b from scale-rotated-ellipse

Percentage Accurate: 0.0% → 29.4%

Time: 51.9s

Alternatives: 6

Speedup: 9.6×

• could not determine a ground truth

  1. a = -1.5122580878142512e+229
  2. b = -6.079667380915416e-289
  3. angle = 2.6940957356263927e-114
  4. x-scale = 9.383711592115368e+295
  5. y-scale = 4.803960450026458e-305

Specification

Math

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) - \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (-
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) - sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Initial Program: 0.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) - \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (-
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) - sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Alternative 1: 29.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\left(0.5 + 0.5\right) \cdot \left|a\right|, \left|a\right|, \left(\left(0.5 - 0.5\right) \cdot \left|b\right|\right) \cdot \left|b\right|\right)\\ t_1 := \left|a\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 1.25 \cdot 10^{+62}:\\ \;\;\;\;\frac{0.25}{\left|b\right|} \cdot \frac{\left(\left(\frac{\sqrt{\left(8 \cdot \left(0.5 - \mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, \sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)\right)\right) \cdot {\left(\left|b\right|\right)}^{4}}}{\left|y-scale\right|} \cdot y-scale\right) \cdot y-scale\right) \cdot \left|a\right|}{\left|b\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(0.25 \cdot \frac{\sqrt{\left(\left(t\_0 - \left|t\_0\right|\right) \cdot {t\_1}^{4}\right) \cdot 8}}{\left|x-scale\right|}\right) \cdot \left(x-scale \cdot x-scale\right)}{t\_1}}{t\_1}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (fma
          (* (+ 0.5 0.5) (fabs a))
          (fabs a)
          (* (* (- 0.5 0.5) (fabs b)) (fabs b))))
        (t_1 (* (fabs a) (fabs b))))
   (if (<= (fabs b) 1.25e+62)
     (*
      (/ 0.25 (fabs b))
      (/
       (*
        (*
         (*
          (/
           (sqrt
            (*
             (*
              8.0
              (-
               0.5
               (fma
                (cos (* (* PI angle) 0.011111111111111112))
                0.5
                (sqrt (pow (sin (* (* PI angle) 0.005555555555555556)) 4.0)))))
             (pow (fabs b) 4.0)))
           (fabs y-scale))
          y-scale)
         y-scale)
        (fabs a))
       (fabs b)))
     (/
      (/
       (*
        (*
         0.25
         (/
          (sqrt (* (* (- t_0 (fabs t_0)) (pow t_1 4.0)) 8.0))
          (fabs x-scale)))
        (* x-scale x-scale))
       t_1)
      t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fma(((0.5 + 0.5) * fabs(a)), fabs(a), (((0.5 - 0.5) * fabs(b)) * fabs(b)));
	double t_1 = fabs(a) * fabs(b);
	double tmp;
	if (fabs(b) <= 1.25e+62) {
		tmp = (0.25 / fabs(b)) * (((((sqrt(((8.0 * (0.5 - fma(cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5, sqrt(pow(sin(((((double) M_PI) * angle) * 0.005555555555555556)), 4.0))))) * pow(fabs(b), 4.0))) / fabs(y_45_scale)) * y_45_scale) * y_45_scale) * fabs(a)) / fabs(b));
	} else {
		tmp = (((0.25 * (sqrt((((t_0 - fabs(t_0)) * pow(t_1, 4.0)) * 8.0)) / fabs(x_45_scale))) * (x_45_scale * x_45_scale)) / t_1) / t_1;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = fma(Float64(Float64(0.5 + 0.5) * abs(a)), abs(a), Float64(Float64(Float64(0.5 - 0.5) * abs(b)) * abs(b)))
	t_1 = Float64(abs(a) * abs(b))
	tmp = 0.0
	if (abs(b) <= 1.25e+62)
		tmp = Float64(Float64(0.25 / abs(b)) * Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(8.0 * Float64(0.5 - fma(cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5, sqrt((sin(Float64(Float64(pi * angle) * 0.005555555555555556)) ^ 4.0))))) * (abs(b) ^ 4.0))) / abs(y_45_scale)) * y_45_scale) * y_45_scale) * abs(a)) / abs(b)));
	else
		tmp = Float64(Float64(Float64(Float64(0.25 * Float64(sqrt(Float64(Float64(Float64(t_0 - abs(t_0)) * (t_1 ^ 4.0)) * 8.0)) / abs(x_45_scale))) * Float64(x_45_scale * x_45_scale)) / t_1) / t_1);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(0.5 + 0.5), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision] + N[(N[(N[(0.5 - 0.5), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[a], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.25e+62], N[(N[(0.25 / N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[Sqrt[N[(N[(8.0 * N[(0.5 - N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * 0.5 + N[Sqrt[N[Power[N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.25 * N[(N[Sqrt[N[(N[(N[(t$95$0 - N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, 4.0], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.25000000000000007e62

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.5%

      \[\leadsto \color{blue}{0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]

   7. Applied rewrites 3.9%

      \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]

   9. Applied rewrites 20.0%

      \[\leadsto \frac{0.25}{b} \cdot \color{blue}{\frac{\left(\frac{\sqrt{8 \cdot \left(\left(\left(0.5 - \cos \left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 2\right) \cdot 0.5\right) - \sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {b}^{4}\right)}}{\left|y-scale\right|} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot a}{b}} \]

   11. Applied rewrites 24.7%

       \[\leadsto \frac{0.25}{b} \cdot \frac{\left(\left(\frac{\sqrt{\left(8 \cdot \left(0.5 - \mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, \sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)\right)\right) \cdot {b}^{4}}}{\left|y-scale\right|} \cdot y-scale\right) \cdot y-scale\right) \cdot a}{b} \]

   if 1.25000000000000007e62 < b

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) - \sqrt{{\left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}^{2}}\right)\right)}{{x-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   4. Applied rewrites 0.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) - \sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}}\right)\right)}{{x-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   6. Applied rewrites 2.4%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right), a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 2.2%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right), a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 1.9%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   13. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   14. Step-by-step derivation

   15. Applied rewrites 2.1%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   16. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   17. Step-by-step derivation

   18. Applied rewrites 2.4%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   20. Applied rewrites 15.9%

       \[\leadsto \frac{\frac{\left(0.25 \cdot \frac{\sqrt{\left(\left(\mathsf{fma}\left(\left(0.5 + 0.5\right) \cdot a, a, \left(\left(0.5 - 0.5\right) \cdot b\right) \cdot b\right) - \left|\mathsf{fma}\left(\left(0.5 + 0.5\right) \cdot a, a, \left(\left(0.5 - 0.5\right) \cdot b\right) \cdot b\right)\right|\right) \cdot {\left(a \cdot b\right)}^{4}\right) \cdot 8}}{\left|x-scale\right|}\right) \cdot \left(x-scale \cdot x-scale\right)}{a \cdot b}}{\color{blue}{a \cdot b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 26.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} t_0 := a \cdot \left|b\right|\\ t_1 := \mathsf{fma}\left(\left(0.5 + 0.5\right) \cdot a, a, \left(\left(0.5 - 0.5\right) \cdot \left|b\right|\right) \cdot \left|b\right|\right)\\ \mathbf{if}\;\left|b\right| \leq 6.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{0.25}{\left|b\right|} \cdot \frac{\left(\left(\frac{\sqrt{8 \cdot \left({\left(\left|b\right|\right)}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5} - \sqrt{{\pi}^{4} \cdot 9.525986892242036 \cdot 10^{-10}}\right)\right)}}{\left|y-scale\right|} \cdot angle\right) \cdot \left(y-scale \cdot y-scale\right)\right) \cdot a}{\left|b\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(0.25 \cdot \frac{\sqrt{\left(\left(t\_1 - \left|t\_1\right|\right) \cdot {t\_0}^{4}\right) \cdot 8}}{\left|x-scale\right|}\right) \cdot \left(x-scale \cdot x-scale\right)}{t\_0}}{t\_0}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (fabs b)))
        (t_1 (fma (* (+ 0.5 0.5) a) a (* (* (- 0.5 0.5) (fabs b)) (fabs b)))))
   (if (<= (fabs b) 6.5e+54)
     (*
      (/ 0.25 (fabs b))
      (/
       (*
        (*
         (*
          (/
           (sqrt
            (*
             8.0
             (*
              (pow (fabs b) 4.0)
              (-
               (* (* PI PI) 3.08641975308642e-5)
               (sqrt (* (pow PI 4.0) 9.525986892242036e-10))))))
           (fabs y-scale))
          angle)
         (* y-scale y-scale))
        a)
       (fabs b)))
     (/
      (/
       (*
        (*
         0.25
         (/
          (sqrt (* (* (- t_1 (fabs t_1)) (pow t_0 4.0)) 8.0))
          (fabs x-scale)))
        (* x-scale x-scale))
       t_0)
      t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * fabs(b);
	double t_1 = fma(((0.5 + 0.5) * a), a, (((0.5 - 0.5) * fabs(b)) * fabs(b)));
	double tmp;
	if (fabs(b) <= 6.5e+54) {
		tmp = (0.25 / fabs(b)) * (((((sqrt((8.0 * (pow(fabs(b), 4.0) * (((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5) - sqrt((pow(((double) M_PI), 4.0) * 9.525986892242036e-10)))))) / fabs(y_45_scale)) * angle) * (y_45_scale * y_45_scale)) * a) / fabs(b));
	} else {
		tmp = (((0.25 * (sqrt((((t_1 - fabs(t_1)) * pow(t_0, 4.0)) * 8.0)) / fabs(x_45_scale))) * (x_45_scale * x_45_scale)) / t_0) / t_0;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * abs(b))
	t_1 = fma(Float64(Float64(0.5 + 0.5) * a), a, Float64(Float64(Float64(0.5 - 0.5) * abs(b)) * abs(b)))
	tmp = 0.0
	if (abs(b) <= 6.5e+54)
		tmp = Float64(Float64(0.25 / abs(b)) * Float64(Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((abs(b) ^ 4.0) * Float64(Float64(Float64(pi * pi) * 3.08641975308642e-5) - sqrt(Float64((pi ^ 4.0) * 9.525986892242036e-10)))))) / abs(y_45_scale)) * angle) * Float64(y_45_scale * y_45_scale)) * a) / abs(b)));
	else
		tmp = Float64(Float64(Float64(Float64(0.25 * Float64(sqrt(Float64(Float64(Float64(t_1 - abs(t_1)) * (t_0 ^ 4.0)) * 8.0)) / abs(x_45_scale))) * Float64(x_45_scale * x_45_scale)) / t_0) / t_0);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 + 0.5), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(0.5 - 0.5), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 6.5e+54], N[(N[(0.25 / N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] - N[Sqrt[N[(N[Power[Pi, 4.0], $MachinePrecision] * 9.525986892242036e-10), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.25 * N[(N[Sqrt[N[(N[(N[(t$95$1 - N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 6.5e54

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.5%

      \[\leadsto \color{blue}{0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]

   7. Applied rewrites 3.9%

      \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

   10. Applied rewrites 4.0%

       \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

   12. Applied rewrites 21.3%

       \[\leadsto \frac{0.25}{b} \cdot \color{blue}{\frac{\left(\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5} - \sqrt{{\pi}^{4} \cdot 9.525986892242036 \cdot 10^{-10}}\right)\right)}}{\left|y-scale\right|} \cdot angle\right) \cdot \left(y-scale \cdot y-scale\right)\right) \cdot a}{b}} \]

   if 6.5e54 < b

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) - \sqrt{{\left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}^{2}}\right)\right)}{{x-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   4. Applied rewrites 0.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) - \sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}}\right)\right)}{{x-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   6. Applied rewrites 2.4%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right), a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 2.2%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right), a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 1.9%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   13. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   14. Step-by-step derivation

   15. Applied rewrites 2.1%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   16. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   17. Step-by-step derivation

   18. Applied rewrites 2.4%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   20. Applied rewrites 15.9%

       \[\leadsto \frac{\frac{\left(0.25 \cdot \frac{\sqrt{\left(\left(\mathsf{fma}\left(\left(0.5 + 0.5\right) \cdot a, a, \left(\left(0.5 - 0.5\right) \cdot b\right) \cdot b\right) - \left|\mathsf{fma}\left(\left(0.5 + 0.5\right) \cdot a, a, \left(\left(0.5 - 0.5\right) \cdot b\right) \cdot b\right)\right|\right) \cdot {\left(a \cdot b\right)}^{4}\right) \cdot 8}}{\left|x-scale\right|}\right) \cdot \left(x-scale \cdot x-scale\right)}{a \cdot b}}{\color{blue}{a \cdot b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 26.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} t_0 := a \cdot \left|b\right|\\ t_1 := \mathsf{fma}\left(\left(0.5 + 0.5\right) \cdot a, a, \left(\left(0.5 - 0.5\right) \cdot \left|b\right|\right) \cdot \left|b\right|\right)\\ \mathbf{if}\;\left|b\right| \leq 5.1 \cdot 10^{+73}:\\ \;\;\;\;\frac{0.25}{\left|b\right|} \cdot \frac{\left(\left(\frac{\sqrt{8 \cdot \left({\left(\left|b\right|\right)}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5} - \sqrt{{\pi}^{4} \cdot 9.525986892242036 \cdot 10^{-10}}\right)\right)}}{\left|y-scale\right|} \cdot angle\right) \cdot \left(y-scale \cdot y-scale\right)\right) \cdot a}{\left|b\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \left(\left(\frac{\sqrt{\left(\left(t\_1 - \left|t\_1\right|\right) \cdot {t\_0}^{4}\right) \cdot 8}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale\right)}{t\_0 \cdot t\_0}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (fabs b)))
        (t_1 (fma (* (+ 0.5 0.5) a) a (* (* (- 0.5 0.5) (fabs b)) (fabs b)))))
   (if (<= (fabs b) 5.1e+73)
     (*
      (/ 0.25 (fabs b))
      (/
       (*
        (*
         (*
          (/
           (sqrt
            (*
             8.0
             (*
              (pow (fabs b) 4.0)
              (-
               (* (* PI PI) 3.08641975308642e-5)
               (sqrt (* (pow PI 4.0) 9.525986892242036e-10))))))
           (fabs y-scale))
          angle)
         (* y-scale y-scale))
        a)
       (fabs b)))
     (/
      (*
       0.25
       (*
        (*
         (/ (sqrt (* (* (- t_1 (fabs t_1)) (pow t_0 4.0)) 8.0)) (fabs x-scale))
         x-scale)
        x-scale))
      (* t_0 t_0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * fabs(b);
	double t_1 = fma(((0.5 + 0.5) * a), a, (((0.5 - 0.5) * fabs(b)) * fabs(b)));
	double tmp;
	if (fabs(b) <= 5.1e+73) {
		tmp = (0.25 / fabs(b)) * (((((sqrt((8.0 * (pow(fabs(b), 4.0) * (((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5) - sqrt((pow(((double) M_PI), 4.0) * 9.525986892242036e-10)))))) / fabs(y_45_scale)) * angle) * (y_45_scale * y_45_scale)) * a) / fabs(b));
	} else {
		tmp = (0.25 * (((sqrt((((t_1 - fabs(t_1)) * pow(t_0, 4.0)) * 8.0)) / fabs(x_45_scale)) * x_45_scale) * x_45_scale)) / (t_0 * t_0);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * abs(b))
	t_1 = fma(Float64(Float64(0.5 + 0.5) * a), a, Float64(Float64(Float64(0.5 - 0.5) * abs(b)) * abs(b)))
	tmp = 0.0
	if (abs(b) <= 5.1e+73)
		tmp = Float64(Float64(0.25 / abs(b)) * Float64(Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((abs(b) ^ 4.0) * Float64(Float64(Float64(pi * pi) * 3.08641975308642e-5) - sqrt(Float64((pi ^ 4.0) * 9.525986892242036e-10)))))) / abs(y_45_scale)) * angle) * Float64(y_45_scale * y_45_scale)) * a) / abs(b)));
	else
		tmp = Float64(Float64(0.25 * Float64(Float64(Float64(sqrt(Float64(Float64(Float64(t_1 - abs(t_1)) * (t_0 ^ 4.0)) * 8.0)) / abs(x_45_scale)) * x_45_scale) * x_45_scale)) / Float64(t_0 * t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 + 0.5), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(0.5 - 0.5), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 5.1e+73], N[(N[(0.25 / N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] - N[Sqrt[N[(N[Power[Pi, 4.0], $MachinePrecision] * 9.525986892242036e-10), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(N[(N[(N[Sqrt[N[(N[(N[(t$95$1 - N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.10000000000000024e73

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.5%

      \[\leadsto \color{blue}{0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]

   7. Applied rewrites 3.9%

      \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

   10. Applied rewrites 4.0%

       \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

   12. Applied rewrites 21.3%

       \[\leadsto \frac{0.25}{b} \cdot \color{blue}{\frac{\left(\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5} - \sqrt{{\pi}^{4} \cdot 9.525986892242036 \cdot 10^{-10}}\right)\right)}}{\left|y-scale\right|} \cdot angle\right) \cdot \left(y-scale \cdot y-scale\right)\right) \cdot a}{b}} \]

   if 5.10000000000000024e73 < b

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) - \sqrt{{\left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}^{2}}\right)\right)}{{x-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   4. Applied rewrites 0.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) - \sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}}\right)\right)}{{x-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   6. Applied rewrites 2.4%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right), a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 2.2%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right), a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 1.9%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   13. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   14. Step-by-step derivation

   15. Applied rewrites 2.1%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   16. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   17. Step-by-step derivation

   18. Applied rewrites 2.4%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   19. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot \color{blue}{b}\right) \cdot \left(a \cdot b\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   20. Applied rewrites 10.0%

       \[\leadsto \frac{0.25 \cdot \left(\left(\frac{\sqrt{\left(\left(\mathsf{fma}\left(\left(0.5 + 0.5\right) \cdot a, a, \left(\left(0.5 - 0.5\right) \cdot b\right) \cdot b\right) - \left|\mathsf{fma}\left(\left(0.5 + 0.5\right) \cdot a, a, \left(\left(0.5 - 0.5\right) \cdot b\right) \cdot b\right)\right|\right) \cdot {\left(a \cdot b\right)}^{4}\right) \cdot 8}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale\right)}{\left(a \cdot \color{blue}{b}\right) \cdot \left(a \cdot b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 25.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} t_0 := a \cdot \left|b\right|\\ t_1 := \mathsf{fma}\left(\left(0.5 + 0.5\right) \cdot a, a, \left(\left(0.5 - 0.5\right) \cdot \left|b\right|\right) \cdot \left|b\right|\right)\\ \mathbf{if}\;\left|b\right| \leq 5.1 \cdot 10^{+73}:\\ \;\;\;\;\frac{0.25}{\left|b\right|} \cdot \frac{\left(\left(\frac{\sqrt{8 \cdot \left({\left(\left|b\right|\right)}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5} - \sqrt{{\pi}^{4} \cdot 9.525986892242036 \cdot 10^{-10}}\right)\right)}}{\left|y-scale\right|} \cdot angle\right) \cdot \left(y-scale \cdot y-scale\right)\right) \cdot a}{\left|b\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{\left(\left(t\_1 - \left|t\_1\right|\right) \cdot {t\_0}^{4}\right) \cdot 8}}{\left|x-scale\right|} \cdot \left(x-scale \cdot x-scale\right)\right) \cdot 0.25}{t\_0 \cdot t\_0}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (fabs b)))
        (t_1 (fma (* (+ 0.5 0.5) a) a (* (* (- 0.5 0.5) (fabs b)) (fabs b)))))
   (if (<= (fabs b) 5.1e+73)
     (*
      (/ 0.25 (fabs b))
      (/
       (*
        (*
         (*
          (/
           (sqrt
            (*
             8.0
             (*
              (pow (fabs b) 4.0)
              (-
               (* (* PI PI) 3.08641975308642e-5)
               (sqrt (* (pow PI 4.0) 9.525986892242036e-10))))))
           (fabs y-scale))
          angle)
         (* y-scale y-scale))
        a)
       (fabs b)))
     (/
      (*
       (*
        (/ (sqrt (* (* (- t_1 (fabs t_1)) (pow t_0 4.0)) 8.0)) (fabs x-scale))
        (* x-scale x-scale))
       0.25)
      (* t_0 t_0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * fabs(b);
	double t_1 = fma(((0.5 + 0.5) * a), a, (((0.5 - 0.5) * fabs(b)) * fabs(b)));
	double tmp;
	if (fabs(b) <= 5.1e+73) {
		tmp = (0.25 / fabs(b)) * (((((sqrt((8.0 * (pow(fabs(b), 4.0) * (((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5) - sqrt((pow(((double) M_PI), 4.0) * 9.525986892242036e-10)))))) / fabs(y_45_scale)) * angle) * (y_45_scale * y_45_scale)) * a) / fabs(b));
	} else {
		tmp = (((sqrt((((t_1 - fabs(t_1)) * pow(t_0, 4.0)) * 8.0)) / fabs(x_45_scale)) * (x_45_scale * x_45_scale)) * 0.25) / (t_0 * t_0);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * abs(b))
	t_1 = fma(Float64(Float64(0.5 + 0.5) * a), a, Float64(Float64(Float64(0.5 - 0.5) * abs(b)) * abs(b)))
	tmp = 0.0
	if (abs(b) <= 5.1e+73)
		tmp = Float64(Float64(0.25 / abs(b)) * Float64(Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((abs(b) ^ 4.0) * Float64(Float64(Float64(pi * pi) * 3.08641975308642e-5) - sqrt(Float64((pi ^ 4.0) * 9.525986892242036e-10)))))) / abs(y_45_scale)) * angle) * Float64(y_45_scale * y_45_scale)) * a) / abs(b)));
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(t_1 - abs(t_1)) * (t_0 ^ 4.0)) * 8.0)) / abs(x_45_scale)) * Float64(x_45_scale * x_45_scale)) * 0.25) / Float64(t_0 * t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 + 0.5), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(0.5 - 0.5), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 5.1e+73], N[(N[(0.25 / N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] - N[Sqrt[N[(N[Power[Pi, 4.0], $MachinePrecision] * 9.525986892242036e-10), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[N[(N[(N[(t$95$1 - N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.10000000000000024e73

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.5%

      \[\leadsto \color{blue}{0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]

   7. Applied rewrites 3.9%

      \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

   10. Applied rewrites 4.0%

       \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

   12. Applied rewrites 21.3%

       \[\leadsto \frac{0.25}{b} \cdot \color{blue}{\frac{\left(\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5} - \sqrt{{\pi}^{4} \cdot 9.525986892242036 \cdot 10^{-10}}\right)\right)}}{\left|y-scale\right|} \cdot angle\right) \cdot \left(y-scale \cdot y-scale\right)\right) \cdot a}{b}} \]

   if 5.10000000000000024e73 < b

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) - \sqrt{{\left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}^{2}}\right)\right)}{{x-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   4. Applied rewrites 0.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) - \sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}}\right)\right)}{{x-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   6. Applied rewrites 2.4%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right), a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 2.2%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right), a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 1.9%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   13. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   14. Step-by-step derivation

   15. Applied rewrites 2.1%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]

   16. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   17. Step-by-step derivation

   18. Applied rewrites 2.4%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(0.5 + 0.5, a \cdot a, \left(0.5 - 0.5\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \]
   19. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot b\right)} \cdot \left(a \cdot b\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right) - \left|\mathsf{fma}\left(\frac{1}{2} + \frac{1}{2}, a \cdot a, \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)\right|\right)}{x-scale \cdot x-scale} \cdot 8} \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{1}{4}}{\color{blue}{\left(a \cdot b\right)} \cdot \left(a \cdot b\right)} \]

   20. Applied rewrites 8.5%

       \[\leadsto \frac{\left(\frac{\sqrt{\left(\left(\mathsf{fma}\left(\left(0.5 + 0.5\right) \cdot a, a, \left(\left(0.5 - 0.5\right) \cdot b\right) \cdot b\right) - \left|\mathsf{fma}\left(\left(0.5 + 0.5\right) \cdot a, a, \left(\left(0.5 - 0.5\right) \cdot b\right) \cdot b\right)\right|\right) \cdot {\left(a \cdot b\right)}^{4}\right) \cdot 8}}{\left|x-scale\right|} \cdot \left(x-scale \cdot x-scale\right)\right) \cdot 0.25}{\color{blue}{\left(a \cdot b\right)} \cdot \left(a \cdot b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 21.3% accurate, 9.5× speedup

Math

\[\frac{0.25}{b} \cdot \frac{\left(\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5} - \sqrt{{\pi}^{4} \cdot 9.525986892242036 \cdot 10^{-10}}\right)\right)}}{\left|y-scale\right|} \cdot angle\right) \cdot \left(y-scale \cdot y-scale\right)\right) \cdot a}{b} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  (/ 0.25 b)
  (/
   (*
    (*
     (*
      (/
       (sqrt
        (*
         8.0
         (*
          (pow b 4.0)
          (-
           (* (* PI PI) 3.08641975308642e-5)
           (sqrt (* (pow PI 4.0) 9.525986892242036e-10))))))
       (fabs y-scale))
      angle)
     (* y-scale y-scale))
    a)
   b)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 / b) * (((((sqrt((8.0 * (pow(b, 4.0) * (((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5) - sqrt((pow(((double) M_PI), 4.0) * 9.525986892242036e-10)))))) / fabs(y_45_scale)) * angle) * (y_45_scale * y_45_scale)) * a) / b);
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 / b) * (((((Math.sqrt((8.0 * (Math.pow(b, 4.0) * (((Math.PI * Math.PI) * 3.08641975308642e-5) - Math.sqrt((Math.pow(Math.PI, 4.0) * 9.525986892242036e-10)))))) / Math.abs(y_45_scale)) * angle) * (y_45_scale * y_45_scale)) * a) / b);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (0.25 / b) * (((((math.sqrt((8.0 * (math.pow(b, 4.0) * (((math.pi * math.pi) * 3.08641975308642e-5) - math.sqrt((math.pow(math.pi, 4.0) * 9.525986892242036e-10)))))) / math.fabs(y_45_scale)) * angle) * (y_45_scale * y_45_scale)) * a) / b)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(0.25 / b) * Float64(Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((b ^ 4.0) * Float64(Float64(Float64(pi * pi) * 3.08641975308642e-5) - sqrt(Float64((pi ^ 4.0) * 9.525986892242036e-10)))))) / abs(y_45_scale)) * angle) * Float64(y_45_scale * y_45_scale)) * a) / b))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (0.25 / b) * (((((sqrt((8.0 * ((b ^ 4.0) * (((pi * pi) * 3.08641975308642e-5) - sqrt(((pi ^ 4.0) * 9.525986892242036e-10)))))) / abs(y_45_scale)) * angle) * (y_45_scale * y_45_scale)) * a) / b);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(0.25 / b), $MachinePrecision] * N[(N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[b, 4.0], $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] - N[Sqrt[N[(N[Power[Pi, 4.0], $MachinePrecision] * 9.525986892242036e-10), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0%

   \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

2. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

4. Applied rewrites 0.5%

   \[\leadsto \color{blue}{0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

5. Taylor expanded in x-scale around 0

   \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]

7. Applied rewrites 3.9%

   \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]

8. Taylor expanded in angle around 0

   \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

10. Applied rewrites 4.0%

    \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

12. Applied rewrites 21.3%

    \[\leadsto \frac{0.25}{b} \cdot \color{blue}{\frac{\left(\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5} - \sqrt{{\pi}^{4} \cdot 9.525986892242036 \cdot 10^{-10}}\right)\right)}}{\left|y-scale\right|} \cdot angle\right) \cdot \left(y-scale \cdot y-scale\right)\right) \cdot a}{b}} \]
13. Add Preprocessing

Alternative 6: 9.3% accurate, 9.6× speedup

Math

\[\frac{0.25 \cdot \left(\left(\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5} - \sqrt{{\pi}^{4} \cdot 9.525986892242036 \cdot 10^{-10}}\right)\right)}}{\left|y-scale\right|} \cdot angle\right) \cdot \left(y-scale \cdot y-scale\right)\right) \cdot a\right)}{b \cdot b} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/
  (*
   0.25
   (*
    (*
     (*
      (/
       (sqrt
        (*
         8.0
         (*
          (pow b 4.0)
          (-
           (* (* PI PI) 3.08641975308642e-5)
           (sqrt (* (pow PI 4.0) 9.525986892242036e-10))))))
       (fabs y-scale))
      angle)
     (* y-scale y-scale))
    a))
  (* b b)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 * ((((sqrt((8.0 * (pow(b, 4.0) * (((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5) - sqrt((pow(((double) M_PI), 4.0) * 9.525986892242036e-10)))))) / fabs(y_45_scale)) * angle) * (y_45_scale * y_45_scale)) * a)) / (b * b);
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 * ((((Math.sqrt((8.0 * (Math.pow(b, 4.0) * (((Math.PI * Math.PI) * 3.08641975308642e-5) - Math.sqrt((Math.pow(Math.PI, 4.0) * 9.525986892242036e-10)))))) / Math.abs(y_45_scale)) * angle) * (y_45_scale * y_45_scale)) * a)) / (b * b);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (0.25 * ((((math.sqrt((8.0 * (math.pow(b, 4.0) * (((math.pi * math.pi) * 3.08641975308642e-5) - math.sqrt((math.pow(math.pi, 4.0) * 9.525986892242036e-10)))))) / math.fabs(y_45_scale)) * angle) * (y_45_scale * y_45_scale)) * a)) / (b * b)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(0.25 * Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((b ^ 4.0) * Float64(Float64(Float64(pi * pi) * 3.08641975308642e-5) - sqrt(Float64((pi ^ 4.0) * 9.525986892242036e-10)))))) / abs(y_45_scale)) * angle) * Float64(y_45_scale * y_45_scale)) * a)) / Float64(b * b))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (0.25 * ((((sqrt((8.0 * ((b ^ 4.0) * (((pi * pi) * 3.08641975308642e-5) - sqrt(((pi ^ 4.0) * 9.525986892242036e-10)))))) / abs(y_45_scale)) * angle) * (y_45_scale * y_45_scale)) * a)) / (b * b);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(0.25 * N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[b, 4.0], $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] - N[Sqrt[N[(N[Power[Pi, 4.0], $MachinePrecision] * 9.525986892242036e-10), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0%

   \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

2. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

4. Applied rewrites 0.5%

   \[\leadsto \color{blue}{0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

5. Taylor expanded in x-scale around 0

   \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]

7. Applied rewrites 3.9%

   \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]

8. Taylor expanded in angle around 0

   \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

10. Applied rewrites 4.0%

    \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}\right)}{{y-scale}^{2}}}\right)\right)}{{b}^{2}} \]

12. Applied rewrites 9.3%

    \[\leadsto \frac{0.25 \cdot \left(\left(\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5} - \sqrt{{\pi}^{4} \cdot 9.525986892242036 \cdot 10^{-10}}\right)\right)}}{\left|y-scale\right|} \cdot angle\right) \cdot \left(y-scale \cdot y-scale\right)\right) \cdot a\right)}{\color{blue}{b \cdot b}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025170 
(FPCore (a b angle x-scale y-scale)
  :name "b from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (- (+ (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) 2.0) (pow (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) 2.0))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))