b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 18.6%

Time: 40.0s

Alternatives: 10

Speedup: 12.5×

• could not determine a ground truth

  1. a = 1.584872285272454e-296
  2. b = -3.4796614612407818e+75
  3. angle = 6.677302656456952e+24
  4. x-scale = 2.227302681372153e+307
  5. y-scale = 4.3801847881507214e-300

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: 18.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(b\_m \cdot a\_m\right) \cdot \left(b\_m \cdot \left(-a\_m\right)\right)\\ t_1 := \frac{4 \cdot t\_0}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ t_2 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_3 := {t\_2}^{2} - \sqrt{{t\_2}^{4}}\\ \mathbf{if}\;b\_m \leq 6.2 \cdot 10^{-162}:\\ \;\;\;\;0.25 \cdot \left(\frac{a\_m}{b\_m} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \frac{\sqrt{8 \cdot \left({b\_m}^{4} \cdot t\_3\right)}}{x-scale}}{b\_m}\right)\\ \mathbf{elif}\;b\_m \leq 5.5 \cdot 10^{+136}:\\ \;\;\;\;0.25 \cdot \frac{a\_m \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\left(b\_m \cdot b\_m\right) \cdot \sqrt{8 \cdot t\_3}}{x-scale}\right)}{b\_m \cdot b\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_1\right) \cdot t\_0\right) \cdot \frac{a\_m \cdot a\_m - \sqrt{{a\_m}^{4}}}{y-scale \cdot y-scale}}}{t\_1}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* b_m a_m) (* b_m (- a_m))))
        (t_1 (/ (* 4.0 t_0) (pow (* x-scale y-scale) 2.0)))
        (t_2 (cos (* 0.005555555555555556 (* angle PI))))
        (t_3 (- (pow t_2 2.0) (sqrt (pow t_2 4.0)))))
   (if (<= b_m 6.2e-162)
     (*
      0.25
      (*
       (/ a_m b_m)
       (/
        (*
         (* x-scale x-scale)
         (/ (sqrt (* 8.0 (* (pow b_m 4.0) t_3))) x-scale))
        b_m)))
     (if (<= b_m 5.5e+136)
       (*
        0.25
        (/
         (*
          a_m
          (*
           (* x-scale x-scale)
           (/ (* (* b_m b_m) (sqrt (* 8.0 t_3))) x-scale)))
         (* b_m b_m)))
       (/
        (-
         (sqrt
          (*
           (* (* 2.0 t_1) t_0)
           (/ (- (* a_m a_m) (sqrt (pow a_m 4.0))) (* y-scale y-scale)))))
        t_1)))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b_m * a_m) * (b_m * -a_m);
	double t_1 = (4.0 * t_0) / pow((x_45_scale * y_45_scale), 2.0);
	double t_2 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_3 = pow(t_2, 2.0) - sqrt(pow(t_2, 4.0));
	double tmp;
	if (b_m <= 6.2e-162) {
		tmp = 0.25 * ((a_m / b_m) * (((x_45_scale * x_45_scale) * (sqrt((8.0 * (pow(b_m, 4.0) * t_3))) / x_45_scale)) / b_m));
	} else if (b_m <= 5.5e+136) {
		tmp = 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (((b_m * b_m) * sqrt((8.0 * t_3))) / x_45_scale))) / (b_m * b_m));
	} else {
		tmp = -sqrt((((2.0 * t_1) * t_0) * (((a_m * a_m) - sqrt(pow(a_m, 4.0))) / (y_45_scale * y_45_scale)))) / t_1;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b_m * a_m) * (b_m * -a_m);
	double t_1 = (4.0 * t_0) / Math.pow((x_45_scale * y_45_scale), 2.0);
	double t_2 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	double t_3 = Math.pow(t_2, 2.0) - Math.sqrt(Math.pow(t_2, 4.0));
	double tmp;
	if (b_m <= 6.2e-162) {
		tmp = 0.25 * ((a_m / b_m) * (((x_45_scale * x_45_scale) * (Math.sqrt((8.0 * (Math.pow(b_m, 4.0) * t_3))) / x_45_scale)) / b_m));
	} else if (b_m <= 5.5e+136) {
		tmp = 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (((b_m * b_m) * Math.sqrt((8.0 * t_3))) / x_45_scale))) / (b_m * b_m));
	} else {
		tmp = -Math.sqrt((((2.0 * t_1) * t_0) * (((a_m * a_m) - Math.sqrt(Math.pow(a_m, 4.0))) / (y_45_scale * y_45_scale)))) / t_1;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle, x_45_scale, y_45_scale):
	t_0 = (b_m * a_m) * (b_m * -a_m)
	t_1 = (4.0 * t_0) / math.pow((x_45_scale * y_45_scale), 2.0)
	t_2 = math.cos((0.005555555555555556 * (angle * math.pi)))
	t_3 = math.pow(t_2, 2.0) - math.sqrt(math.pow(t_2, 4.0))
	tmp = 0
	if b_m <= 6.2e-162:
		tmp = 0.25 * ((a_m / b_m) * (((x_45_scale * x_45_scale) * (math.sqrt((8.0 * (math.pow(b_m, 4.0) * t_3))) / x_45_scale)) / b_m))
	elif b_m <= 5.5e+136:
		tmp = 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (((b_m * b_m) * math.sqrt((8.0 * t_3))) / x_45_scale))) / (b_m * b_m))
	else:
		tmp = -math.sqrt((((2.0 * t_1) * t_0) * (((a_m * a_m) - math.sqrt(math.pow(a_m, 4.0))) / (y_45_scale * y_45_scale)))) / t_1
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b_m * a_m) * Float64(b_m * Float64(-a_m)))
	t_1 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	t_2 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_3 = Float64((t_2 ^ 2.0) - sqrt((t_2 ^ 4.0)))
	tmp = 0.0
	if (b_m <= 6.2e-162)
		tmp = Float64(0.25 * Float64(Float64(a_m / b_m) * Float64(Float64(Float64(x_45_scale * x_45_scale) * Float64(sqrt(Float64(8.0 * Float64((b_m ^ 4.0) * t_3))) / x_45_scale)) / b_m)));
	elseif (b_m <= 5.5e+136)
		tmp = Float64(0.25 * Float64(Float64(a_m * Float64(Float64(x_45_scale * x_45_scale) * Float64(Float64(Float64(b_m * b_m) * sqrt(Float64(8.0 * t_3))) / x_45_scale))) / Float64(b_m * b_m)));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_1) * t_0) * Float64(Float64(Float64(a_m * a_m) - sqrt((a_m ^ 4.0))) / Float64(y_45_scale * y_45_scale))))) / t_1);
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (b_m * a_m) * (b_m * -a_m);
	t_1 = (4.0 * t_0) / ((x_45_scale * y_45_scale) ^ 2.0);
	t_2 = cos((0.005555555555555556 * (angle * pi)));
	t_3 = (t_2 ^ 2.0) - sqrt((t_2 ^ 4.0));
	tmp = 0.0;
	if (b_m <= 6.2e-162)
		tmp = 0.25 * ((a_m / b_m) * (((x_45_scale * x_45_scale) * (sqrt((8.0 * ((b_m ^ 4.0) * t_3))) / x_45_scale)) / b_m));
	elseif (b_m <= 5.5e+136)
		tmp = 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (((b_m * b_m) * sqrt((8.0 * t_3))) / x_45_scale))) / (b_m * b_m));
	else
		tmp = -sqrt((((2.0 * t_1) * t_0) * (((a_m * a_m) - sqrt((a_m ^ 4.0))) / (y_45_scale * y_45_scale)))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * t$95$0), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[t$95$2, 2.0], $MachinePrecision] - N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 6.2e-162], N[(0.25 * N[(N[(a$95$m / b$95$m), $MachinePrecision] * N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[Power[b$95$m, 4.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5.5e+136], N[(0.25 * N[(N[(a$95$m * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[Sqrt[N[Power[a$95$m, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 6.1999999999999997e-162

   1. Initial program 0.1%

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

   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%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 4.3%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 8.9%

       \[\leadsto 0.25 \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\sqrt{8 \cdot \left({b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)\right)}}{x-scale}\right)}{b \cdot b} \]
   11. Step-by-step derivation

   12. Applied rewrites 15.4%

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

   if 6.1999999999999997e-162 < b < 5.50000000000000039e136

   1. Initial program 0.1%

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

   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%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 4.3%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 8.9%

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

   11. Taylor expanded in b around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

   13. Applied rewrites 10.8%

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

   if 5.50000000000000039e136 < b

   1. Initial program 0.1%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 1.1

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

   7. Applied rewrites 1.1%

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

   8. Taylor expanded in angle around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      5. lower-pow.f64 1.9

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   10. Applied rewrites 1.9%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{\color{blue}{y-scale} \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 11.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(b\_m \cdot a\_m\right) \cdot \left(b\_m \cdot \left(-a\_m\right)\right)\\ t_1 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_2 := \frac{4 \cdot t\_0}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \mathbf{if}\;b\_m \leq 5.5 \cdot 10^{+136}:\\ \;\;\;\;0.25 \cdot \frac{a\_m \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\left(b\_m \cdot b\_m\right) \cdot \sqrt{8 \cdot \left({t\_1}^{2} - \sqrt{{t\_1}^{4}}\right)}}{x-scale}\right)}{b\_m \cdot b\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_2\right) \cdot t\_0\right) \cdot \frac{a\_m \cdot a\_m - \sqrt{{a\_m}^{4}}}{y-scale \cdot y-scale}}}{t\_2}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* b_m a_m) (* b_m (- a_m))))
        (t_1 (cos (* 0.005555555555555556 (* angle PI))))
        (t_2 (/ (* 4.0 t_0) (pow (* x-scale y-scale) 2.0))))
   (if (<= b_m 5.5e+136)
     (*
      0.25
      (/
       (*
        a_m
        (*
         (* x-scale x-scale)
         (/
          (* (* b_m b_m) (sqrt (* 8.0 (- (pow t_1 2.0) (sqrt (pow t_1 4.0))))))
          x-scale)))
       (* b_m b_m)))
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_2) t_0)
         (/ (- (* a_m a_m) (sqrt (pow a_m 4.0))) (* y-scale y-scale)))))
      t_2))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b_m * a_m) * (b_m * -a_m);
	double t_1 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_2 = (4.0 * t_0) / pow((x_45_scale * y_45_scale), 2.0);
	double tmp;
	if (b_m <= 5.5e+136) {
		tmp = 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (((b_m * b_m) * sqrt((8.0 * (pow(t_1, 2.0) - sqrt(pow(t_1, 4.0)))))) / x_45_scale))) / (b_m * b_m));
	} else {
		tmp = -sqrt((((2.0 * t_2) * t_0) * (((a_m * a_m) - sqrt(pow(a_m, 4.0))) / (y_45_scale * y_45_scale)))) / t_2;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b_m * a_m) * (b_m * -a_m);
	double t_1 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	double t_2 = (4.0 * t_0) / Math.pow((x_45_scale * y_45_scale), 2.0);
	double tmp;
	if (b_m <= 5.5e+136) {
		tmp = 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (((b_m * b_m) * Math.sqrt((8.0 * (Math.pow(t_1, 2.0) - Math.sqrt(Math.pow(t_1, 4.0)))))) / x_45_scale))) / (b_m * b_m));
	} else {
		tmp = -Math.sqrt((((2.0 * t_2) * t_0) * (((a_m * a_m) - Math.sqrt(Math.pow(a_m, 4.0))) / (y_45_scale * y_45_scale)))) / t_2;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle, x_45_scale, y_45_scale):
	t_0 = (b_m * a_m) * (b_m * -a_m)
	t_1 = math.cos((0.005555555555555556 * (angle * math.pi)))
	t_2 = (4.0 * t_0) / math.pow((x_45_scale * y_45_scale), 2.0)
	tmp = 0
	if b_m <= 5.5e+136:
		tmp = 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (((b_m * b_m) * math.sqrt((8.0 * (math.pow(t_1, 2.0) - math.sqrt(math.pow(t_1, 4.0)))))) / x_45_scale))) / (b_m * b_m))
	else:
		tmp = -math.sqrt((((2.0 * t_2) * t_0) * (((a_m * a_m) - math.sqrt(math.pow(a_m, 4.0))) / (y_45_scale * y_45_scale)))) / t_2
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b_m * a_m) * Float64(b_m * Float64(-a_m)))
	t_1 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_2 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	tmp = 0.0
	if (b_m <= 5.5e+136)
		tmp = Float64(0.25 * Float64(Float64(a_m * Float64(Float64(x_45_scale * x_45_scale) * Float64(Float64(Float64(b_m * b_m) * sqrt(Float64(8.0 * Float64((t_1 ^ 2.0) - sqrt((t_1 ^ 4.0)))))) / x_45_scale))) / Float64(b_m * b_m)));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_2) * t_0) * Float64(Float64(Float64(a_m * a_m) - sqrt((a_m ^ 4.0))) / Float64(y_45_scale * y_45_scale))))) / t_2);
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (b_m * a_m) * (b_m * -a_m);
	t_1 = cos((0.005555555555555556 * (angle * pi)));
	t_2 = (4.0 * t_0) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = 0.0;
	if (b_m <= 5.5e+136)
		tmp = 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (((b_m * b_m) * sqrt((8.0 * ((t_1 ^ 2.0) - sqrt((t_1 ^ 4.0)))))) / x_45_scale))) / (b_m * b_m));
	else
		tmp = -sqrt((((2.0 * t_2) * t_0) * (((a_m * a_m) - sqrt((a_m ^ 4.0))) / (y_45_scale * y_45_scale)))) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * t$95$0), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 5.5e+136], N[(0.25 * N[(N[(a$95$m * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[Power[t$95$1, 2.0], $MachinePrecision] - N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[Sqrt[N[Power[a$95$m, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.50000000000000039e136

   1. Initial program 0.1%

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

   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%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 4.3%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 8.9%

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

   11. Taylor expanded in b around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

   13. Applied rewrites 10.8%

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

   if 5.50000000000000039e136 < b

   1. Initial program 0.1%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 1.1

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

   7. Applied rewrites 1.1%

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

   8. Taylor expanded in angle around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      5. lower-pow.f64 1.9

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   10. Applied rewrites 1.9%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{\color{blue}{y-scale} \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 9.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(b\_m \cdot a\_m\right) \cdot \left(b\_m \cdot \left(-a\_m\right)\right)\\ t_1 := \frac{4 \cdot t\_0}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \mathbf{if}\;b\_m \leq 1.6 \cdot 10^{+93}:\\ \;\;\;\;0.25 \cdot \frac{a\_m \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\sqrt{8 \cdot \left({b\_m}^{4} \cdot \left(1 - \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)\right)}}{x-scale}\right)}{b\_m \cdot b\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_1\right) \cdot t\_0\right) \cdot \frac{a\_m \cdot a\_m - \sqrt{{a\_m}^{4}}}{y-scale \cdot y-scale}}}{t\_1}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* b_m a_m) (* b_m (- a_m))))
        (t_1 (/ (* 4.0 t_0) (pow (* x-scale y-scale) 2.0))))
   (if (<= b_m 1.6e+93)
     (*
      0.25
      (/
       (*
        a_m
        (*
         (* x-scale x-scale)
         (/
          (sqrt
           (*
            8.0
            (*
             (pow b_m 4.0)
             (-
              1.0
              (sqrt (pow (cos (* 0.005555555555555556 (* angle PI))) 4.0))))))
          x-scale)))
       (* b_m b_m)))
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_1) t_0)
         (/ (- (* a_m a_m) (sqrt (pow a_m 4.0))) (* y-scale y-scale)))))
      t_1))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b_m * a_m) * (b_m * -a_m);
	double t_1 = (4.0 * t_0) / pow((x_45_scale * y_45_scale), 2.0);
	double tmp;
	if (b_m <= 1.6e+93) {
		tmp = 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (sqrt((8.0 * (pow(b_m, 4.0) * (1.0 - sqrt(pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 4.0)))))) / x_45_scale))) / (b_m * b_m));
	} else {
		tmp = -sqrt((((2.0 * t_1) * t_0) * (((a_m * a_m) - sqrt(pow(a_m, 4.0))) / (y_45_scale * y_45_scale)))) / t_1;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b_m * a_m) * (b_m * -a_m);
	double t_1 = (4.0 * t_0) / Math.pow((x_45_scale * y_45_scale), 2.0);
	double tmp;
	if (b_m <= 1.6e+93) {
		tmp = 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (Math.sqrt((8.0 * (Math.pow(b_m, 4.0) * (1.0 - Math.sqrt(Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 4.0)))))) / x_45_scale))) / (b_m * b_m));
	} else {
		tmp = -Math.sqrt((((2.0 * t_1) * t_0) * (((a_m * a_m) - Math.sqrt(Math.pow(a_m, 4.0))) / (y_45_scale * y_45_scale)))) / t_1;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle, x_45_scale, y_45_scale):
	t_0 = (b_m * a_m) * (b_m * -a_m)
	t_1 = (4.0 * t_0) / math.pow((x_45_scale * y_45_scale), 2.0)
	tmp = 0
	if b_m <= 1.6e+93:
		tmp = 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (math.sqrt((8.0 * (math.pow(b_m, 4.0) * (1.0 - math.sqrt(math.pow(math.cos((0.005555555555555556 * (angle * math.pi))), 4.0)))))) / x_45_scale))) / (b_m * b_m))
	else:
		tmp = -math.sqrt((((2.0 * t_1) * t_0) * (((a_m * a_m) - math.sqrt(math.pow(a_m, 4.0))) / (y_45_scale * y_45_scale)))) / t_1
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b_m * a_m) * Float64(b_m * Float64(-a_m)))
	t_1 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	tmp = 0.0
	if (b_m <= 1.6e+93)
		tmp = Float64(0.25 * Float64(Float64(a_m * Float64(Float64(x_45_scale * x_45_scale) * Float64(sqrt(Float64(8.0 * Float64((b_m ^ 4.0) * Float64(1.0 - sqrt((cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 4.0)))))) / x_45_scale))) / Float64(b_m * b_m)));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_1) * t_0) * Float64(Float64(Float64(a_m * a_m) - sqrt((a_m ^ 4.0))) / Float64(y_45_scale * y_45_scale))))) / t_1);
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (b_m * a_m) * (b_m * -a_m);
	t_1 = (4.0 * t_0) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = 0.0;
	if (b_m <= 1.6e+93)
		tmp = 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (sqrt((8.0 * ((b_m ^ 4.0) * (1.0 - sqrt((cos((0.005555555555555556 * (angle * pi))) ^ 4.0)))))) / x_45_scale))) / (b_m * b_m));
	else
		tmp = -sqrt((((2.0 * t_1) * t_0) * (((a_m * a_m) - sqrt((a_m ^ 4.0))) / (y_45_scale * y_45_scale)))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * t$95$0), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.6e+93], N[(0.25 * N[(N[(a$95$m * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(1.0 - N[Sqrt[N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[Sqrt[N[Power[a$95$m, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001e93

   1. Initial program 0.1%

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

   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%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 4.3%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 8.9%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 8.4%

       \[\leadsto 0.25 \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\sqrt{8 \cdot \left({b}^{4} \cdot \left(1 - \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)\right)}}{x-scale}\right)}{b \cdot b} \]

   if 1.6000000000000001e93 < b

   1. Initial program 0.1%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 1.1

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

   7. Applied rewrites 1.1%

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

   8. Taylor expanded in angle around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      5. lower-pow.f64 1.9

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{y-scale \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   10. Applied rewrites 1.9%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{a \cdot a - \sqrt{{a}^{4}}}{\color{blue}{y-scale} \cdot y-scale}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 8.4% accurate, 7.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ 0.25 \cdot \frac{a\_m \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\sqrt{8 \cdot \left({b\_m}^{4} \cdot \left(1 - \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)\right)}}{x-scale}\right)}{b\_m \cdot b\_m} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (*
  0.25
  (/
   (*
    a_m
    (*
     (* x-scale x-scale)
     (/
      (sqrt
       (*
        8.0
        (*
         (pow b_m 4.0)
         (-
          1.0
          (sqrt (pow (cos (* 0.005555555555555556 (* angle PI))) 4.0))))))
      x-scale)))
   (* b_m b_m))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (sqrt((8.0 * (pow(b_m, 4.0) * (1.0 - sqrt(pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 4.0)))))) / x_45_scale))) / (b_m * b_m));
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (Math.sqrt((8.0 * (Math.pow(b_m, 4.0) * (1.0 - Math.sqrt(Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 4.0)))))) / x_45_scale))) / (b_m * b_m));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle, x_45_scale, y_45_scale):
	return 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (math.sqrt((8.0 * (math.pow(b_m, 4.0) * (1.0 - math.sqrt(math.pow(math.cos((0.005555555555555556 * (angle * math.pi))), 4.0)))))) / x_45_scale))) / (b_m * b_m))

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	return Float64(0.25 * Float64(Float64(a_m * Float64(Float64(x_45_scale * x_45_scale) * Float64(sqrt(Float64(8.0 * Float64((b_m ^ 4.0) * Float64(1.0 - sqrt((cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 4.0)))))) / x_45_scale))) / Float64(b_m * b_m)))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.25 * ((a_m * ((x_45_scale * x_45_scale) * (sqrt((8.0 * ((b_m ^ 4.0) * (1.0 - sqrt((cos((0.005555555555555556 * (angle * pi))) ^ 4.0)))))) / x_45_scale))) / (b_m * b_m));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(0.25 * N[(N[(a$95$m * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(1.0 - N[Sqrt[N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

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%

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

5. Taylor expanded in y-scale around 0

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

   1. lower-sqrt.f64 N/A

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

7. Applied rewrites 4.3%

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

8. Taylor expanded in x-scale around 0

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

   1. lower-/.f64 N/A

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

10. Applied rewrites 8.9%

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

11. Taylor expanded in angle around 0

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

13. Applied rewrites 8.4%

    \[\leadsto 0.25 \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\sqrt{8 \cdot \left({b}^{4} \cdot \left(1 - \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)\right)}}{x-scale}\right)}{b \cdot b} \]
14. Add Preprocessing

Alternative 5: 4.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 7.6 \cdot 10^{-209}:\\ \;\;\;\;0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \frac{\sqrt{8 \cdot \frac{{b\_m}^{4} \cdot \left(\frac{b\_m \cdot b\_m}{x-scale \cdot x-scale} - \sqrt{\frac{{b\_m}^{4}}{{x-scale}^{4}}}\right)}{x-scale \cdot x-scale}}}{y-scale}\right)}{b\_m \cdot b\_m}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b\_m}^{4} \cdot \left(b\_m \cdot b\_m - \sqrt{{b\_m}^{4}}\right)}{y-scale \cdot y-scale}}}{b\_m \cdot b\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 7.6e-209)
   (*
    0.25
    (/
     (*
      (* x-scale x-scale)
      (*
       (* y-scale y-scale)
       (/
        (sqrt
         (*
          8.0
          (/
           (*
            (pow b_m 4.0)
            (-
             (/ (* b_m b_m) (* x-scale x-scale))
             (sqrt (/ (pow b_m 4.0) (pow x-scale 4.0)))))
           (* x-scale x-scale))))
        y-scale)))
     (* b_m b_m)))
   (*
    0.25
    (/
     (*
      (* y-scale y-scale)
      (sqrt
       (*
        8.0
        (/
         (* (pow b_m 4.0) (- (* b_m b_m) (sqrt (pow b_m 4.0))))
         (* y-scale y-scale)))))
     (* b_m b_m)))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 7.6e-209) {
		tmp = 0.25 * (((x_45_scale * x_45_scale) * ((y_45_scale * y_45_scale) * (sqrt((8.0 * ((pow(b_m, 4.0) * (((b_m * b_m) / (x_45_scale * x_45_scale)) - sqrt((pow(b_m, 4.0) / pow(x_45_scale, 4.0))))) / (x_45_scale * x_45_scale)))) / y_45_scale))) / (b_m * b_m));
	} else {
		tmp = 0.25 * (((y_45_scale * y_45_scale) * sqrt((8.0 * ((pow(b_m, 4.0) * ((b_m * b_m) - sqrt(pow(b_m, 4.0)))) / (y_45_scale * y_45_scale))))) / (b_m * b_m));
	}
	return tmp;
}

Fortran

a_m =     private
b_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a_m, b_m, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (y_45scale <= 7.6d-209) then
        tmp = 0.25d0 * (((x_45scale * x_45scale) * ((y_45scale * y_45scale) * (sqrt((8.0d0 * (((b_m ** 4.0d0) * (((b_m * b_m) / (x_45scale * x_45scale)) - sqrt(((b_m ** 4.0d0) / (x_45scale ** 4.0d0))))) / (x_45scale * x_45scale)))) / y_45scale))) / (b_m * b_m))
    else
        tmp = 0.25d0 * (((y_45scale * y_45scale) * sqrt((8.0d0 * (((b_m ** 4.0d0) * ((b_m * b_m) - sqrt((b_m ** 4.0d0)))) / (y_45scale * y_45scale))))) / (b_m * b_m))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 7.6e-209) {
		tmp = 0.25 * (((x_45_scale * x_45_scale) * ((y_45_scale * y_45_scale) * (Math.sqrt((8.0 * ((Math.pow(b_m, 4.0) * (((b_m * b_m) / (x_45_scale * x_45_scale)) - Math.sqrt((Math.pow(b_m, 4.0) / Math.pow(x_45_scale, 4.0))))) / (x_45_scale * x_45_scale)))) / y_45_scale))) / (b_m * b_m));
	} else {
		tmp = 0.25 * (((y_45_scale * y_45_scale) * Math.sqrt((8.0 * ((Math.pow(b_m, 4.0) * ((b_m * b_m) - Math.sqrt(Math.pow(b_m, 4.0)))) / (y_45_scale * y_45_scale))))) / (b_m * b_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if y_45_scale <= 7.6e-209:
		tmp = 0.25 * (((x_45_scale * x_45_scale) * ((y_45_scale * y_45_scale) * (math.sqrt((8.0 * ((math.pow(b_m, 4.0) * (((b_m * b_m) / (x_45_scale * x_45_scale)) - math.sqrt((math.pow(b_m, 4.0) / math.pow(x_45_scale, 4.0))))) / (x_45_scale * x_45_scale)))) / y_45_scale))) / (b_m * b_m))
	else:
		tmp = 0.25 * (((y_45_scale * y_45_scale) * math.sqrt((8.0 * ((math.pow(b_m, 4.0) * ((b_m * b_m) - math.sqrt(math.pow(b_m, 4.0)))) / (y_45_scale * y_45_scale))))) / (b_m * b_m))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 7.6e-209)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale * x_45_scale) * Float64(Float64(y_45_scale * y_45_scale) * Float64(sqrt(Float64(8.0 * Float64(Float64((b_m ^ 4.0) * Float64(Float64(Float64(b_m * b_m) / Float64(x_45_scale * x_45_scale)) - sqrt(Float64((b_m ^ 4.0) / (x_45_scale ^ 4.0))))) / Float64(x_45_scale * x_45_scale)))) / y_45_scale))) / Float64(b_m * b_m)));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(y_45_scale * y_45_scale) * sqrt(Float64(8.0 * Float64(Float64((b_m ^ 4.0) * Float64(Float64(b_m * b_m) - sqrt((b_m ^ 4.0)))) / Float64(y_45_scale * y_45_scale))))) / Float64(b_m * b_m)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (y_45_scale <= 7.6e-209)
		tmp = 0.25 * (((x_45_scale * x_45_scale) * ((y_45_scale * y_45_scale) * (sqrt((8.0 * (((b_m ^ 4.0) * (((b_m * b_m) / (x_45_scale * x_45_scale)) - sqrt(((b_m ^ 4.0) / (x_45_scale ^ 4.0))))) / (x_45_scale * x_45_scale)))) / y_45_scale))) / (b_m * b_m));
	else
		tmp = 0.25 * (((y_45_scale * y_45_scale) * sqrt((8.0 * (((b_m ^ 4.0) * ((b_m * b_m) - sqrt((b_m ^ 4.0)))) / (y_45_scale * y_45_scale))))) / (b_m * b_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 7.6e-209], N[(0.25 * N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(N[Power[b$95$m, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[Sqrt[N[Power[b$95$m, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.5999999999999998e-209

   1. Initial program 0.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\left(\frac{a \cdot a}{y-scale \cdot y-scale} + \frac{b \cdot b}{x-scale \cdot x-scale}\right) - \sqrt{{\left(\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right)}^{2}}\right)\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

   5. Taylor expanded in a around -inf

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

   7. Applied rewrites 0.6%

      \[\leadsto -0.25 \cdot \color{blue}{\frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 0.6%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{b \cdot b}{x-scale \cdot x-scale} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)}{b \cdot b}} \]

   11. Taylor expanded in y-scale around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \frac{\sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2}}}}{y-scale}\right)}{b \cdot b} \]

   13. Applied rewrites 2.4%

       \[\leadsto 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \frac{\sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{b \cdot b}{x-scale \cdot x-scale} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{x-scale \cdot x-scale}}}{y-scale}\right)}{b \cdot b} \]

   if 7.5999999999999998e-209 < y-scale

   1. Initial program 0.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\left(\frac{a \cdot a}{y-scale \cdot y-scale} + \frac{b \cdot b}{x-scale \cdot x-scale}\right) - \sqrt{{\left(\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right)}^{2}}\right)\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

   5. Taylor expanded in a around -inf

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

   7. Applied rewrites 0.6%

      \[\leadsto -0.25 \cdot \color{blue}{\frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 0.6%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{b \cdot b}{x-scale \cdot x-scale} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)}{b \cdot b}} \]

   11. Taylor expanded in x-scale around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

       2. pow2 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

       6. lower-/.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

   13. Applied rewrites 3.9%

       \[\leadsto 0.25 \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(b \cdot b - \sqrt{{b}^{4}}\right)}{y-scale \cdot y-scale}}}{b \cdot b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 3.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 7.6 \cdot 10^{-209}:\\ \;\;\;\;0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{{b\_m}^{4} \cdot \left(\frac{b\_m \cdot b\_m}{x-scale \cdot x-scale} - \sqrt{\frac{{b\_m}^{4}}{{x-scale}^{4}}}\right)}{x-scale \cdot x-scale}}\right)}{b\_m \cdot b\_m}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b\_m}^{4} \cdot \left(b\_m \cdot b\_m - \sqrt{{b\_m}^{4}}\right)}{y-scale \cdot y-scale}}}{b\_m \cdot b\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 7.6e-209)
   (*
    0.25
    (/
     (*
      (* x-scale x-scale)
      (*
       y-scale
       (sqrt
        (*
         8.0
         (/
          (*
           (pow b_m 4.0)
           (-
            (/ (* b_m b_m) (* x-scale x-scale))
            (sqrt (/ (pow b_m 4.0) (pow x-scale 4.0)))))
          (* x-scale x-scale))))))
     (* b_m b_m)))
   (*
    0.25
    (/
     (*
      (* y-scale y-scale)
      (sqrt
       (*
        8.0
        (/
         (* (pow b_m 4.0) (- (* b_m b_m) (sqrt (pow b_m 4.0))))
         (* y-scale y-scale)))))
     (* b_m b_m)))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 7.6e-209) {
		tmp = 0.25 * (((x_45_scale * x_45_scale) * (y_45_scale * sqrt((8.0 * ((pow(b_m, 4.0) * (((b_m * b_m) / (x_45_scale * x_45_scale)) - sqrt((pow(b_m, 4.0) / pow(x_45_scale, 4.0))))) / (x_45_scale * x_45_scale)))))) / (b_m * b_m));
	} else {
		tmp = 0.25 * (((y_45_scale * y_45_scale) * sqrt((8.0 * ((pow(b_m, 4.0) * ((b_m * b_m) - sqrt(pow(b_m, 4.0)))) / (y_45_scale * y_45_scale))))) / (b_m * b_m));
	}
	return tmp;
}

Fortran

a_m =     private
b_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a_m, b_m, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (y_45scale <= 7.6d-209) then
        tmp = 0.25d0 * (((x_45scale * x_45scale) * (y_45scale * sqrt((8.0d0 * (((b_m ** 4.0d0) * (((b_m * b_m) / (x_45scale * x_45scale)) - sqrt(((b_m ** 4.0d0) / (x_45scale ** 4.0d0))))) / (x_45scale * x_45scale)))))) / (b_m * b_m))
    else
        tmp = 0.25d0 * (((y_45scale * y_45scale) * sqrt((8.0d0 * (((b_m ** 4.0d0) * ((b_m * b_m) - sqrt((b_m ** 4.0d0)))) / (y_45scale * y_45scale))))) / (b_m * b_m))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 7.6e-209) {
		tmp = 0.25 * (((x_45_scale * x_45_scale) * (y_45_scale * Math.sqrt((8.0 * ((Math.pow(b_m, 4.0) * (((b_m * b_m) / (x_45_scale * x_45_scale)) - Math.sqrt((Math.pow(b_m, 4.0) / Math.pow(x_45_scale, 4.0))))) / (x_45_scale * x_45_scale)))))) / (b_m * b_m));
	} else {
		tmp = 0.25 * (((y_45_scale * y_45_scale) * Math.sqrt((8.0 * ((Math.pow(b_m, 4.0) * ((b_m * b_m) - Math.sqrt(Math.pow(b_m, 4.0)))) / (y_45_scale * y_45_scale))))) / (b_m * b_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if y_45_scale <= 7.6e-209:
		tmp = 0.25 * (((x_45_scale * x_45_scale) * (y_45_scale * math.sqrt((8.0 * ((math.pow(b_m, 4.0) * (((b_m * b_m) / (x_45_scale * x_45_scale)) - math.sqrt((math.pow(b_m, 4.0) / math.pow(x_45_scale, 4.0))))) / (x_45_scale * x_45_scale)))))) / (b_m * b_m))
	else:
		tmp = 0.25 * (((y_45_scale * y_45_scale) * math.sqrt((8.0 * ((math.pow(b_m, 4.0) * ((b_m * b_m) - math.sqrt(math.pow(b_m, 4.0)))) / (y_45_scale * y_45_scale))))) / (b_m * b_m))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 7.6e-209)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale * x_45_scale) * Float64(y_45_scale * sqrt(Float64(8.0 * Float64(Float64((b_m ^ 4.0) * Float64(Float64(Float64(b_m * b_m) / Float64(x_45_scale * x_45_scale)) - sqrt(Float64((b_m ^ 4.0) / (x_45_scale ^ 4.0))))) / Float64(x_45_scale * x_45_scale)))))) / Float64(b_m * b_m)));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(y_45_scale * y_45_scale) * sqrt(Float64(8.0 * Float64(Float64((b_m ^ 4.0) * Float64(Float64(b_m * b_m) - sqrt((b_m ^ 4.0)))) / Float64(y_45_scale * y_45_scale))))) / Float64(b_m * b_m)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (y_45_scale <= 7.6e-209)
		tmp = 0.25 * (((x_45_scale * x_45_scale) * (y_45_scale * sqrt((8.0 * (((b_m ^ 4.0) * (((b_m * b_m) / (x_45_scale * x_45_scale)) - sqrt(((b_m ^ 4.0) / (x_45_scale ^ 4.0))))) / (x_45_scale * x_45_scale)))))) / (b_m * b_m));
	else
		tmp = 0.25 * (((y_45_scale * y_45_scale) * sqrt((8.0 * (((b_m ^ 4.0) * ((b_m * b_m) - sqrt((b_m ^ 4.0)))) / (y_45_scale * y_45_scale))))) / (b_m * b_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 7.6e-209], N[(0.25 * N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * N[Sqrt[N[(8.0 * N[(N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(N[Power[b$95$m, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[Sqrt[N[Power[b$95$m, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.5999999999999998e-209

   1. Initial program 0.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\left(\frac{a \cdot a}{y-scale \cdot y-scale} + \frac{b \cdot b}{x-scale \cdot x-scale}\right) - \sqrt{{\left(\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right)}^{2}}\right)\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

   5. Taylor expanded in a around -inf

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

   7. Applied rewrites 0.6%

      \[\leadsto -0.25 \cdot \color{blue}{\frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 0.6%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{b \cdot b}{x-scale \cdot x-scale} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)}{b \cdot b}} \]

   11. Taylor expanded in y-scale around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2}}}\right)}{b \cdot b} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2}}}\right)}{b \cdot b} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2}}}\right)}{b \cdot b} \]

   13. Applied rewrites 2.6%

       \[\leadsto 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{b \cdot b}{x-scale \cdot x-scale} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{x-scale \cdot x-scale}}\right)}{b \cdot b} \]

   if 7.5999999999999998e-209 < y-scale

   1. Initial program 0.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\left(\frac{a \cdot a}{y-scale \cdot y-scale} + \frac{b \cdot b}{x-scale \cdot x-scale}\right) - \sqrt{{\left(\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right)}^{2}}\right)\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

   5. Taylor expanded in a around -inf

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

   7. Applied rewrites 0.6%

      \[\leadsto -0.25 \cdot \color{blue}{\frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 0.6%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{b \cdot b}{x-scale \cdot x-scale} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)}{b \cdot b}} \]

   11. Taylor expanded in x-scale around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

       2. pow2 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

       6. lower-/.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

   13. Applied rewrites 3.9%

       \[\leadsto 0.25 \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(b \cdot b - \sqrt{{b}^{4}}\right)}{y-scale \cdot y-scale}}}{b \cdot b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 3.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ 0.25 \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b\_m}^{4} \cdot \left(b\_m \cdot b\_m - \sqrt{{b\_m}^{4}}\right)}{y-scale \cdot y-scale}}}{b\_m \cdot b\_m} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (*
  0.25
  (/
   (*
    (* y-scale y-scale)
    (sqrt
     (*
      8.0
      (/
       (* (pow b_m 4.0) (- (* b_m b_m) (sqrt (pow b_m 4.0))))
       (* y-scale y-scale)))))
   (* b_m b_m))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * (((y_45_scale * y_45_scale) * sqrt((8.0 * ((pow(b_m, 4.0) * ((b_m * b_m) - sqrt(pow(b_m, 4.0)))) / (y_45_scale * y_45_scale))))) / (b_m * b_m));
}

Fortran

a_m =     private
b_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a_m, b_m, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = 0.25d0 * (((y_45scale * y_45scale) * sqrt((8.0d0 * (((b_m ** 4.0d0) * ((b_m * b_m) - sqrt((b_m ** 4.0d0)))) / (y_45scale * y_45scale))))) / (b_m * b_m))
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * (((y_45_scale * y_45_scale) * Math.sqrt((8.0 * ((Math.pow(b_m, 4.0) * ((b_m * b_m) - Math.sqrt(Math.pow(b_m, 4.0)))) / (y_45_scale * y_45_scale))))) / (b_m * b_m));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle, x_45_scale, y_45_scale):
	return 0.25 * (((y_45_scale * y_45_scale) * math.sqrt((8.0 * ((math.pow(b_m, 4.0) * ((b_m * b_m) - math.sqrt(math.pow(b_m, 4.0)))) / (y_45_scale * y_45_scale))))) / (b_m * b_m))

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	return Float64(0.25 * Float64(Float64(Float64(y_45_scale * y_45_scale) * sqrt(Float64(8.0 * Float64(Float64((b_m ^ 4.0) * Float64(Float64(b_m * b_m) - sqrt((b_m ^ 4.0)))) / Float64(y_45_scale * y_45_scale))))) / Float64(b_m * b_m)))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.25 * (((y_45_scale * y_45_scale) * sqrt((8.0 * (((b_m ^ 4.0) * ((b_m * b_m) - sqrt((b_m ^ 4.0)))) / (y_45_scale * y_45_scale))))) / (b_m * b_m));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(0.25 * N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[Sqrt[N[Power[b$95$m, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 0.1%

   \[\leadsto \color{blue}{0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\left(\frac{a \cdot a}{y-scale \cdot y-scale} + \frac{b \cdot b}{x-scale \cdot x-scale}\right) - \sqrt{{\left(\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right)}^{2}}\right)\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

5. Taylor expanded in a around -inf

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

7. Applied rewrites 0.6%

   \[\leadsto -0.25 \cdot \color{blue}{\frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

8. Taylor expanded in a around 0

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

10. Applied rewrites 0.6%

    \[\leadsto 0.25 \cdot \color{blue}{\frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{b \cdot b}{x-scale \cdot x-scale} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)}{b \cdot b}} \]

11. Taylor expanded in x-scale around 0

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

    1. lower-*.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

    2. pow2 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

    4. lower-sqrt.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

    6. lower-/.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left({b}^{2} - \sqrt{{b}^{4}}\right)}{{y-scale}^{2}}}}{b \cdot b} \]

13. Applied rewrites 3.9%

    \[\leadsto 0.25 \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(b \cdot b - \sqrt{{b}^{4}}\right)}{y-scale \cdot y-scale}}}{b \cdot b} \]
14. Add Preprocessing

Alternative 8: 2.0% accurate, 12.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ -0.25 \cdot \left(a\_m \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{\frac{1}{y-scale \cdot y-scale} - \sqrt{{y-scale}^{-4}}}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (*
  -0.25
  (*
   a_m
   (*
    (* x-scale x-scale)
    (*
     (* y-scale y-scale)
     (sqrt
      (*
       8.0
       (/
        (- (/ 1.0 (* y-scale y-scale)) (sqrt (pow y-scale -4.0)))
        (* (* x-scale x-scale) (* y-scale y-scale))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return -0.25 * (a_m * ((x_45_scale * x_45_scale) * ((y_45_scale * y_45_scale) * sqrt((8.0 * (((1.0 / (y_45_scale * y_45_scale)) - sqrt(pow(y_45_scale, -4.0))) / ((x_45_scale * x_45_scale) * (y_45_scale * y_45_scale))))))));
}

Fortran

a_m =     private
b_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a_m, b_m, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (-0.25d0) * (a_m * ((x_45scale * x_45scale) * ((y_45scale * y_45scale) * sqrt((8.0d0 * (((1.0d0 / (y_45scale * y_45scale)) - sqrt((y_45scale ** (-4.0d0)))) / ((x_45scale * x_45scale) * (y_45scale * y_45scale))))))))
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return -0.25 * (a_m * ((x_45_scale * x_45_scale) * ((y_45_scale * y_45_scale) * Math.sqrt((8.0 * (((1.0 / (y_45_scale * y_45_scale)) - Math.sqrt(Math.pow(y_45_scale, -4.0))) / ((x_45_scale * x_45_scale) * (y_45_scale * y_45_scale))))))));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle, x_45_scale, y_45_scale):
	return -0.25 * (a_m * ((x_45_scale * x_45_scale) * ((y_45_scale * y_45_scale) * math.sqrt((8.0 * (((1.0 / (y_45_scale * y_45_scale)) - math.sqrt(math.pow(y_45_scale, -4.0))) / ((x_45_scale * x_45_scale) * (y_45_scale * y_45_scale))))))))

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	return Float64(-0.25 * Float64(a_m * Float64(Float64(x_45_scale * x_45_scale) * Float64(Float64(y_45_scale * y_45_scale) * sqrt(Float64(8.0 * Float64(Float64(Float64(1.0 / Float64(y_45_scale * y_45_scale)) - sqrt((y_45_scale ^ -4.0))) / Float64(Float64(x_45_scale * x_45_scale) * Float64(y_45_scale * y_45_scale)))))))))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m, angle, x_45_scale, y_45_scale)
	tmp = -0.25 * (a_m * ((x_45_scale * x_45_scale) * ((y_45_scale * y_45_scale) * sqrt((8.0 * (((1.0 / (y_45_scale * y_45_scale)) - sqrt((y_45_scale ^ -4.0))) / ((x_45_scale * x_45_scale) * (y_45_scale * y_45_scale))))))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(-0.25 * N[(a$95$m * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[(1.0 / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[Power[y$45$scale, -4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 0.1%

   \[\leadsto \color{blue}{0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\left(\frac{a \cdot a}{y-scale \cdot y-scale} + \frac{b \cdot b}{x-scale \cdot x-scale}\right) - \sqrt{{\left(\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right)}^{2}}\right)\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

5. Taylor expanded in a around -inf

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

7. Applied rewrites 0.6%

   \[\leadsto -0.25 \cdot \color{blue}{\frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

8. Taylor expanded in b around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

10. Applied rewrites 2.0%

    \[\leadsto -0.25 \cdot \left(a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \color{blue}{\left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{\frac{1}{y-scale \cdot y-scale} - \sqrt{{y-scale}^{-4}}}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)}\right)\right) \]
11. Add Preprocessing

Alternative 9: 0.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ 0.25 \cdot \frac{a\_m \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b\_m}^{4} \cdot 0}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)\right)}{b\_m \cdot b\_m} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (*
  0.25
  (/
   (*
    a_m
    (*
     (* x-scale x-scale)
     (*
      (* y-scale y-scale)
      (sqrt
       (*
        8.0
        (/
         (* (pow b_m 4.0) 0.0)
         (* (* x-scale x-scale) (* y-scale y-scale))))))))
   (* b_m b_m))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * ((a_m * ((x_45_scale * x_45_scale) * ((y_45_scale * y_45_scale) * sqrt((8.0 * ((pow(b_m, 4.0) * 0.0) / ((x_45_scale * x_45_scale) * (y_45_scale * y_45_scale)))))))) / (b_m * b_m));
}

Fortran

a_m =     private
b_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a_m, b_m, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = 0.25d0 * ((a_m * ((x_45scale * x_45scale) * ((y_45scale * y_45scale) * sqrt((8.0d0 * (((b_m ** 4.0d0) * 0.0d0) / ((x_45scale * x_45scale) * (y_45scale * y_45scale)))))))) / (b_m * b_m))
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * ((a_m * ((x_45_scale * x_45_scale) * ((y_45_scale * y_45_scale) * Math.sqrt((8.0 * ((Math.pow(b_m, 4.0) * 0.0) / ((x_45_scale * x_45_scale) * (y_45_scale * y_45_scale)))))))) / (b_m * b_m));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle, x_45_scale, y_45_scale):
	return 0.25 * ((a_m * ((x_45_scale * x_45_scale) * ((y_45_scale * y_45_scale) * math.sqrt((8.0 * ((math.pow(b_m, 4.0) * 0.0) / ((x_45_scale * x_45_scale) * (y_45_scale * y_45_scale)))))))) / (b_m * b_m))

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	return Float64(0.25 * Float64(Float64(a_m * Float64(Float64(x_45_scale * x_45_scale) * Float64(Float64(y_45_scale * y_45_scale) * sqrt(Float64(8.0 * Float64(Float64((b_m ^ 4.0) * 0.0) / Float64(Float64(x_45_scale * x_45_scale) * Float64(y_45_scale * y_45_scale)))))))) / Float64(b_m * b_m)))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.25 * ((a_m * ((x_45_scale * x_45_scale) * ((y_45_scale * y_45_scale) * sqrt((8.0 * (((b_m ^ 4.0) * 0.0) / ((x_45_scale * x_45_scale) * (y_45_scale * y_45_scale)))))))) / (b_m * b_m));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(0.25 * N[(N[(a$95$m * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[b$95$m, 4.0], $MachinePrecision] * 0.0), $MachinePrecision] / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

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%

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

5. Taylor expanded in angle around 0

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 0.7

      \[\leadsto 0.25 \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)\right)}{b \cdot b} \]

7. Applied rewrites 0.7%

   \[\leadsto 0.25 \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)\right)}{b \cdot b} \]

8. Taylor expanded in y-scale around 0

   \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot 0}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)\right)}{b \cdot b} \]
9. Step-by-step derivation

10. Applied rewrites 0.9%

    \[\leadsto 0.25 \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot 0}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)\right)}{b \cdot b} \]
11. Add Preprocessing

Alternative 10: 0.0% accurate, 20.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ -0.25 \cdot \frac{a\_m \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left({y-scale}^{21} \cdot \mathsf{NaN}\right)\right)}{b\_m \cdot b\_m} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (*
  -0.25
  (/ (* a_m (* (* x-scale x-scale) (* (pow y-scale 21.0) NAN))) (* b_m b_m))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return -0.25 * ((a_m * ((x_45_scale * x_45_scale) * (pow(y_45_scale, 21.0) * ((double) NAN)))) / (b_m * b_m));
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return -0.25 * ((a_m * ((x_45_scale * x_45_scale) * (Math.pow(y_45_scale, 21.0) * Double.NaN))) / (b_m * b_m));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle, x_45_scale, y_45_scale):
	return -0.25 * ((a_m * ((x_45_scale * x_45_scale) * (math.pow(y_45_scale, 21.0) * math.nan))) / (b_m * b_m))

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	return Float64(-0.25 * Float64(Float64(a_m * Float64(Float64(x_45_scale * x_45_scale) * Float64((y_45_scale ^ 21.0) * NaN))) / Float64(b_m * b_m)))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m, angle, x_45_scale, y_45_scale)
	tmp = -0.25 * ((a_m * ((x_45_scale * x_45_scale) * ((y_45_scale ^ 21.0) * NaN))) / (b_m * b_m));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(-0.25 * N[(N[(a$95$m * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[Power[y$45$scale, 21.0], $MachinePrecision] * Indeterminate), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 0.1%

   \[\leadsto \color{blue}{0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\left(\frac{a \cdot a}{y-scale \cdot y-scale} + \frac{b \cdot b}{x-scale \cdot x-scale}\right) - \sqrt{{\left(\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right)}^{2}}\right)\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

5. Taylor expanded in a around -inf

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

7. Applied rewrites 0.6%

   \[\leadsto -0.25 \cdot \color{blue}{\frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

8. Taylor expanded in y-scale around 0

   \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{21} \cdot \mathsf{NAN}\left(\right)\right)\right)}{{b}^{\color{blue}{2}}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{21} \cdot \mathsf{NAN}\left(\right)\right)\right)}{{b}^{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{21} \cdot \mathsf{NAN}\left(\right)\right)\right)}{{b}^{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{21} \cdot \mathsf{NAN}\left(\right)\right)\right)}{{b}^{2}} \]

   4. pow2 N/A

      \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left({y-scale}^{21} \cdot \mathsf{NAN}\left(\right)\right)\right)}{{b}^{2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left({y-scale}^{21} \cdot \mathsf{NAN}\left(\right)\right)\right)}{{b}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left({y-scale}^{21} \cdot \mathsf{NAN}\left(\right)\right)\right)}{{b}^{2}} \]

   7. lower-pow.f64 N/A

      \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left({y-scale}^{21} \cdot \mathsf{NAN}\left(\right)\right)\right)}{{b}^{2}} \]

   8. lower-NAN.f64 N/A

      \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left({y-scale}^{21} \cdot \mathsf{NaN}\right)\right)}{{b}^{2}} \]

   9. pow2 N/A

      \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left({y-scale}^{21} \cdot \mathsf{NaN}\right)\right)}{b \cdot b} \]

   10. lift-*.f64 0.0

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left({y-scale}^{21} \cdot \mathsf{NaN}\right)\right)}{b \cdot b} \]

10. Applied rewrites 0.0%

    \[\leadsto -0.25 \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left({y-scale}^{21} \cdot \mathsf{NaN}\right)\right)}{b \cdot \color{blue}{b}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025138 
(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))))