b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 22.4%

Time: 38.3s

Alternatives: 7

Speedup: 14.3×

• could not determine a ground truth

  1. a = 2.926877616639182e-307
  2. b = -1.1452315642515501e+173
  3. angle = 9.939348403897322e+278
  4. x-scale = 1.5247587244146376e+297
  5. y-scale = -1.2068651571379866e-142

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: 22.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \left(b \cdot a\_m\right) \cdot \left(b \cdot \left(-a\_m\right)\right)\\ t_1 := \frac{4 \cdot t\_0}{{\left(x-scale\_m \cdot y-scale\right)}^{2}}\\ \mathbf{if}\;x-scale\_m \leq 1.85 \cdot 10^{+155}:\\ \;\;\;\;0.25 \cdot \frac{a\_m \cdot \left({x-scale\_m}^{2} \cdot \sqrt{8 \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}\right)}{\left|x-scale\_m\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_1\right) \cdot t\_0\right) \cdot \frac{{b}^{2} - \sqrt{{b}^{4}}}{{x-scale\_m}^{2}}}}{t\_1}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (* (* b a_m) (* b (- a_m))))
        (t_1 (/ (* 4.0 t_0) (pow (* x-scale_m y-scale) 2.0))))
   (if (<= x-scale_m 1.85e+155)
     (*
      0.25
      (/
       (*
        a_m
        (*
         (pow x-scale_m 2.0)
         (sqrt
          (*
           8.0
           (-
            (+ 0.5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
            (sqrt (pow (cos (* 0.005555555555555556 (* angle PI))) 4.0)))))))
       (fabs x-scale_m)))
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_1) t_0)
         (/ (- (pow b 2.0) (sqrt (pow b 4.0))) (pow x-scale_m 2.0)))))
      t_1))))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (b * a_m) * (b * -a_m);
	double t_1 = (4.0 * t_0) / pow((x_45_scale_m * y_45_scale), 2.0);
	double tmp;
	if (x_45_scale_m <= 1.85e+155) {
		tmp = 0.25 * ((a_m * (pow(x_45_scale_m, 2.0) * sqrt((8.0 * ((0.5 + (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))) - sqrt(pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 4.0))))))) / fabs(x_45_scale_m));
	} else {
		tmp = -sqrt((((2.0 * t_1) * t_0) * ((pow(b, 2.0) - sqrt(pow(b, 4.0))) / pow(x_45_scale_m, 2.0)))) / t_1;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (b * a_m) * (b * -a_m);
	double t_1 = (4.0 * t_0) / Math.pow((x_45_scale_m * y_45_scale), 2.0);
	double tmp;
	if (x_45_scale_m <= 1.85e+155) {
		tmp = 0.25 * ((a_m * (Math.pow(x_45_scale_m, 2.0) * Math.sqrt((8.0 * ((0.5 + (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))) - Math.sqrt(Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 4.0))))))) / Math.abs(x_45_scale_m));
	} else {
		tmp = -Math.sqrt((((2.0 * t_1) * t_0) * ((Math.pow(b, 2.0) - Math.sqrt(Math.pow(b, 4.0))) / Math.pow(x_45_scale_m, 2.0)))) / t_1;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale):
	t_0 = (b * a_m) * (b * -a_m)
	t_1 = (4.0 * t_0) / math.pow((x_45_scale_m * y_45_scale), 2.0)
	tmp = 0
	if x_45_scale_m <= 1.85e+155:
		tmp = 0.25 * ((a_m * (math.pow(x_45_scale_m, 2.0) * math.sqrt((8.0 * ((0.5 + (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))) - math.sqrt(math.pow(math.cos((0.005555555555555556 * (angle * math.pi))), 4.0))))))) / math.fabs(x_45_scale_m))
	else:
		tmp = -math.sqrt((((2.0 * t_1) * t_0) * ((math.pow(b, 2.0) - math.sqrt(math.pow(b, 4.0))) / math.pow(x_45_scale_m, 2.0)))) / t_1
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(b * a_m) * Float64(b * Float64(-a_m)))
	t_1 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale_m * y_45_scale) ^ 2.0))
	tmp = 0.0
	if (x_45_scale_m <= 1.85e+155)
		tmp = Float64(0.25 * Float64(Float64(a_m * Float64((x_45_scale_m ^ 2.0) * sqrt(Float64(8.0 * Float64(Float64(0.5 + Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))) - sqrt((cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 4.0))))))) / abs(x_45_scale_m)));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_1) * t_0) * Float64(Float64((b ^ 2.0) - sqrt((b ^ 4.0))) / (x_45_scale_m ^ 2.0))))) / t_1);
	end
	return tmp
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
function tmp_2 = code(a_m, b, angle, x_45_scale_m, y_45_scale)
	t_0 = (b * a_m) * (b * -a_m);
	t_1 = (4.0 * t_0) / ((x_45_scale_m * y_45_scale) ^ 2.0);
	tmp = 0.0;
	if (x_45_scale_m <= 1.85e+155)
		tmp = 0.25 * ((a_m * ((x_45_scale_m ^ 2.0) * sqrt((8.0 * ((0.5 + (0.5 * cos((0.011111111111111112 * (angle * pi))))) - sqrt((cos((0.005555555555555556 * (angle * pi))) ^ 4.0))))))) / abs(x_45_scale_m));
	else
		tmp = -sqrt((((2.0 * t_1) * t_0) * (((b ^ 2.0) - sqrt((b ^ 4.0))) / (x_45_scale_m ^ 2.0)))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(b * a$95$m), $MachinePrecision] * N[(b * (-a$95$m)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * t$95$0), $MachinePrecision] / N[Power[N[(x$45$scale$95$m * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 1.85e+155], N[(0.25 * N[(N[(a$95$m * N[(N[Power[x$45$scale$95$m, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(0.5 + N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[Power[b, 2.0], $MachinePrecision] - N[Sqrt[N[Power[b, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.8499999999999999e155

   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

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

   4. Applied rewrites 0.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 4.2%

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

   9. Applied rewrites 17.7%

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

   10. Taylor expanded in b around 0

       \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}\right)}{\left|x-scale\right|} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}\right)}{\left|x-scale\right|} \]

   12. Applied rewrites 21.9%

       \[\leadsto 0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}\right)}{\left|x-scale\right|} \]

   if 1.8499999999999999e155 < x-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

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

   4. Applied rewrites 0.2%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 2.1

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

   7. Applied rewrites 2.1%

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

Alternative 2: 21.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{a\_m \cdot \left(x-scale\_m \cdot x-scale\_m\right)}{b}\\ \mathbf{if}\;b \leq 1.15 \cdot 10^{+61}:\\ \;\;\;\;0.25 \cdot \left(t\_0 \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(1 - \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {b}^{4}\right)}}{\left|x-scale\_m\right|}}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(t\_0 \cdot \frac{b \cdot \sqrt{8 \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}}{\left|x-scale\_m\right|}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (/ (* a_m (* x-scale_m x-scale_m)) b)))
   (if (<= b 1.15e+61)
     (*
      0.25
      (*
       t_0
       (/
        (/
         (sqrt
          (*
           8.0
           (*
            (-
             1.0
             (sqrt (pow (cos (* (* PI angle) 0.005555555555555556)) 4.0)))
            (pow b 4.0))))
         (fabs x-scale_m))
        b)))
     (*
      0.25
      (*
       t_0
       (/
        (*
         b
         (sqrt
          (*
           8.0
           (-
            (+ 0.5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
            (sqrt (pow (cos (* 0.005555555555555556 (* angle PI))) 4.0))))))
        (fabs x-scale_m)))))))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (a_m * (x_45_scale_m * x_45_scale_m)) / b;
	double tmp;
	if (b <= 1.15e+61) {
		tmp = 0.25 * (t_0 * ((sqrt((8.0 * ((1.0 - sqrt(pow(cos(((((double) M_PI) * angle) * 0.005555555555555556)), 4.0))) * pow(b, 4.0)))) / fabs(x_45_scale_m)) / b));
	} else {
		tmp = 0.25 * (t_0 * ((b * sqrt((8.0 * ((0.5 + (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))) - sqrt(pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 4.0)))))) / fabs(x_45_scale_m)));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (a_m * (x_45_scale_m * x_45_scale_m)) / b;
	double tmp;
	if (b <= 1.15e+61) {
		tmp = 0.25 * (t_0 * ((Math.sqrt((8.0 * ((1.0 - Math.sqrt(Math.pow(Math.cos(((Math.PI * angle) * 0.005555555555555556)), 4.0))) * Math.pow(b, 4.0)))) / Math.abs(x_45_scale_m)) / b));
	} else {
		tmp = 0.25 * (t_0 * ((b * Math.sqrt((8.0 * ((0.5 + (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))) - Math.sqrt(Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 4.0)))))) / Math.abs(x_45_scale_m)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale):
	t_0 = (a_m * (x_45_scale_m * x_45_scale_m)) / b
	tmp = 0
	if b <= 1.15e+61:
		tmp = 0.25 * (t_0 * ((math.sqrt((8.0 * ((1.0 - math.sqrt(math.pow(math.cos(((math.pi * angle) * 0.005555555555555556)), 4.0))) * math.pow(b, 4.0)))) / math.fabs(x_45_scale_m)) / b))
	else:
		tmp = 0.25 * (t_0 * ((b * math.sqrt((8.0 * ((0.5 + (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))) - math.sqrt(math.pow(math.cos((0.005555555555555556 * (angle * math.pi))), 4.0)))))) / math.fabs(x_45_scale_m)))
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(a_m * Float64(x_45_scale_m * x_45_scale_m)) / b)
	tmp = 0.0
	if (b <= 1.15e+61)
		tmp = Float64(0.25 * Float64(t_0 * Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(1.0 - sqrt((cos(Float64(Float64(pi * angle) * 0.005555555555555556)) ^ 4.0))) * (b ^ 4.0)))) / abs(x_45_scale_m)) / b)));
	else
		tmp = Float64(0.25 * Float64(t_0 * Float64(Float64(b * sqrt(Float64(8.0 * Float64(Float64(0.5 + Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))) - sqrt((cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 4.0)))))) / abs(x_45_scale_m))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
function tmp_2 = code(a_m, b, angle, x_45_scale_m, y_45_scale)
	t_0 = (a_m * (x_45_scale_m * x_45_scale_m)) / b;
	tmp = 0.0;
	if (b <= 1.15e+61)
		tmp = 0.25 * (t_0 * ((sqrt((8.0 * ((1.0 - sqrt((cos(((pi * angle) * 0.005555555555555556)) ^ 4.0))) * (b ^ 4.0)))) / abs(x_45_scale_m)) / b));
	else
		tmp = 0.25 * (t_0 * ((b * sqrt((8.0 * ((0.5 + (0.5 * cos((0.011111111111111112 * (angle * pi))))) - sqrt((cos((0.005555555555555556 * (angle * pi))) ^ 4.0)))))) / abs(x_45_scale_m)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(a$95$m * N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[b, 1.15e+61], N[(0.25 * N[(t$95$0 * N[(N[(N[Sqrt[N[(8.0 * N[(N[(1.0 - N[Sqrt[N[Power[N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale$95$m], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(t$95$0 * N[(N[(b * N[Sqrt[N[(8.0 * N[(N[(0.5 + N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15e61

   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

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

   4. Applied rewrites 0.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 4.2%

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

   9. Applied rewrites 17.7%

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

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 18.8%

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

   if 1.15e61 < 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

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

   4. Applied rewrites 0.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 4.2%

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

   9. Applied rewrites 17.7%

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

   10. Taylor expanded in b around 0

       \[\leadsto \frac{1}{4} \cdot \left(\frac{a \cdot \left(x-scale \cdot x-scale\right)}{b} \cdot \frac{b \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}}{\left|x-scale\right|}\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\frac{a \cdot \left(x-scale \cdot x-scale\right)}{b} \cdot \frac{b \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}}{\left|x-scale\right|}\right) \]

   12. Applied rewrites 19.6%

       \[\leadsto 0.25 \cdot \left(\frac{a \cdot \left(x-scale \cdot x-scale\right)}{b} \cdot \frac{b \cdot \sqrt{8 \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}}{\left|x-scale\right|}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 20.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ 0.25 \cdot \frac{a\_m \cdot \left({x-scale\_m}^{2} \cdot \sqrt{8 \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}\right)}{\left|x-scale\_m\right|} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b angle x-scale_m y-scale)
 :precision binary64
 (*
  0.25
  (/
   (*
    a_m
    (*
     (pow x-scale_m 2.0)
     (sqrt
      (*
       8.0
       (-
        (+ 0.5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
        (sqrt (pow (cos (* 0.005555555555555556 (* angle PI))) 4.0)))))))
   (fabs x-scale_m))))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return 0.25 * ((a_m * (pow(x_45_scale_m, 2.0) * sqrt((8.0 * ((0.5 + (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))) - sqrt(pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 4.0))))))) / fabs(x_45_scale_m));
}

Java

a_m = Math.abs(a);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return 0.25 * ((a_m * (Math.pow(x_45_scale_m, 2.0) * Math.sqrt((8.0 * ((0.5 + (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))) - Math.sqrt(Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 4.0))))))) / Math.abs(x_45_scale_m));
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale):
	return 0.25 * ((a_m * (math.pow(x_45_scale_m, 2.0) * math.sqrt((8.0 * ((0.5 + (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))) - math.sqrt(math.pow(math.cos((0.005555555555555556 * (angle * math.pi))), 4.0))))))) / math.fabs(x_45_scale_m))

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale)
	return Float64(0.25 * Float64(Float64(a_m * Float64((x_45_scale_m ^ 2.0) * sqrt(Float64(8.0 * Float64(Float64(0.5 + Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))) - sqrt((cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 4.0))))))) / abs(x_45_scale_m)))
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
function tmp = code(a_m, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.25 * ((a_m * ((x_45_scale_m ^ 2.0) * sqrt((8.0 * ((0.5 + (0.5 * cos((0.011111111111111112 * (angle * pi))))) - sqrt((cos((0.005555555555555556 * (angle * pi))) ^ 4.0))))))) / abs(x_45_scale_m));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := N[(0.25 * N[(N[(a$95$m * N[(N[Power[x$45$scale$95$m, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(0.5 + N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale$95$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 y-scale around 0

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

4. Applied rewrites 0.6%

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

5. Taylor expanded in a around inf

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 4.2%

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

9. Applied rewrites 17.7%

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

10. Taylor expanded in b around 0

    \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}\right)}{\left|x-scale\right|} \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}\right)}{\left|x-scale\right|} \]

12. Applied rewrites 21.9%

    \[\leadsto 0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}\right)}{\left|x-scale\right|} \]
13. Add Preprocessing

Alternative 4: 19.9% accurate, 6.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.04 \cdot 10^{+127}:\\ \;\;\;\;0.25 \cdot \left(\frac{a\_m \cdot \left(x-scale\_m \cdot x-scale\_m\right)}{b} \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(1 - \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {b}^{4}\right)}}{\left|x-scale\_m\right|}}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \left(\frac{\sqrt{8 \cdot \left({\left(a\_m \cdot b\right)}^{4} \cdot \left(a\_m \cdot a\_m - \sqrt{{a\_m}^{4}}\right)\right)}}{\left|x-scale\_m\right|} \cdot \left(x-scale\_m \cdot x-scale\_m\right)\right)}{a\_m \cdot \left(\left(a\_m \cdot b\right) \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b angle x-scale_m y-scale)
 :precision binary64
 (if (<= b 1.04e+127)
   (*
    0.25
    (*
     (/ (* a_m (* x-scale_m x-scale_m)) b)
     (/
      (/
       (sqrt
        (*
         8.0
         (*
          (- 1.0 (sqrt (pow (cos (* (* PI angle) 0.005555555555555556)) 4.0)))
          (pow b 4.0))))
       (fabs x-scale_m))
      b)))
   (/
    (*
     0.25
     (*
      (/
       (sqrt
        (* 8.0 (* (pow (* a_m b) 4.0) (- (* a_m a_m) (sqrt (pow a_m 4.0))))))
       (fabs x-scale_m))
      (* x-scale_m x-scale_m)))
    (* a_m (* (* a_m b) b)))))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b <= 1.04e+127) {
		tmp = 0.25 * (((a_m * (x_45_scale_m * x_45_scale_m)) / b) * ((sqrt((8.0 * ((1.0 - sqrt(pow(cos(((((double) M_PI) * angle) * 0.005555555555555556)), 4.0))) * pow(b, 4.0)))) / fabs(x_45_scale_m)) / b));
	} else {
		tmp = (0.25 * ((sqrt((8.0 * (pow((a_m * b), 4.0) * ((a_m * a_m) - sqrt(pow(a_m, 4.0)))))) / fabs(x_45_scale_m)) * (x_45_scale_m * x_45_scale_m))) / (a_m * ((a_m * b) * b));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b <= 1.04e+127) {
		tmp = 0.25 * (((a_m * (x_45_scale_m * x_45_scale_m)) / b) * ((Math.sqrt((8.0 * ((1.0 - Math.sqrt(Math.pow(Math.cos(((Math.PI * angle) * 0.005555555555555556)), 4.0))) * Math.pow(b, 4.0)))) / Math.abs(x_45_scale_m)) / b));
	} else {
		tmp = (0.25 * ((Math.sqrt((8.0 * (Math.pow((a_m * b), 4.0) * ((a_m * a_m) - Math.sqrt(Math.pow(a_m, 4.0)))))) / Math.abs(x_45_scale_m)) * (x_45_scale_m * x_45_scale_m))) / (a_m * ((a_m * b) * b));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale):
	tmp = 0
	if b <= 1.04e+127:
		tmp = 0.25 * (((a_m * (x_45_scale_m * x_45_scale_m)) / b) * ((math.sqrt((8.0 * ((1.0 - math.sqrt(math.pow(math.cos(((math.pi * angle) * 0.005555555555555556)), 4.0))) * math.pow(b, 4.0)))) / math.fabs(x_45_scale_m)) / b))
	else:
		tmp = (0.25 * ((math.sqrt((8.0 * (math.pow((a_m * b), 4.0) * ((a_m * a_m) - math.sqrt(math.pow(a_m, 4.0)))))) / math.fabs(x_45_scale_m)) * (x_45_scale_m * x_45_scale_m))) / (a_m * ((a_m * b) * b))
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (b <= 1.04e+127)
		tmp = Float64(0.25 * Float64(Float64(Float64(a_m * Float64(x_45_scale_m * x_45_scale_m)) / b) * Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(1.0 - sqrt((cos(Float64(Float64(pi * angle) * 0.005555555555555556)) ^ 4.0))) * (b ^ 4.0)))) / abs(x_45_scale_m)) / b)));
	else
		tmp = Float64(Float64(0.25 * Float64(Float64(sqrt(Float64(8.0 * Float64((Float64(a_m * b) ^ 4.0) * Float64(Float64(a_m * a_m) - sqrt((a_m ^ 4.0)))))) / abs(x_45_scale_m)) * Float64(x_45_scale_m * x_45_scale_m))) / Float64(a_m * Float64(Float64(a_m * b) * b)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
function tmp_2 = code(a_m, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0;
	if (b <= 1.04e+127)
		tmp = 0.25 * (((a_m * (x_45_scale_m * x_45_scale_m)) / b) * ((sqrt((8.0 * ((1.0 - sqrt((cos(((pi * angle) * 0.005555555555555556)) ^ 4.0))) * (b ^ 4.0)))) / abs(x_45_scale_m)) / b));
	else
		tmp = (0.25 * ((sqrt((8.0 * (((a_m * b) ^ 4.0) * ((a_m * a_m) - sqrt((a_m ^ 4.0)))))) / abs(x_45_scale_m)) * (x_45_scale_m * x_45_scale_m))) / (a_m * ((a_m * b) * b));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[b, 1.04e+127], N[(0.25 * N[(N[(N[(a$95$m * N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * N[(N[(N[Sqrt[N[(8.0 * N[(N[(1.0 - N[Sqrt[N[Power[N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale$95$m], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[N[(a$95$m * b), $MachinePrecision], 4.0], $MachinePrecision] * N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[Sqrt[N[Power[a$95$m, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale$95$m], $MachinePrecision]), $MachinePrecision] * N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * N[(N[(a$95$m * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.04e127

   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

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

   4. Applied rewrites 0.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 4.2%

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

   9. Applied rewrites 17.7%

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

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 18.8%

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

   if 1.04e127 < 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

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

   4. Applied rewrites 0.6%

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

   5. Taylor expanded in angle around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 0.6

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

   7. Applied rewrites 0.6%

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

   9. Applied rewrites 2.4%

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

       1. lift-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-pow.f64 2.4

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. associate-*l* N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lower-*.f64 6.5

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

   11. Applied rewrites 6.5%

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

Alternative 5: 6.5% accurate, 9.9× speedup

Math

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

FPCore

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

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return (0.25 * ((sqrt((8.0 * (pow((a_m * b), 4.0) * ((a_m * a_m) - sqrt(pow(a_m, 4.0)))))) / fabs(x_45_scale_m)) * (x_45_scale_m * x_45_scale_m))) / (a_m * ((a_m * b) * b));
}

Fortran

a_m =     private
x-scale_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, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = (0.25d0 * ((sqrt((8.0d0 * (((a_m * b) ** 4.0d0) * ((a_m * a_m) - sqrt((a_m ** 4.0d0)))))) / abs(x_45scale_m)) * (x_45scale_m * x_45scale_m))) / (a_m * ((a_m * b) * b))
end function

Java

a_m = Math.abs(a);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return (0.25 * ((Math.sqrt((8.0 * (Math.pow((a_m * b), 4.0) * ((a_m * a_m) - Math.sqrt(Math.pow(a_m, 4.0)))))) / Math.abs(x_45_scale_m)) * (x_45_scale_m * x_45_scale_m))) / (a_m * ((a_m * b) * b));
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale):
	return (0.25 * ((math.sqrt((8.0 * (math.pow((a_m * b), 4.0) * ((a_m * a_m) - math.sqrt(math.pow(a_m, 4.0)))))) / math.fabs(x_45_scale_m)) * (x_45_scale_m * x_45_scale_m))) / (a_m * ((a_m * b) * b))

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale)
	return Float64(Float64(0.25 * Float64(Float64(sqrt(Float64(8.0 * Float64((Float64(a_m * b) ^ 4.0) * Float64(Float64(a_m * a_m) - sqrt((a_m ^ 4.0)))))) / abs(x_45_scale_m)) * Float64(x_45_scale_m * x_45_scale_m))) / Float64(a_m * Float64(Float64(a_m * b) * b)))
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
function tmp = code(a_m, b, angle, x_45_scale_m, y_45_scale)
	tmp = (0.25 * ((sqrt((8.0 * (((a_m * b) ^ 4.0) * ((a_m * a_m) - sqrt((a_m ^ 4.0)))))) / abs(x_45_scale_m)) * (x_45_scale_m * x_45_scale_m))) / (a_m * ((a_m * b) * b));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := N[(N[(0.25 * N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[N[(a$95$m * b), $MachinePrecision], 4.0], $MachinePrecision] * N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[Sqrt[N[Power[a$95$m, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale$95$m], $MachinePrecision]), $MachinePrecision] * N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * N[(N[(a$95$m * b), $MachinePrecision] * b), $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 y-scale around 0

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

4. Applied rewrites 0.6%

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

5. Taylor expanded in angle around 0

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

   1. lower--.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-pow.f64 0.6

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

7. Applied rewrites 0.6%

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

9. Applied rewrites 2.4%

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

    1. lift-*.f64 N/A

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

    2. pow2 N/A

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

    3. lift-pow.f64 2.4

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

    4. lower-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-*l* N/A

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

    7. lift-pow.f64 N/A

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

    8. pow2 N/A

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

    9. associate-*l* N/A

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

    10. lift-*.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. lower-*.f64 6.5

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

11. Applied rewrites 6.5%

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

Alternative 6: 3.2% accurate, 9.9× speedup

Math

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

FPCore

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

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return 0.25 * ((((sqrt(((((a_m * a_m) - sqrt(pow(a_m, 4.0))) * pow((a_m * b), 4.0)) * 8.0)) / fabs(x_45_scale_m)) * x_45_scale_m) * x_45_scale_m) / (((a_m * a_m) * b) * b));
}

Fortran

a_m =     private
x-scale_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, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = 0.25d0 * ((((sqrt(((((a_m * a_m) - sqrt((a_m ** 4.0d0))) * ((a_m * b) ** 4.0d0)) * 8.0d0)) / abs(x_45scale_m)) * x_45scale_m) * x_45scale_m) / (((a_m * a_m) * b) * b))
end function

Java

a_m = Math.abs(a);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return 0.25 * ((((Math.sqrt(((((a_m * a_m) - Math.sqrt(Math.pow(a_m, 4.0))) * Math.pow((a_m * b), 4.0)) * 8.0)) / Math.abs(x_45_scale_m)) * x_45_scale_m) * x_45_scale_m) / (((a_m * a_m) * b) * b));
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale):
	return 0.25 * ((((math.sqrt(((((a_m * a_m) - math.sqrt(math.pow(a_m, 4.0))) * math.pow((a_m * b), 4.0)) * 8.0)) / math.fabs(x_45_scale_m)) * x_45_scale_m) * x_45_scale_m) / (((a_m * a_m) * b) * b))

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale)
	return Float64(0.25 * Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(a_m * a_m) - sqrt((a_m ^ 4.0))) * (Float64(a_m * b) ^ 4.0)) * 8.0)) / abs(x_45_scale_m)) * x_45_scale_m) * x_45_scale_m) / Float64(Float64(Float64(a_m * a_m) * b) * b)))
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
function tmp = code(a_m, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.25 * ((((sqrt(((((a_m * a_m) - sqrt((a_m ^ 4.0))) * ((a_m * b) ^ 4.0)) * 8.0)) / abs(x_45_scale_m)) * x_45_scale_m) * x_45_scale_m) / (((a_m * a_m) * b) * b));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := N[(0.25 * N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[Sqrt[N[Power[a$95$m, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[(a$95$m * b), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale$95$m], $MachinePrecision]), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision] / N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] * b), $MachinePrecision] * b), $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 y-scale around 0

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

4. Applied rewrites 0.6%

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

5. Taylor expanded in angle around 0

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

   1. lower--.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-pow.f64 0.6

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

7. Applied rewrites 0.6%

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

9. Applied rewrites 2.4%

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

    1. lift-/.f64 N/A

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

11. Applied rewrites 3.1%

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

Alternative 7: 3.1% accurate, 14.3× speedup

Math

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

FPCore

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

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return (0.25 * ((sqrt((8.0 * (pow((a_m * b), 4.0) * 0.0))) / fabs(x_45_scale_m)) * (x_45_scale_m * x_45_scale_m))) / ((a_m * a_m) * (b * b));
}

Fortran

a_m =     private
x-scale_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, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = (0.25d0 * ((sqrt((8.0d0 * (((a_m * b) ** 4.0d0) * 0.0d0))) / abs(x_45scale_m)) * (x_45scale_m * x_45scale_m))) / ((a_m * a_m) * (b * b))
end function

Java

a_m = Math.abs(a);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return (0.25 * ((Math.sqrt((8.0 * (Math.pow((a_m * b), 4.0) * 0.0))) / Math.abs(x_45_scale_m)) * (x_45_scale_m * x_45_scale_m))) / ((a_m * a_m) * (b * b));
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale):
	return (0.25 * ((math.sqrt((8.0 * (math.pow((a_m * b), 4.0) * 0.0))) / math.fabs(x_45_scale_m)) * (x_45_scale_m * x_45_scale_m))) / ((a_m * a_m) * (b * b))

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale)
	return Float64(Float64(0.25 * Float64(Float64(sqrt(Float64(8.0 * Float64((Float64(a_m * b) ^ 4.0) * 0.0))) / abs(x_45_scale_m)) * Float64(x_45_scale_m * x_45_scale_m))) / Float64(Float64(a_m * a_m) * Float64(b * b)))
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
function tmp = code(a_m, b, angle, x_45_scale_m, y_45_scale)
	tmp = (0.25 * ((sqrt((8.0 * (((a_m * b) ^ 4.0) * 0.0))) / abs(x_45_scale_m)) * (x_45_scale_m * x_45_scale_m))) / ((a_m * a_m) * (b * b));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := N[(N[(0.25 * N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[N[(a$95$m * b), $MachinePrecision], 4.0], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale$95$m], $MachinePrecision]), $MachinePrecision] * N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[(b * b), $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 y-scale around 0

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

4. Applied rewrites 0.6%

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

5. Taylor expanded in angle around 0

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

   1. lower--.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-pow.f64 0.6

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

7. Applied rewrites 0.6%

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

9. Applied rewrites 2.4%

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

10. Taylor expanded in a around 0

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

12. Applied rewrites 3.2%

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

Reproduce

herbie shell --seed 2025154 
(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))))