b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 37.8%

Time: 2.1min

Alternatives: 4

Speedup: 484.7×

• could not determine a ground truth

  1. a = -3.303058424525601e+110
  2. b = -2.210639077852027e-142
  3. angle = -2.6242403069369786e+218
  4. x-scale = -6.621526326551168e-274
  5. y-scale = 7.110420005492658e+127

Specification

Math

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

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: 0.1% accurate, 1.0× speedup

Math

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

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: 37.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := {\sin t\_0}^{2}\\ t_2 := \frac{t\_1}{y-scale\_m \cdot y-scale\_m}\\ t_3 := {\cos t\_0}^{2}\\ \mathbf{if}\;y-scale\_m \leq 2.35 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(b\_m \cdot 0.25\right) \cdot \left(x-scale\_m \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\frac{t\_3}{x-scale\_m \cdot x-scale\_m} + -0.5 \cdot \frac{2 \cdot \frac{t\_3 \cdot t\_1}{x-scale\_m \cdot x-scale\_m}}{t\_1}}\\ \mathbf{elif}\;y-scale\_m \leq 2.1 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(b\_m \cdot 0.25\right) \cdot \left(\sqrt{8} \cdot \left(y-scale\_m \cdot x-scale\_m\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(-0.5, \frac{2 \cdot \left(t\_3 \cdot t\_2\right)}{t\_3}, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(x-scale\_m \cdot \left(a\_m \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (pow (sin t_0) 2.0))
        (t_2 (/ t_1 (* y-scale_m y-scale_m)))
        (t_3 (pow (cos t_0) 2.0)))
   (if (<= y-scale_m 2.35e-162)
     (*
      (* (* b_m 0.25) (* x-scale_m (* y-scale_m (sqrt 8.0))))
      (sqrt
       (+
        (/ t_3 (* x-scale_m x-scale_m))
        (* -0.5 (/ (* 2.0 (/ (* t_3 t_1) (* x-scale_m x-scale_m))) t_1)))))
     (if (<= y-scale_m 2.1e-27)
       (*
        (* (* b_m 0.25) (* (sqrt 8.0) (* y-scale_m x-scale_m)))
        (sqrt (fma -0.5 (/ (* 2.0 (* t_3 t_2)) t_3) t_2)))
       (* 4.0 (* x-scale_m (* a_m 0.25)))))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = pow(sin(t_0), 2.0);
	double t_2 = t_1 / (y_45_scale_m * y_45_scale_m);
	double t_3 = pow(cos(t_0), 2.0);
	double tmp;
	if (y_45_scale_m <= 2.35e-162) {
		tmp = ((b_m * 0.25) * (x_45_scale_m * (y_45_scale_m * sqrt(8.0)))) * sqrt(((t_3 / (x_45_scale_m * x_45_scale_m)) + (-0.5 * ((2.0 * ((t_3 * t_1) / (x_45_scale_m * x_45_scale_m))) / t_1))));
	} else if (y_45_scale_m <= 2.1e-27) {
		tmp = ((b_m * 0.25) * (sqrt(8.0) * (y_45_scale_m * x_45_scale_m))) * sqrt(fma(-0.5, ((2.0 * (t_3 * t_2)) / t_3), t_2));
	} else {
		tmp = 4.0 * (x_45_scale_m * (a_m * 0.25));
	}
	return tmp;
}

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0) ^ 2.0
	t_2 = Float64(t_1 / Float64(y_45_scale_m * y_45_scale_m))
	t_3 = cos(t_0) ^ 2.0
	tmp = 0.0
	if (y_45_scale_m <= 2.35e-162)
		tmp = Float64(Float64(Float64(b_m * 0.25) * Float64(x_45_scale_m * Float64(y_45_scale_m * sqrt(8.0)))) * sqrt(Float64(Float64(t_3 / Float64(x_45_scale_m * x_45_scale_m)) + Float64(-0.5 * Float64(Float64(2.0 * Float64(Float64(t_3 * t_1) / Float64(x_45_scale_m * x_45_scale_m))) / t_1)))));
	elseif (y_45_scale_m <= 2.1e-27)
		tmp = Float64(Float64(Float64(b_m * 0.25) * Float64(sqrt(8.0) * Float64(y_45_scale_m * x_45_scale_m))) * sqrt(fma(-0.5, Float64(Float64(2.0 * Float64(t_3 * t_2)) / t_3), t_2)));
	else
		tmp = Float64(4.0 * Float64(x_45_scale_m * Float64(a_m * 0.25)));
	end
	return tmp
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Cos[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 2.35e-162], N[(N[(N[(b$95$m * 0.25), $MachinePrecision] * N[(x$45$scale$95$m * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(t$95$3 / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(2.0 * N[(N[(t$95$3 * t$95$1), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 2.1e-27], N[(N[(N[(b$95$m * 0.25), $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[(y$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-0.5 * N[(N[(2.0 * N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(x$45$scale$95$m * N[(a$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 2.35e-162

   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. Add Preprocessing

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 1.9%

      \[\leadsto \left(0.25 \cdot \left(\left(b \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\left(\frac{{\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}{y-scale \cdot y-scale}\right) - \sqrt{\mathsf{fma}\left(4, {\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2} \cdot \frac{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}, {\left(\frac{{1}^{2}}{x-scale \cdot x-scale} - \frac{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}\right)}} \]

   9. Taylor expanded in y-scale around 0

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

   11. Applied rewrites 11.3%

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

   if 2.35e-162 < y-scale < 2.10000000000000015e-27

   1. Initial program 0.2%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 11.6%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 25.5%

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

   if 2.10000000000000015e-27 < y-scale

   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. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right)} \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\color{blue}{\left(x-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\left(x-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      8. lower-sqrt.f64 31.9

         \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

   5. Applied rewrites 31.9%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\left(\left(a \cdot 0.25\right) \cdot x-scale\right) \cdot 4} \]
3. Recombined 3 regimes into one program.

4. Final simplification 18.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.35 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(b \cdot 0.25\right) \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + -0.5 \cdot \frac{2 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\\ \mathbf{elif}\;y-scale \leq 2.1 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(b \cdot 0.25\right) \cdot \left(\sqrt{8} \cdot \left(y-scale \cdot x-scale\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(-0.5, \frac{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(x-scale \cdot \left(a \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 40.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \frac{{\sin t\_0}^{2}}{y-scale\_m \cdot y-scale\_m}\\ t_2 := {\cos t\_0}^{2}\\ \mathbf{if}\;y-scale\_m \leq 1.8 \cdot 10^{-160}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(y-scale\_m \cdot \sqrt{8}\right) \cdot \left(a\_m \cdot x-scale\_m\right)\right)\right) \cdot \sqrt{\frac{0.5}{x-scale\_m \cdot x-scale\_m} + \frac{-0.5}{x-scale\_m \cdot x-scale\_m}}\\ \mathbf{elif}\;y-scale\_m \leq 2.1 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(b\_m \cdot 0.25\right) \cdot \left(\sqrt{8} \cdot \left(y-scale\_m \cdot x-scale\_m\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(-0.5, \frac{2 \cdot \left(t\_2 \cdot t\_1\right)}{t\_2}, t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(x-scale\_m \cdot \left(a\_m \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (/ (pow (sin t_0) 2.0) (* y-scale_m y-scale_m)))
        (t_2 (pow (cos t_0) 2.0)))
   (if (<= y-scale_m 1.8e-160)
     (*
      (* 0.25 (* (* y-scale_m (sqrt 8.0)) (* a_m x-scale_m)))
      (sqrt
       (+ (/ 0.5 (* x-scale_m x-scale_m)) (/ -0.5 (* x-scale_m x-scale_m)))))
     (if (<= y-scale_m 2.1e-27)
       (*
        (* (* b_m 0.25) (* (sqrt 8.0) (* y-scale_m x-scale_m)))
        (sqrt (fma -0.5 (/ (* 2.0 (* t_2 t_1)) t_2) t_1)))
       (* 4.0 (* x-scale_m (* a_m 0.25)))))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = pow(sin(t_0), 2.0) / (y_45_scale_m * y_45_scale_m);
	double t_2 = pow(cos(t_0), 2.0);
	double tmp;
	if (y_45_scale_m <= 1.8e-160) {
		tmp = (0.25 * ((y_45_scale_m * sqrt(8.0)) * (a_m * x_45_scale_m))) * sqrt(((0.5 / (x_45_scale_m * x_45_scale_m)) + (-0.5 / (x_45_scale_m * x_45_scale_m))));
	} else if (y_45_scale_m <= 2.1e-27) {
		tmp = ((b_m * 0.25) * (sqrt(8.0) * (y_45_scale_m * x_45_scale_m))) * sqrt(fma(-0.5, ((2.0 * (t_2 * t_1)) / t_2), t_1));
	} else {
		tmp = 4.0 * (x_45_scale_m * (a_m * 0.25));
	}
	return tmp;
}

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64((sin(t_0) ^ 2.0) / Float64(y_45_scale_m * y_45_scale_m))
	t_2 = cos(t_0) ^ 2.0
	tmp = 0.0
	if (y_45_scale_m <= 1.8e-160)
		tmp = Float64(Float64(0.25 * Float64(Float64(y_45_scale_m * sqrt(8.0)) * Float64(a_m * x_45_scale_m))) * sqrt(Float64(Float64(0.5 / Float64(x_45_scale_m * x_45_scale_m)) + Float64(-0.5 / Float64(x_45_scale_m * x_45_scale_m)))));
	elseif (y_45_scale_m <= 2.1e-27)
		tmp = Float64(Float64(Float64(b_m * 0.25) * Float64(sqrt(8.0) * Float64(y_45_scale_m * x_45_scale_m))) * sqrt(fma(-0.5, Float64(Float64(2.0 * Float64(t_2 * t_1)) / t_2), t_1)));
	else
		tmp = Float64(4.0 * Float64(x_45_scale_m * Float64(a_m * 0.25)));
	end
	return tmp
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Cos[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 1.8e-160], N[(N[(0.25 * N[(N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(a$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(0.5 / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 2.1e-27], N[(N[(N[(b$95$m * 0.25), $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[(y$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-0.5 * N[(N[(2.0 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(x$45$scale$95$m * N[(a$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 1.7999999999999999e-160

   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. Add Preprocessing

   4. Applied rewrites 0.2%

      \[\leadsto \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 \color{blue}{\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}{x-scale \cdot x-scale} + \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}{y-scale \cdot y-scale} - \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a, a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)\right)\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}}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(a \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{{y-scale}^{2}}\right)\right) - \left(\sqrt{\frac{{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{{x-scale}^{2}}\right)}\right)} \]

   7. Applied rewrites 3.5%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\frac{0.5}{x-scale \cdot x-scale} + \left(\mathsf{fma}\left(0.5, \frac{\cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)}{y-scale \cdot y-scale}, \frac{0.5}{y-scale \cdot y-scale}\right) - \mathsf{fma}\left(0.5, \frac{\cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)}{x-scale \cdot x-scale}, \sqrt{\frac{{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)}^{2}}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} + {\left(\frac{0.5}{x-scale \cdot x-scale} - \mathsf{fma}\left(0.5, \frac{\cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)}{x-scale \cdot x-scale} + \frac{\cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)}{y-scale \cdot y-scale}, \frac{0.5}{y-scale \cdot y-scale}\right)\right)}^{2}}\right)\right)}} \]

   8. Taylor expanded in y-scale around inf

      \[\leadsto \left(\frac{1}{4} \cdot \left(\left(a \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{x-scale \cdot x-scale} + \frac{\frac{-1}{2}}{{x-scale}^{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 13.0%

       \[\leadsto \left(0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\frac{0.5}{x-scale \cdot x-scale} + \frac{-0.5}{x-scale \cdot x-scale}} \]

   if 1.7999999999999999e-160 < y-scale < 2.10000000000000015e-27

   1. Initial program 0.2%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 11.6%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 25.5%

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

   if 2.10000000000000015e-27 < y-scale

   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. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right)} \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\color{blue}{\left(x-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\left(x-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      8. lower-sqrt.f64 31.9

         \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

   5. Applied rewrites 31.9%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\left(\left(a \cdot 0.25\right) \cdot x-scale\right) \cdot 4} \]
3. Recombined 3 regimes into one program.

4. Final simplification 19.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.8 \cdot 10^{-160}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot x-scale\right)\right)\right) \cdot \sqrt{\frac{0.5}{x-scale \cdot x-scale} + \frac{-0.5}{x-scale \cdot x-scale}}\\ \mathbf{elif}\;y-scale \leq 2.1 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(b \cdot 0.25\right) \cdot \left(\sqrt{8} \cdot \left(y-scale \cdot x-scale\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(-0.5, \frac{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(x-scale \cdot \left(a \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 38.8% accurate, 33.4× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 7.2 \cdot 10^{-73}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(y-scale\_m \cdot \sqrt{8}\right) \cdot \left(a\_m \cdot x-scale\_m\right)\right)\right) \cdot \sqrt{\frac{0.5}{x-scale\_m \cdot x-scale\_m} + \frac{-0.5}{x-scale\_m \cdot x-scale\_m}}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(x-scale\_m \cdot \left(a\_m \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 7.2e-73)
   (*
    (* 0.25 (* (* y-scale_m (sqrt 8.0)) (* a_m x-scale_m)))
    (sqrt
     (+ (/ 0.5 (* x-scale_m x-scale_m)) (/ -0.5 (* x-scale_m x-scale_m)))))
   (* 4.0 (* x-scale_m (* a_m 0.25)))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 7.2e-73) {
		tmp = (0.25 * ((y_45_scale_m * sqrt(8.0)) * (a_m * x_45_scale_m))) * sqrt(((0.5 / (x_45_scale_m * x_45_scale_m)) + (-0.5 / (x_45_scale_m * x_45_scale_m))));
	} else {
		tmp = 4.0 * (x_45_scale_m * (a_m * 0.25));
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (y_45scale_m <= 7.2d-73) then
        tmp = (0.25d0 * ((y_45scale_m * sqrt(8.0d0)) * (a_m * x_45scale_m))) * sqrt(((0.5d0 / (x_45scale_m * x_45scale_m)) + ((-0.5d0) / (x_45scale_m * x_45scale_m))))
    else
        tmp = 4.0d0 * (x_45scale_m * (a_m * 0.25d0))
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 7.2e-73) {
		tmp = (0.25 * ((y_45_scale_m * Math.sqrt(8.0)) * (a_m * x_45_scale_m))) * Math.sqrt(((0.5 / (x_45_scale_m * x_45_scale_m)) + (-0.5 / (x_45_scale_m * x_45_scale_m))));
	} else {
		tmp = 4.0 * (x_45_scale_m * (a_m * 0.25));
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 7.2e-73:
		tmp = (0.25 * ((y_45_scale_m * math.sqrt(8.0)) * (a_m * x_45_scale_m))) * math.sqrt(((0.5 / (x_45_scale_m * x_45_scale_m)) + (-0.5 / (x_45_scale_m * x_45_scale_m))))
	else:
		tmp = 4.0 * (x_45_scale_m * (a_m * 0.25))
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 7.2e-73)
		tmp = Float64(Float64(0.25 * Float64(Float64(y_45_scale_m * sqrt(8.0)) * Float64(a_m * x_45_scale_m))) * sqrt(Float64(Float64(0.5 / Float64(x_45_scale_m * x_45_scale_m)) + Float64(-0.5 / Float64(x_45_scale_m * x_45_scale_m)))));
	else
		tmp = Float64(4.0 * Float64(x_45_scale_m * Float64(a_m * 0.25)));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 7.2e-73)
		tmp = (0.25 * ((y_45_scale_m * sqrt(8.0)) * (a_m * x_45_scale_m))) * sqrt(((0.5 / (x_45_scale_m * x_45_scale_m)) + (-0.5 / (x_45_scale_m * x_45_scale_m))));
	else
		tmp = 4.0 * (x_45_scale_m * (a_m * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 7.2e-73], N[(N[(0.25 * N[(N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(a$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(0.5 / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(x$45$scale$95$m * N[(a$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.1999999999999999e-73

   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. Add Preprocessing

   4. Applied rewrites 0.2%

      \[\leadsto \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 \color{blue}{\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}{x-scale \cdot x-scale} + \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}{y-scale \cdot y-scale} - \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a, a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)\right)\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}}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(a \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{{y-scale}^{2}}\right)\right) - \left(\sqrt{\frac{{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{{x-scale}^{2}}\right)}\right)} \]

   7. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\frac{0.5}{x-scale \cdot x-scale} + \left(\mathsf{fma}\left(0.5, \frac{\cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)}{y-scale \cdot y-scale}, \frac{0.5}{y-scale \cdot y-scale}\right) - \mathsf{fma}\left(0.5, \frac{\cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)}{x-scale \cdot x-scale}, \sqrt{\frac{{\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)}^{2}}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} + {\left(\frac{0.5}{x-scale \cdot x-scale} - \mathsf{fma}\left(0.5, \frac{\cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)}{x-scale \cdot x-scale} + \frac{\cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)}{y-scale \cdot y-scale}, \frac{0.5}{y-scale \cdot y-scale}\right)\right)}^{2}}\right)\right)}} \]

   8. Taylor expanded in y-scale around inf

      \[\leadsto \left(\frac{1}{4} \cdot \left(\left(a \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{x-scale \cdot x-scale} + \frac{\frac{-1}{2}}{{x-scale}^{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 13.2%

       \[\leadsto \left(0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\frac{0.5}{x-scale \cdot x-scale} + \frac{-0.5}{x-scale \cdot x-scale}} \]

   if 7.1999999999999999e-73 < y-scale

   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. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right)} \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\color{blue}{\left(x-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\left(x-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      8. lower-sqrt.f64 29.1

         \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

   5. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 29.3%

      \[\leadsto \color{blue}{\left(\left(a \cdot 0.25\right) \cdot x-scale\right) \cdot 4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 7.2 \cdot 10^{-73}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot x-scale\right)\right)\right) \cdot \sqrt{\frac{0.5}{x-scale \cdot x-scale} + \frac{-0.5}{x-scale \cdot x-scale}}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(x-scale \cdot \left(a \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.3% accurate, 484.7× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ a\_m \cdot x-scale\_m \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (* a_m x-scale_m))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return a_m * x_45_scale_m;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = a_m * x_45scale_m
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return a_m * x_45_scale_m;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return a_m * x_45_scale_m

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(a_m * x_45_scale_m)
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = a_m * x_45_scale_m;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(a$95$m * x$45$scale$95$m), $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. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right)} \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\color{blue}{\left(x-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\left(x-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

   8. lower-sqrt.f64 22.8

      \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

5. Applied rewrites 22.8%

   \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 22.9%

   \[\leadsto \color{blue}{\left(\left(a \cdot 0.25\right) \cdot x-scale\right) \cdot 4} \]
8. Step-by-step derivation

9. Applied rewrites 22.9%

   \[\leadsto x-scale \cdot \color{blue}{a} \]

10. Final simplification 22.9%

    \[\leadsto a \cdot x-scale \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(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))))