI angle)))))
   (if (<= (/ angle 180.0) 1e+92)
     (* -0.011111111111111112 (* PI (* angle (sqrt (* a (* a (* a a)))))))
     (*
      (sin (* 2.0 (* PI (* angle 0.005555555555555556))))
      (* (+ b_m a) (- b_m a))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 4e+24) {
		tmp = (b_m + a) * ((b_m - a) * sin((0.011111111111111112 * (((double) M_PI) * angle))));
	} else if ((angle / 180.0) <= 1e+92) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * sqrt((a * (a * (a * a))))));
	} else {
		tmp = sin((2.0 * (((double) M_PI) * (angle * 0.005555555555555556)))) * ((b_m + a) * (b_m - a));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 4e+24) {
		tmp = (b_m + a) * ((b_m - a) * Math.sin((0.011111111111111112 * (Math.PI * angle))));
	} else if ((angle / 180.0) <= 1e+92) {
		tmp = -0.011111111111111112 * (Math.PI * (angle * Math.sqrt((a * (a * (a * a))))));
	} else {
		tmp = Math.sin((2.0 * (Math.PI * (angle * 0.005555555555555556)))) * ((b_m + a) * (b_m - a));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if (angle / 180.0) <= 4e+24:
		tmp = (b_m + a) * ((b_m - a) * math.sin((0.011111111111111112 * (math.pi * angle))))
	elif (angle / 180.0) <= 1e+92:
		tmp = -0.011111111111111112 * (math.pi * (angle * math.sqrt((a * (a * (a * a))))))
	else:
		tmp = math.sin((2.0 * (math.pi * (angle * 0.005555555555555556)))) * ((b_m + a) * (b_m - a))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 4e+24)
		tmp = Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * sin(Float64(0.011111111111111112 * Float64(pi * angle)))));
	elseif (Float64(angle / 180.0) <= 1e+92)
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * sqrt(Float64(a * Float64(a * Float64(a * a)))))));
	else
		tmp = Float64(sin(Float64(2.0 * Float64(pi * Float64(angle * 0.005555555555555556)))) * Float64(Float64(b_m + a) * Float64(b_m - a)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 4e+24)
		tmp = (b_m + a) * ((b_m - a) * sin((0.011111111111111112 * (pi * angle))));
	elseif ((angle / 180.0) <= 1e+92)
		tmp = -0.011111111111111112 * (pi * (angle * sqrt((a * (a * (a * a))))));
	else
		tmp = sin((2.0 * (pi * (angle * 0.005555555555555556)))) * ((b_m + a) * (b_m - a));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 4e+24], N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+92], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[Sqrt[N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(2.0 * N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.9999999999999999e24

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 77.0%

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

   if 3.9999999999999999e24 < (/.f64 angle #s(literal 180 binary64)) < 1e92

   1. Initial program 27.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 49.0

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

   5. Applied rewrites 49.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 49.3%

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

   10. Applied rewrites 49.3%

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

   if 1e92 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 35.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 41.7

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

      19. lift-/.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 41.2

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

   4. Applied rewrites 41.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

   6. Applied rewrites 45.5%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+24}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+92}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \sqrt{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+24}:\\ \;\;\;\;\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+92}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \sqrt{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 4e+24)
   (* (+ b_m a) (* (- b_m a) (sin (* 0.011111111111111112 (* PI angle)))))
   (if (<= (/ angle 180.0) 1e+92)
     (* -0.011111111111111112 (* PI (* angle (sqrt (* a (* a (* a a)))))))
     (*
      (* (+ b_m a) (- b_m a))
      (sin (* 2.0 (* angle (* PI 0.005555555555555556))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 4e+24) {
		tmp = (b_m + a) * ((b_m - a) * sin((0.011111111111111112 * (((double) M_PI) * angle))));
	} else if ((angle / 180.0) <= 1e+92) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * sqrt((a * (a * (a * a))))));
	} else {
		tmp = ((b_m + a) * (b_m - a)) * sin((2.0 * (angle * (((double) M_PI) * 0.005555555555555556))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 4e+24) {
		tmp = (b_m + a) * ((b_m - a) * Math.sin((0.011111111111111112 * (Math.PI * angle))));
	} else if ((angle / 180.0) <= 1e+92) {
		tmp = -0.011111111111111112 * (Math.PI * (angle * Math.sqrt((a * (a * (a * a))))));
	} else {
		tmp = ((b_m + a) * (b_m - a)) * Math.sin((2.0 * (angle * (Math.PI * 0.005555555555555556))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if (angle / 180.0) <= 4e+24:
		tmp = (b_m + a) * ((b_m - a) * math.sin((0.011111111111111112 * (math.pi * angle))))
	elif (angle / 180.0) <= 1e+92:
		tmp = -0.011111111111111112 * (math.pi * (angle * math.sqrt((a * (a * (a * a))))))
	else:
		tmp = ((b_m + a) * (b_m - a)) * math.sin((2.0 * (angle * (math.pi * 0.005555555555555556))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 4e+24)
		tmp = Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * sin(Float64(0.011111111111111112 * Float64(pi * angle)))));
	elseif (Float64(angle / 180.0) <= 1e+92)
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * sqrt(Float64(a * Float64(a * Float64(a * a)))))));
	else
		tmp = Float64(Float64(Float64(b_m + a) * Float64(b_m - a)) * sin(Float64(2.0 * Float64(angle * Float64(pi * 0.005555555555555556)))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 4e+24)
		tmp = (b_m + a) * ((b_m - a) * sin((0.011111111111111112 * (pi * angle))));
	elseif ((angle / 180.0) <= 1e+92)
		tmp = -0.011111111111111112 * (pi * (angle * sqrt((a * (a * (a * a))))));
	else
		tmp = ((b_m + a) * (b_m - a)) * sin((2.0 * (angle * (pi * 0.005555555555555556))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 4e+24], N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+92], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[Sqrt[N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.9999999999999999e24

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 77.0%

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

   if 3.9999999999999999e24 < (/.f64 angle #s(literal 180 binary64)) < 1e92

   1. Initial program 27.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 49.0

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

   5. Applied rewrites 49.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 49.3%

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

   10. Applied rewrites 49.3%

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

   if 1e92 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 35.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. lift--.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 6.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

   6. Applied rewrites 43.8%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+24}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+92}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \sqrt{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+67}:\\ \;\;\;\;\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 1e+67)
   (* (+ b_m a) (* (- b_m a) (* angle (* PI 0.011111111111111112))))
   (* (sin (* 0.011111111111111112 (* PI angle))) (* (+ b_m a) (- b_m a)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 1e+67) {
		tmp = (b_m + a) * ((b_m - a) * (angle * (((double) M_PI) * 0.011111111111111112)));
	} else {
		tmp = sin((0.011111111111111112 * (((double) M_PI) * angle))) * ((b_m + a) * (b_m - a));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 1e+67) {
		tmp = (b_m + a) * ((b_m - a) * (angle * (Math.PI * 0.011111111111111112)));
	} else {
		tmp = Math.sin((0.011111111111111112 * (Math.PI * angle))) * ((b_m + a) * (b_m - a));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if (angle / 180.0) <= 1e+67:
		tmp = (b_m + a) * ((b_m - a) * (angle * (math.pi * 0.011111111111111112)))
	else:
		tmp = math.sin((0.011111111111111112 * (math.pi * angle))) * ((b_m + a) * (b_m - a))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1e+67)
		tmp = Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * Float64(angle * Float64(pi * 0.011111111111111112))));
	else
		tmp = Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle))) * Float64(Float64(b_m + a) * Float64(b_m - a)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 1e+67)
		tmp = (b_m + a) * ((b_m - a) * (angle * (pi * 0.011111111111111112)));
	else
		tmp = sin((0.011111111111111112 * (pi * angle))) * ((b_m + a) * (b_m - a));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+67], N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999983e66

   1. Initial program 59.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      13. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      16. lower--.f64 62.9

          \[\leadsto \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} \]

   if 9.99999999999999983e66 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 33.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

      13. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

      16. lower--.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

   4. Applied rewrites 39.2%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+67}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 4 \cdot 10^{+218}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (pow a 2.0) 4e+218)
   (* (* angle (* PI 0.011111111111111112)) (* (+ b_m a) (- b_m a)))
   (* -0.011111111111111112 (* PI (* a (* a angle))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (pow(a, 2.0) <= 4e+218) {
		tmp = (angle * (((double) M_PI) * 0.011111111111111112)) * ((b_m + a) * (b_m - a));
	} else {
		tmp = -0.011111111111111112 * (((double) M_PI) * (a * (a * angle)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if (Math.pow(a, 2.0) <= 4e+218) {
		tmp = (angle * (Math.PI * 0.011111111111111112)) * ((b_m + a) * (b_m - a));
	} else {
		tmp = -0.011111111111111112 * (Math.PI * (a * (a * angle)));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if math.pow(a, 2.0) <= 4e+218:
		tmp = (angle * (math.pi * 0.011111111111111112)) * ((b_m + a) * (b_m - a))
	else:
		tmp = -0.011111111111111112 * (math.pi * (a * (a * angle)))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if ((a ^ 2.0) <= 4e+218)
		tmp = Float64(Float64(angle * Float64(pi * 0.011111111111111112)) * Float64(Float64(b_m + a) * Float64(b_m - a)));
	else
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(a * Float64(a * angle))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if ((a ^ 2.0) <= 4e+218)
		tmp = (angle * (pi * 0.011111111111111112)) * ((b_m + a) * (b_m - a));
	else
		tmp = -0.011111111111111112 * (pi * (a * (a * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 4e+218], N[(N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(Pi * N[(a * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 4.00000000000000033e218

   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. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{90}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      13. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      16. lower--.f64 55.5

          \[\leadsto \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   if 4.00000000000000033e218 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 42.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{90}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      13. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      16. lower--.f64 53.8

          \[\leadsto \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 53.8%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.0%

      \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.9%

       \[\leadsto -0.011111111111111112 \cdot \left(\left(\left(a \cdot angle\right) \cdot a\right) \cdot \pi\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 4 \cdot 10^{+218}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 1.5 \cdot 10^{+78}:\\ \;\;\;\;\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(b\_m + a\right) \cdot \left(\left(b\_m \cdot 2\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 1.5e+78)
   (* (+ b_m a) (* (- b_m a) (* angle (* PI 0.011111111111111112))))
   (*
    1.0
    (*
     (+ b_m a)
     (*
      (* b_m 2.0)
      (*
       angle
       (fma
        -2.8577960676726107e-8
        (* (* angle angle) (* PI (* PI PI)))
        (* PI 0.005555555555555556))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 1.5e+78) {
		tmp = (b_m + a) * ((b_m - a) * (angle * (((double) M_PI) * 0.011111111111111112)));
	} else {
		tmp = 1.0 * ((b_m + a) * ((b_m * 2.0) * (angle * fma(-2.8577960676726107e-8, ((angle * angle) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.005555555555555556)))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1.5e+78)
		tmp = Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * Float64(angle * Float64(pi * 0.011111111111111112))));
	else
		tmp = Float64(1.0 * Float64(Float64(b_m + a) * Float64(Float64(b_m * 2.0) * Float64(angle * fma(-2.8577960676726107e-8, Float64(Float64(angle * angle) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.005555555555555556))))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1.5e+78], N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m * 2.0), $MachinePrecision] * N[(angle * N[(-2.8577960676726107e-8 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.49999999999999991e78

   1. Initial program 58.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{90}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      13. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      16. lower--.f64 62.3

          \[\leadsto \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} \]

   if 1.49999999999999991e78 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 32.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      10. difference-of-squares N/A

          \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      16. lower--.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      17. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      18. lower-*.f64 40.3

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      19. lift-/.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      20. div-inv N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      21. lower-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      22. metadata-eval 39.8

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right)\right)\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 36.7%

      \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in b around inf

      \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(2 \cdot \left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}\right) \cdot 1 \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot b\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \cdot 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot b\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \cdot 1 \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\color{blue}{\left(2 \cdot b\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot 1 \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(2 \cdot b\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot 1 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(2 \cdot b\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot 1 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(2 \cdot b\right) \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \cdot 1 \]

      7. lower-PI.f64 24.8

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(2 \cdot b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)\right)\right) \cdot 1 \]

   10. Applied rewrites 24.8%

       \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \cdot 1 \]

   11. Taylor expanded in angle around 0

       \[\leadsto \left(\left(b + a\right) \cdot \left(\left(2 \cdot b\right) \cdot \left(angle \cdot \color{blue}{\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 34.1%

       \[\leadsto \left(\left(b + a\right) \cdot \left(\left(2 \cdot b\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.005555555555555556\right)}\right)\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 1.5 \cdot 10^{+78}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(b + a\right) \cdot \left(\left(b \cdot 2\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 62.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot t\_0\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+244}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \sqrt{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI 0.011111111111111112))))
   (if (<= (/ angle 180.0) 2e+54)
     (* (+ b_m a) (* (- b_m a) t_0))
     (if (<= (/ angle 180.0) 5e+244)
       (* -0.011111111111111112 (* PI (* angle (sqrt (* a (* a (* a a)))))))
       (* t_0 (* (+ b_m a) (- b_m a)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = angle * (((double) M_PI) * 0.011111111111111112);
	double tmp;
	if ((angle / 180.0) <= 2e+54) {
		tmp = (b_m + a) * ((b_m - a) * t_0);
	} else if ((angle / 180.0) <= 5e+244) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * sqrt((a * (a * (a * a))))));
	} else {
		tmp = t_0 * ((b_m + a) * (b_m - a));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = angle * (Math.PI * 0.011111111111111112);
	double tmp;
	if ((angle / 180.0) <= 2e+54) {
		tmp = (b_m + a) * ((b_m - a) * t_0);
	} else if ((angle / 180.0) <= 5e+244) {
		tmp = -0.011111111111111112 * (Math.PI * (angle * Math.sqrt((a * (a * (a * a))))));
	} else {
		tmp = t_0 * ((b_m + a) * (b_m - a));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = angle * (math.pi * 0.011111111111111112)
	tmp = 0
	if (angle / 180.0) <= 2e+54:
		tmp = (b_m + a) * ((b_m - a) * t_0)
	elif (angle / 180.0) <= 5e+244:
		tmp = -0.011111111111111112 * (math.pi * (angle * math.sqrt((a * (a * (a * a))))))
	else:
		tmp = t_0 * ((b_m + a) * (b_m - a))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(angle * Float64(pi * 0.011111111111111112))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e+54)
		tmp = Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * t_0));
	elseif (Float64(angle / 180.0) <= 5e+244)
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * sqrt(Float64(a * Float64(a * Float64(a * a)))))));
	else
		tmp = Float64(t_0 * Float64(Float64(b_m + a) * Float64(b_m - a)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = angle * (pi * 0.011111111111111112);
	tmp = 0.0;
	if ((angle / 180.0) <= 2e+54)
		tmp = (b_m + a) * ((b_m - a) * t_0);
	elseif ((angle / 180.0) <= 5e+244)
		tmp = -0.011111111111111112 * (pi * (angle * sqrt((a * (a * (a * a))))));
	else
		tmp = t_0 * ((b_m + a) * (b_m - a));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+54], N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+244], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[Sqrt[N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.0000000000000002e54

   1. Initial program 60.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{90}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      13. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      16. lower--.f64 63.0

          \[\leadsto \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 63.0%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.9%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} \]

   if 2.0000000000000002e54 < (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000022e244

   1. Initial program 30.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{90}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      13. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      16. lower--.f64 26.8

          \[\leadsto \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 26.8%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.6%

      \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.8%

       \[\leadsto -0.011111111111111112 \cdot \left(\left(\sqrt{a \cdot \left(a \cdot \left(a \cdot a\right)\right)} \cdot angle\right) \cdot \pi\right) \]

   if 5.00000000000000022e244 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 40.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{90}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      13. difference-of-squares N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      15. lower-+.f64 N/A

          \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      16. lower--.f64 37.7

          \[\leadsto \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+244}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \sqrt{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.4% accurate, 16.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (* (+ b_m a) (* (- b_m a) (* angle (* PI 0.011111111111111112)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	return (b_m + a) * ((b_m - a) * (angle * (((double) M_PI) * 0.011111111111111112)));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	return (b_m + a) * ((b_m - a) * (angle * (Math.PI * 0.011111111111111112)));
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	return (b_m + a) * ((b_m - a) * (angle * (math.pi * 0.011111111111111112)))

Julia

b_m = abs(b)
function code(a, b_m, angle)
	return Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * Float64(angle * Float64(pi * 0.011111111111111112))))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle)
	tmp = (b_m + a) * ((b_m - a) * (angle * (pi * 0.011111111111111112)));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

   3. *-commutative N/A

      \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   10. lower-PI.f64 N/A

       \[\leadsto \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{90}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   11. unpow2 N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

   12. unpow2 N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

   13. difference-of-squares N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   15. lower-+.f64 N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

   16. lower--.f64 54.9

       \[\leadsto \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

5. Applied rewrites 54.9%

   \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 62.9%

   \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} \]
8. Add Preprocessing

Alternative 19: 38.5% accurate, 21.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ -0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (* -0.011111111111111112 (* a (* PI (* a angle)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	return -0.011111111111111112 * (a * (((double) M_PI) * (a * angle)));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	return -0.011111111111111112 * (a * (Math.PI * (a * angle)));
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	return -0.011111111111111112 * (a * (math.pi * (a * angle)))

Julia

b_m = abs(b)
function code(a, b_m, angle)
	return Float64(-0.011111111111111112 * Float64(a * Float64(pi * Float64(a * angle))))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle)
	tmp = -0.011111111111111112 * (a * (pi * (a * angle)));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := N[(-0.011111111111111112 * N[(a * N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

   3. *-commutative N/A

      \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   10. lower-PI.f64 N/A

       \[\leadsto \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{90}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   11. unpow2 N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

   12. unpow2 N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

   13. difference-of-squares N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   15. lower-+.f64 N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

   16. lower--.f64 54.9

       \[\leadsto \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

5. Applied rewrites 54.9%

   \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

6. Taylor expanded in b around 0

   \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 37.5%

   \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
9. Step-by-step derivation

10. Applied rewrites 40.4%

    \[\leadsto -0.011111111111111112 \cdot \left(a \cdot \left(\left(a \cdot angle\right) \cdot \color{blue}{\pi}\right)\right) \]

11. Final simplification 40.4%

    \[\leadsto -0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right) \]
12. Add Preprocessing

Alternative 20: 38.5% accurate, 21.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ -0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (* -0.011111111111111112 (* a (* a (* PI angle)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	return -0.011111111111111112 * (a * (a * (((double) M_PI) * angle)));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	return -0.011111111111111112 * (a * (a * (Math.PI * angle)));
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	return -0.011111111111111112 * (a * (a * (math.pi * angle)))

Julia

b_m = abs(b)
function code(a, b_m, angle)
	return Float64(-0.011111111111111112 * Float64(a * Float64(a * Float64(pi * angle))))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle)
	tmp = -0.011111111111111112 * (a * (a * (pi * angle)));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := N[(-0.011111111111111112 * N[(a * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

   3. *-commutative N/A

      \[\leadsto angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   10. lower-PI.f64 N/A

       \[\leadsto \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{90}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   11. unpow2 N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

   12. unpow2 N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

   13. difference-of-squares N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   15. lower-+.f64 N/A

       \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

   16. lower--.f64 54.9

       \[\leadsto \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

5. Applied rewrites 54.9%

   \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

6. Taylor expanded in b around 0

   \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 37.5%

   \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
9. Step-by-step derivation

10. Applied rewrites 40.4%

    \[\leadsto -0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right)\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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)))))