ab-angle->ABCF B

Percentage Accurate: 54.0% → 67.7%

Time: 18.7s

Alternatives: 29

Speedup: 10.3×

• 31.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{\sqrt{\pi}}{180}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\frac{\sqrt[3]{\pi} \cdot \sqrt{angle\_m}}{{angle\_m}^{-0.25}}}{180}, \frac{\frac{{\pi}^{0.16666666666666666}}{{angle\_m}^{-0.25}}}{\frac{1}{\sqrt{\pi}}}, -\pi \cdot \left(angle\_m \cdot -0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+192}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(t\_0, \frac{\sqrt{\pi}}{\frac{1}{angle\_m}}, \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+221}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(t\_0, \frac{{\pi}^{0.16666666666666666}}{{angle\_m}^{-0.5}} \cdot \left(\sqrt[3]{\pi} \cdot \left(-\sqrt{angle\_m}\right)\right), \left(-\sqrt{\pi}\right) \cdot \left(\sqrt{\pi} \cdot \left(angle\_m \cdot -0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\pi \cdot angle\_m, 180, 180 \cdot \left(\pi \cdot angle\_m\right)\right)}{32400}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (/ (sqrt PI) 180.0)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+39)
      (*
       (+ b a)
       (*
        (- b a)
        (sin
         (fma
          (/ (/ (* (cbrt PI) (sqrt angle_m)) (pow angle_m -0.25)) 180.0)
          (/
           (/ (pow PI 0.16666666666666666) (pow angle_m -0.25))
           (/ 1.0 (sqrt PI)))
          (- (* PI (* angle_m -0.005555555555555556)))))))
      (if (<= (/ angle_m 180.0) 1e+192)
        (*
         (+ b a)
         (*
          (- b a)
          (sin
           (fma
            t_0
            (/ (sqrt PI) (/ 1.0 angle_m))
            (* PI (* angle_m 0.005555555555555556))))))
        (if (<= (/ angle_m 180.0) 1e+221)
          (*
           (+ b a)
           (*
            (- b a)
            (sin
             (fma
              t_0
              (*
               (/ (pow PI 0.16666666666666666) (pow angle_m -0.5))
               (* (cbrt PI) (- (sqrt angle_m))))
              (*
               (- (sqrt PI))
               (* (sqrt PI) (* angle_m -0.005555555555555556)))))))
          (*
           (+ b a)
           (*
            (- b a)
            (sin
             (/
              (fma (* PI angle_m) 180.0 (* 180.0 (* PI angle_m)))
              32400.0))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sqrt(((double) M_PI)) / 180.0;
	double tmp;
	if ((angle_m / 180.0) <= 4e+39) {
		tmp = (b + a) * ((b - a) * sin(fma((((cbrt(((double) M_PI)) * sqrt(angle_m)) / pow(angle_m, -0.25)) / 180.0), ((pow(((double) M_PI), 0.16666666666666666) / pow(angle_m, -0.25)) / (1.0 / sqrt(((double) M_PI)))), -(((double) M_PI) * (angle_m * -0.005555555555555556)))));
	} else if ((angle_m / 180.0) <= 1e+192) {
		tmp = (b + a) * ((b - a) * sin(fma(t_0, (sqrt(((double) M_PI)) / (1.0 / angle_m)), (((double) M_PI) * (angle_m * 0.005555555555555556)))));
	} else if ((angle_m / 180.0) <= 1e+221) {
		tmp = (b + a) * ((b - a) * sin(fma(t_0, ((pow(((double) M_PI), 0.16666666666666666) / pow(angle_m, -0.5)) * (cbrt(((double) M_PI)) * -sqrt(angle_m))), (-sqrt(((double) M_PI)) * (sqrt(((double) M_PI)) * (angle_m * -0.005555555555555556))))));
	} else {
		tmp = (b + a) * ((b - a) * sin((fma((((double) M_PI) * angle_m), 180.0, (180.0 * (((double) M_PI) * angle_m))) / 32400.0)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(sqrt(pi) / 180.0)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+39)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(fma(Float64(Float64(Float64(cbrt(pi) * sqrt(angle_m)) / (angle_m ^ -0.25)) / 180.0), Float64(Float64((pi ^ 0.16666666666666666) / (angle_m ^ -0.25)) / Float64(1.0 / sqrt(pi))), Float64(-Float64(pi * Float64(angle_m * -0.005555555555555556)))))));
	elseif (Float64(angle_m / 180.0) <= 1e+192)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(fma(t_0, Float64(sqrt(pi) / Float64(1.0 / angle_m)), Float64(pi * Float64(angle_m * 0.005555555555555556))))));
	elseif (Float64(angle_m / 180.0) <= 1e+221)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(fma(t_0, Float64(Float64((pi ^ 0.16666666666666666) / (angle_m ^ -0.5)) * Float64(cbrt(pi) * Float64(-sqrt(angle_m)))), Float64(Float64(-sqrt(pi)) * Float64(sqrt(pi) * Float64(angle_m * -0.005555555555555556)))))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(fma(Float64(pi * angle_m), 180.0, Float64(180.0 * Float64(pi * angle_m))) / 32400.0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[Sqrt[Pi], $MachinePrecision] / 180.0), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+39], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(N[(N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Sqrt[angle$95$m], $MachinePrecision]), $MachinePrecision] / N[Power[angle$95$m, -0.25], $MachinePrecision]), $MachinePrecision] / 180.0), $MachinePrecision] * N[(N[(N[Power[Pi, 0.16666666666666666], $MachinePrecision] / N[Power[angle$95$m, -0.25], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+192], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(t$95$0 * N[(N[Sqrt[Pi], $MachinePrecision] / N[(1.0 / angle$95$m), $MachinePrecision]), $MachinePrecision] + N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+221], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(t$95$0 * N[(N[(N[Power[Pi, 0.16666666666666666], $MachinePrecision] / N[Power[angle$95$m, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * (-N[Sqrt[angle$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] + N[((-N[Sqrt[Pi], $MachinePrecision]) * N[(N[Sqrt[Pi], $MachinePrecision] * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * 180.0 + N[(180.0 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 32400.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999976e39

   1. Initial program 74.6%

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

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

      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 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-out-- N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. add-sqr-sqrt N/A

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

      20. associate-*l* N/A

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

      21. cancel-sign-sub-inv N/A

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

   6. Applied rewrites 91.4%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \frac{\sqrt{\pi}}{\frac{1}{angle}}, \left(-\sqrt{\pi}\right) \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. unpow-prod-down N/A

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

      8. pow2 N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. inv-pow N/A

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

      13. sqr-pow N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 92.0%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \color{blue}{\frac{\sqrt[3]{\pi}}{{angle}^{-0.5}} \cdot \frac{{\pi}^{0.16666666666666666}}{{angle}^{-0.5}}}, \left(-\sqrt{\pi}\right) \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180} \cdot \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{{angle}^{\frac{-1}{2}}} \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}\right) + \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)}\right) \]

   10. Applied rewrites 94.4%

       \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{\frac{\sqrt[3]{\pi} \cdot \sqrt{angle}}{{angle}^{-0.25}}}{180}, \frac{\frac{{\pi}^{0.16666666666666666}}{{angle}^{-0.25}}}{\frac{1}{\sqrt{\pi}}}, \left(angle \cdot -0.005555555555555556\right) \cdot \left(-\pi\right)\right)\right)}\right) \]

   if 3.99999999999999976e39 < (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000004e192

   1. Initial program 25.4%

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

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

      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 24.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-out N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. associate-*l/ N/A

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

      20. lift-PI.f64 N/A

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

      21. *-un-lft-identity N/A

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

      22. add-sqr-sqrt N/A

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

      23. div-inv N/A

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

      24. times-frac N/A

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

   6. Applied rewrites 41.5%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \frac{\sqrt{\pi}}{\frac{1}{angle}}, \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right) \]

   if 1.00000000000000004e192 < (/.f64 angle #s(literal 180 binary64)) < 1e221

   1. Initial program 28.4%

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

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

      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 41.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-out-- N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. add-sqr-sqrt N/A

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

      20. associate-*l* N/A

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

      21. cancel-sign-sub-inv N/A

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

   6. Applied rewrites 29.6%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \frac{\sqrt{\pi}}{\frac{1}{angle}}, \left(-\sqrt{\pi}\right) \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. unpow-prod-down N/A

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

      8. pow2 N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. inv-pow N/A

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

      13. sqr-pow N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 36.3%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \color{blue}{\frac{\sqrt[3]{\pi}}{{angle}^{-0.5}} \cdot \frac{{\pi}^{0.16666666666666666}}{{angle}^{-0.5}}}, \left(-\sqrt{\pi}\right) \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)\right) \]

   9. Taylor expanded in angle around -inf

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \color{blue}{\left(-1 \cdot \left(\sqrt{angle} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \color{blue}{\left(\mathsf{neg}\left(\sqrt{angle} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

       2. lower-neg.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \color{blue}{\left(\mathsf{neg}\left(\sqrt{angle} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \left(\mathsf{neg}\left(\color{blue}{\sqrt{angle} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}\right)\right) \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \left(\mathsf{neg}\left(\color{blue}{\sqrt{angle}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

       5. lower-cbrt.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \left(\mathsf{neg}\left(\sqrt{angle} \cdot \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}\right)\right) \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

       6. lower-PI.f64 68.1

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \left(-\sqrt{angle} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \frac{{\pi}^{0.16666666666666666}}{{angle}^{-0.5}}, \left(-\sqrt{\pi}\right) \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)\right) \]

   11. Applied rewrites 68.1%

       \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \color{blue}{\left(-\sqrt{angle} \cdot \sqrt[3]{\pi}\right)} \cdot \frac{{\pi}^{0.16666666666666666}}{{angle}^{-0.5}}, \left(-\sqrt{\pi}\right) \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)\right) \]

   if 1e221 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 35.2%

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

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

      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 44.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-out N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. frac-add N/A

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

      12. lower-/.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. metadata-eval 49.3

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\pi \cdot angle, 180, 180 \cdot \left(\pi \cdot angle\right)\right)}{\color{blue}{32400}}\right)\right) \]

   6. Applied rewrites 49.3%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\frac{\mathsf{fma}\left(\pi \cdot angle, 180, 180 \cdot \left(\pi \cdot angle\right)\right)}{32400}\right)}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\frac{\sqrt[3]{\pi} \cdot \sqrt{angle}}{{angle}^{-0.25}}}{180}, \frac{\frac{{\pi}^{0.16666666666666666}}{{angle}^{-0.25}}}{\frac{1}{\sqrt{\pi}}}, -\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+192}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \frac{\sqrt{\pi}}{\frac{1}{angle}}, \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+221}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \frac{{\pi}^{0.16666666666666666}}{{angle}^{-0.5}} \cdot \left(\sqrt[3]{\pi} \cdot \left(-\sqrt{angle}\right)\right), \left(-\sqrt{\pi}\right) \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\pi \cdot angle, 180, 180 \cdot \left(\pi \cdot angle\right)\right)}{32400}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{+205}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \left(\frac{-1}{{angle\_m}^{-0.25}} \cdot \frac{-\sqrt[3]{\pi}}{{angle\_m}^{-0.25}}\right) \cdot \frac{{\pi}^{0.16666666666666666}}{{angle\_m}^{-0.5}}, \left(-\sqrt{\pi}\right) \cdot \left(\sqrt{\pi} \cdot \left(angle\_m \cdot -0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 7.8e+205)
    (*
     (+ b a)
     (*
      (- b a)
      (sin
       (fma
        (/ (sqrt PI) 180.0)
        (*
         (* (/ -1.0 (pow angle_m -0.25)) (/ (- (cbrt PI)) (pow angle_m -0.25)))
         (/ (pow PI 0.16666666666666666) (pow angle_m -0.5)))
        (* (- (sqrt PI)) (* (sqrt PI) (* angle_m -0.005555555555555556)))))))
    (* (+ b a) (* (- b a) (sin (* angle_m (* PI 0.011111111111111112))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 7.8e+205) {
		tmp = (b + a) * ((b - a) * sin(fma((sqrt(((double) M_PI)) / 180.0), (((-1.0 / pow(angle_m, -0.25)) * (-cbrt(((double) M_PI)) / pow(angle_m, -0.25))) * (pow(((double) M_PI), 0.16666666666666666) / pow(angle_m, -0.5))), (-sqrt(((double) M_PI)) * (sqrt(((double) M_PI)) * (angle_m * -0.005555555555555556))))));
	} else {
		tmp = (b + a) * ((b - a) * sin((angle_m * (((double) M_PI) * 0.011111111111111112))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 7.8e+205)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(fma(Float64(sqrt(pi) / 180.0), Float64(Float64(Float64(-1.0 / (angle_m ^ -0.25)) * Float64(Float64(-cbrt(pi)) / (angle_m ^ -0.25))) * Float64((pi ^ 0.16666666666666666) / (angle_m ^ -0.5))), Float64(Float64(-sqrt(pi)) * Float64(sqrt(pi) * Float64(angle_m * -0.005555555555555556)))))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(angle_m * Float64(pi * 0.011111111111111112)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 7.8e+205], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(N[Sqrt[Pi], $MachinePrecision] / 180.0), $MachinePrecision] * N[(N[(N[(-1.0 / N[Power[angle$95$m, -0.25], $MachinePrecision]), $MachinePrecision] * N[((-N[Power[Pi, 1/3], $MachinePrecision]) / N[Power[angle$95$m, -0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[Pi, 0.16666666666666666], $MachinePrecision] / N[Power[angle$95$m, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-N[Sqrt[Pi], $MachinePrecision]) * N[(N[Sqrt[Pi], $MachinePrecision] * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 7.7999999999999997e205

   1. Initial program 55.0%

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

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

      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 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-out-- N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. add-sqr-sqrt N/A

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

      20. associate-*l* N/A

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

      21. cancel-sign-sub-inv N/A

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

   6. Applied rewrites 66.8%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \frac{\sqrt{\pi}}{\frac{1}{angle}}, \left(-\sqrt{\pi}\right) \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. unpow-prod-down N/A

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

      8. pow2 N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. inv-pow N/A

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

      13. sqr-pow N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 66.7%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \color{blue}{\frac{\sqrt[3]{\pi}}{{angle}^{-0.5}} \cdot \frac{{\pi}^{0.16666666666666666}}{{angle}^{-0.5}}}, \left(-\sqrt{\pi}\right) \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \color{blue}{\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{{angle}^{\frac{-1}{2}}}} \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \color{blue}{\frac{\mathsf{neg}\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}{\mathsf{neg}\left({angle}^{\frac{-1}{2}}\right)}} \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

      3. neg-mul-1 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \frac{\color{blue}{-1 \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\mathsf{neg}\left({angle}^{\frac{-1}{2}}\right)} \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \frac{-1 \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(\color{blue}{{angle}^{\frac{-1}{2}}}\right)} \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

      5. sqr-pow N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \frac{-1 \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(\color{blue}{{angle}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {angle}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right)} \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \frac{-1 \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}{\color{blue}{{angle}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \left(\mathsf{neg}\left({angle}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)\right)}} \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

      7. times-frac N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \color{blue}{\left(\frac{-1}{{angle}^{\left(\frac{\frac{-1}{2}}{2}\right)}} \cdot \frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left({angle}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}\right)} \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \color{blue}{\left(\frac{-1}{{angle}^{\left(\frac{\frac{-1}{2}}{2}\right)}} \cdot \frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left({angle}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}\right)} \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \left(\color{blue}{\frac{-1}{{angle}^{\left(\frac{\frac{-1}{2}}{2}\right)}}} \cdot \frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left({angle}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}\right) \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \left(\frac{-1}{\color{blue}{{angle}^{\left(\frac{\frac{-1}{2}}{2}\right)}}} \cdot \frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left({angle}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}\right) \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \left(\frac{-1}{{angle}^{\color{blue}{\frac{-1}{4}}}} \cdot \frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left({angle}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}\right) \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \left(\frac{-1}{{angle}^{\frac{-1}{4}}} \cdot \color{blue}{\frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left({angle}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}}\right) \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \left(\frac{-1}{{angle}^{\frac{-1}{4}}} \cdot \frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{\color{blue}{\mathsf{neg}\left({angle}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}}\right) \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

      14. lower-pow.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}, \left(\frac{-1}{{angle}^{\frac{-1}{4}}} \cdot \frac{\sqrt[3]{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(\color{blue}{{angle}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right)}\right) \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}{{angle}^{\frac{-1}{2}}}, \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \frac{-1}{180}\right)\right)\right)\right)\right) \]

      15. metadata-eval 66.7

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \left(\frac{-1}{{angle}^{-0.25}} \cdot \frac{\sqrt[3]{\pi}}{-{angle}^{\color{blue}{-0.25}}}\right) \cdot \frac{{\pi}^{0.16666666666666666}}{{angle}^{-0.5}}, \left(-\sqrt{\pi}\right) \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)\right) \]

   10. Applied rewrites 66.7%

       \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \color{blue}{\left(\frac{-1}{{angle}^{-0.25}} \cdot \frac{\sqrt[3]{\pi}}{-{angle}^{-0.25}}\right)} \cdot \frac{{\pi}^{0.16666666666666666}}{{angle}^{-0.5}}, \left(-\sqrt{\pi}\right) \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)\right) \]

   if 7.7999999999999997e205 < b

   1. Initial program 43.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 78.4

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

   6. Applied rewrites 78.4%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{+205}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\sqrt{\pi}}{180}, \left(\frac{-1}{{angle}^{-0.25}} \cdot \frac{-\sqrt[3]{\pi}}{{angle}^{-0.25}}\right) \cdot \frac{{\pi}^{0.16666666666666666}}{{angle}^{-0.5}}, \left(-\sqrt{\pi}\right) \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(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)))))