a from scale-rotated-ellipse

Percentage Accurate: 2.8% → 24.4%

Time: 42.6s

Alternatives: 12

Speedup: 6.6×

• 69.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 2.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 24.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|y-scale\right|\right)}^{2}\\ t_1 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_2 := {t\_1}^{4}\\ t_3 := {t\_1}^{2}\\ \mathbf{if}\;\left|y-scale\right| \leq 1.7 \cdot 10^{-138}:\\ \;\;\;\;\frac{0.25}{a} \cdot \frac{\left(\left|b\right| \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\left(\left(0.5 - \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) + \sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {a}^{4}\right)}}{\left|x-scale\right|}}{a}\\ \mathbf{elif}\;\left|y-scale\right| \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot \left(\left|b\right| \cdot \left(t\_0 \cdot \sqrt{8 \cdot \frac{\sqrt{t\_2} + t\_3}{t\_0}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left|b\right| \cdot \left({x-scale}^{2} \cdot \left(\left|y-scale\right| \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{t\_2}{{x-scale}^{4}}} + \frac{t\_3}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs y-scale) 2.0))
        (t_1 (cos (* 0.005555555555555556 (* angle PI))))
        (t_2 (pow t_1 4.0))
        (t_3 (pow t_1 2.0)))
   (if (<= (fabs y-scale) 1.7e-138)
     (*
      (/ 0.25 a)
      (/
       (*
        (* (fabs b) (* x-scale x-scale))
        (/
         (sqrt
          (*
           8.0
           (*
            (+
             (- 0.5 (* (cos (* (* angle (+ PI PI)) 0.005555555555555556)) 0.5))
             (sqrt (pow (sin (* (* angle PI) 0.005555555555555556)) 4.0)))
            (pow a 4.0))))
         (fabs x-scale)))
       a))
     (if (<= (fabs y-scale) 1.35e+154)
       (* 0.25 (* (fabs b) (* t_0 (sqrt (* 8.0 (/ (+ (sqrt t_2) t_3) t_0))))))
       (*
        0.25
        (*
         (fabs b)
         (*
          (pow x-scale 2.0)
          (*
           (fabs y-scale)
           (sqrt
            (*
             8.0
             (/
              (+ (sqrt (/ t_2 (pow x-scale 4.0))) (/ t_3 (pow x-scale 2.0)))
              (pow x-scale 2.0))))))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(y_45_scale), 2.0);
	double t_1 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_2 = pow(t_1, 4.0);
	double t_3 = pow(t_1, 2.0);
	double tmp;
	if (fabs(y_45_scale) <= 1.7e-138) {
		tmp = (0.25 / a) * (((fabs(b) * (x_45_scale * x_45_scale)) * (sqrt((8.0 * (((0.5 - (cos(((angle * (((double) M_PI) + ((double) M_PI))) * 0.005555555555555556)) * 0.5)) + sqrt(pow(sin(((angle * ((double) M_PI)) * 0.005555555555555556)), 4.0))) * pow(a, 4.0)))) / fabs(x_45_scale))) / a);
	} else if (fabs(y_45_scale) <= 1.35e+154) {
		tmp = 0.25 * (fabs(b) * (t_0 * sqrt((8.0 * ((sqrt(t_2) + t_3) / t_0)))));
	} else {
		tmp = 0.25 * (fabs(b) * (pow(x_45_scale, 2.0) * (fabs(y_45_scale) * sqrt((8.0 * ((sqrt((t_2 / pow(x_45_scale, 4.0))) + (t_3 / pow(x_45_scale, 2.0))) / pow(x_45_scale, 2.0)))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.pow(Math.abs(y_45_scale), 2.0);
	double t_1 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	double t_2 = Math.pow(t_1, 4.0);
	double t_3 = Math.pow(t_1, 2.0);
	double tmp;
	if (Math.abs(y_45_scale) <= 1.7e-138) {
		tmp = (0.25 / a) * (((Math.abs(b) * (x_45_scale * x_45_scale)) * (Math.sqrt((8.0 * (((0.5 - (Math.cos(((angle * (Math.PI + Math.PI)) * 0.005555555555555556)) * 0.5)) + Math.sqrt(Math.pow(Math.sin(((angle * Math.PI) * 0.005555555555555556)), 4.0))) * Math.pow(a, 4.0)))) / Math.abs(x_45_scale))) / a);
	} else if (Math.abs(y_45_scale) <= 1.35e+154) {
		tmp = 0.25 * (Math.abs(b) * (t_0 * Math.sqrt((8.0 * ((Math.sqrt(t_2) + t_3) / t_0)))));
	} else {
		tmp = 0.25 * (Math.abs(b) * (Math.pow(x_45_scale, 2.0) * (Math.abs(y_45_scale) * Math.sqrt((8.0 * ((Math.sqrt((t_2 / Math.pow(x_45_scale, 4.0))) + (t_3 / Math.pow(x_45_scale, 2.0))) / Math.pow(x_45_scale, 2.0)))))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pow(math.fabs(y_45_scale), 2.0)
	t_1 = math.cos((0.005555555555555556 * (angle * math.pi)))
	t_2 = math.pow(t_1, 4.0)
	t_3 = math.pow(t_1, 2.0)
	tmp = 0
	if math.fabs(y_45_scale) <= 1.7e-138:
		tmp = (0.25 / a) * (((math.fabs(b) * (x_45_scale * x_45_scale)) * (math.sqrt((8.0 * (((0.5 - (math.cos(((angle * (math.pi + math.pi)) * 0.005555555555555556)) * 0.5)) + math.sqrt(math.pow(math.sin(((angle * math.pi) * 0.005555555555555556)), 4.0))) * math.pow(a, 4.0)))) / math.fabs(x_45_scale))) / a)
	elif math.fabs(y_45_scale) <= 1.35e+154:
		tmp = 0.25 * (math.fabs(b) * (t_0 * math.sqrt((8.0 * ((math.sqrt(t_2) + t_3) / t_0)))))
	else:
		tmp = 0.25 * (math.fabs(b) * (math.pow(x_45_scale, 2.0) * (math.fabs(y_45_scale) * math.sqrt((8.0 * ((math.sqrt((t_2 / math.pow(x_45_scale, 4.0))) + (t_3 / math.pow(x_45_scale, 2.0))) / math.pow(x_45_scale, 2.0)))))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0
	t_1 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_2 = t_1 ^ 4.0
	t_3 = t_1 ^ 2.0
	tmp = 0.0
	if (abs(y_45_scale) <= 1.7e-138)
		tmp = Float64(Float64(0.25 / a) * Float64(Float64(Float64(abs(b) * Float64(x_45_scale * x_45_scale)) * Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(0.5 - Float64(cos(Float64(Float64(angle * Float64(pi + pi)) * 0.005555555555555556)) * 0.5)) + sqrt((sin(Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 4.0))) * (a ^ 4.0)))) / abs(x_45_scale))) / a));
	elseif (abs(y_45_scale) <= 1.35e+154)
		tmp = Float64(0.25 * Float64(abs(b) * Float64(t_0 * sqrt(Float64(8.0 * Float64(Float64(sqrt(t_2) + t_3) / t_0))))));
	else
		tmp = Float64(0.25 * Float64(abs(b) * Float64((x_45_scale ^ 2.0) * Float64(abs(y_45_scale) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(t_2 / (x_45_scale ^ 4.0))) + Float64(t_3 / (x_45_scale ^ 2.0))) / (x_45_scale ^ 2.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0;
	t_1 = cos((0.005555555555555556 * (angle * pi)));
	t_2 = t_1 ^ 4.0;
	t_3 = t_1 ^ 2.0;
	tmp = 0.0;
	if (abs(y_45_scale) <= 1.7e-138)
		tmp = (0.25 / a) * (((abs(b) * (x_45_scale * x_45_scale)) * (sqrt((8.0 * (((0.5 - (cos(((angle * (pi + pi)) * 0.005555555555555556)) * 0.5)) + sqrt((sin(((angle * pi) * 0.005555555555555556)) ^ 4.0))) * (a ^ 4.0)))) / abs(x_45_scale))) / a);
	elseif (abs(y_45_scale) <= 1.35e+154)
		tmp = 0.25 * (abs(b) * (t_0 * sqrt((8.0 * ((sqrt(t_2) + t_3) / t_0)))));
	else
		tmp = 0.25 * (abs(b) * ((x_45_scale ^ 2.0) * (abs(y_45_scale) * sqrt((8.0 * ((sqrt((t_2 / (x_45_scale ^ 4.0))) + (t_3 / (x_45_scale ^ 2.0))) / (x_45_scale ^ 2.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 4.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$1, 2.0], $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.7e-138], N[(N[(0.25 / a), $MachinePrecision] * N[(N[(N[(N[Abs[b], $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[(N[(0.5 - N[(N[Cos[N[(N[(angle * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.35e+154], N[(0.25 * N[(N[Abs[b], $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[t$95$2], $MachinePrecision] + t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[Abs[b], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(t$95$2 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$3 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 1.7000000000000001e-138

   1. Initial program 2.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in y-scale around 0

      \[\leadsto 0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2}}}\right)}{{a}^{2}} \]
   6. Step-by-step derivation

      1. lower-sqrt.64 N/A

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

   7. Applied rewrites 1.6%

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

   9. Applied rewrites 9.1%

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

   if 1.7000000000000001e-138 < y-scale < 1.35000000000000003e154

   1. Initial program 2.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in x-scale around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-pow.64 N/A

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

   10. Applied rewrites 8.0%

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

   if 1.35000000000000003e154 < y-scale

   1. Initial program 2.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in y-scale around inf

      \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

   10. Applied rewrites 5.6%

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

Alternative 2: 23.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|y-scale\right|\right)}^{2}\\ t_1 := \frac{0.25}{a} \cdot \frac{\left(\left|b\right| \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\left(\left(0.5 - \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) + \sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {a}^{4}\right)}}{\left|x-scale\right|}}{a}\\ t_2 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;\left|y-scale\right| \leq 1.7 \cdot 10^{-138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\left|y-scale\right| \leq 3.75 \cdot 10^{+152}:\\ \;\;\;\;0.25 \cdot \left(\left|b\right| \cdot \left(t\_0 \cdot \sqrt{8 \cdot \frac{\sqrt{{t\_2}^{4}} + {t\_2}^{2}}{t\_0}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs y-scale) 2.0))
        (t_1
         (*
          (/ 0.25 a)
          (/
           (*
            (* (fabs b) (* x-scale x-scale))
            (/
             (sqrt
              (*
               8.0
               (*
                (+
                 (-
                  0.5
                  (* (cos (* (* angle (+ PI PI)) 0.005555555555555556)) 0.5))
                 (sqrt (pow (sin (* (* angle PI) 0.005555555555555556)) 4.0)))
                (pow a 4.0))))
             (fabs x-scale)))
           a)))
        (t_2 (cos (* 0.005555555555555556 (* angle PI)))))
   (if (<= (fabs y-scale) 1.7e-138)
     t_1
     (if (<= (fabs y-scale) 3.75e+152)
       (*
        0.25
        (*
         (fabs b)
         (*
          t_0
          (sqrt (* 8.0 (/ (+ (sqrt (pow t_2 4.0)) (pow t_2 2.0)) t_0))))))
       t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(y_45_scale), 2.0);
	double t_1 = (0.25 / a) * (((fabs(b) * (x_45_scale * x_45_scale)) * (sqrt((8.0 * (((0.5 - (cos(((angle * (((double) M_PI) + ((double) M_PI))) * 0.005555555555555556)) * 0.5)) + sqrt(pow(sin(((angle * ((double) M_PI)) * 0.005555555555555556)), 4.0))) * pow(a, 4.0)))) / fabs(x_45_scale))) / a);
	double t_2 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (fabs(y_45_scale) <= 1.7e-138) {
		tmp = t_1;
	} else if (fabs(y_45_scale) <= 3.75e+152) {
		tmp = 0.25 * (fabs(b) * (t_0 * sqrt((8.0 * ((sqrt(pow(t_2, 4.0)) + pow(t_2, 2.0)) / t_0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.pow(Math.abs(y_45_scale), 2.0);
	double t_1 = (0.25 / a) * (((Math.abs(b) * (x_45_scale * x_45_scale)) * (Math.sqrt((8.0 * (((0.5 - (Math.cos(((angle * (Math.PI + Math.PI)) * 0.005555555555555556)) * 0.5)) + Math.sqrt(Math.pow(Math.sin(((angle * Math.PI) * 0.005555555555555556)), 4.0))) * Math.pow(a, 4.0)))) / Math.abs(x_45_scale))) / a);
	double t_2 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	double tmp;
	if (Math.abs(y_45_scale) <= 1.7e-138) {
		tmp = t_1;
	} else if (Math.abs(y_45_scale) <= 3.75e+152) {
		tmp = 0.25 * (Math.abs(b) * (t_0 * Math.sqrt((8.0 * ((Math.sqrt(Math.pow(t_2, 4.0)) + Math.pow(t_2, 2.0)) / t_0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pow(math.fabs(y_45_scale), 2.0)
	t_1 = (0.25 / a) * (((math.fabs(b) * (x_45_scale * x_45_scale)) * (math.sqrt((8.0 * (((0.5 - (math.cos(((angle * (math.pi + math.pi)) * 0.005555555555555556)) * 0.5)) + math.sqrt(math.pow(math.sin(((angle * math.pi) * 0.005555555555555556)), 4.0))) * math.pow(a, 4.0)))) / math.fabs(x_45_scale))) / a)
	t_2 = math.cos((0.005555555555555556 * (angle * math.pi)))
	tmp = 0
	if math.fabs(y_45_scale) <= 1.7e-138:
		tmp = t_1
	elif math.fabs(y_45_scale) <= 3.75e+152:
		tmp = 0.25 * (math.fabs(b) * (t_0 * math.sqrt((8.0 * ((math.sqrt(math.pow(t_2, 4.0)) + math.pow(t_2, 2.0)) / t_0)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0
	t_1 = Float64(Float64(0.25 / a) * Float64(Float64(Float64(abs(b) * Float64(x_45_scale * x_45_scale)) * Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(0.5 - Float64(cos(Float64(Float64(angle * Float64(pi + pi)) * 0.005555555555555556)) * 0.5)) + sqrt((sin(Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 4.0))) * (a ^ 4.0)))) / abs(x_45_scale))) / a))
	t_2 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (abs(y_45_scale) <= 1.7e-138)
		tmp = t_1;
	elseif (abs(y_45_scale) <= 3.75e+152)
		tmp = Float64(0.25 * Float64(abs(b) * Float64(t_0 * sqrt(Float64(8.0 * Float64(Float64(sqrt((t_2 ^ 4.0)) + (t_2 ^ 2.0)) / t_0))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0;
	t_1 = (0.25 / a) * (((abs(b) * (x_45_scale * x_45_scale)) * (sqrt((8.0 * (((0.5 - (cos(((angle * (pi + pi)) * 0.005555555555555556)) * 0.5)) + sqrt((sin(((angle * pi) * 0.005555555555555556)) ^ 4.0))) * (a ^ 4.0)))) / abs(x_45_scale))) / a);
	t_2 = cos((0.005555555555555556 * (angle * pi)));
	tmp = 0.0;
	if (abs(y_45_scale) <= 1.7e-138)
		tmp = t_1;
	elseif (abs(y_45_scale) <= 3.75e+152)
		tmp = 0.25 * (abs(b) * (t_0 * sqrt((8.0 * ((sqrt((t_2 ^ 4.0)) + (t_2 ^ 2.0)) / t_0)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.25 / a), $MachinePrecision] * N[(N[(N[(N[Abs[b], $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[(N[(0.5 - N[(N[Cos[N[(N[(angle * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.7e-138], t$95$1, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 3.75e+152], N[(0.25 * N[(N[Abs[b], $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.7000000000000001e-138 or 3.75000000000000023e152 < y-scale

   1. Initial program 2.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in y-scale around 0

      \[\leadsto 0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2}}}\right)}{{a}^{2}} \]
   6. Step-by-step derivation

      1. lower-sqrt.64 N/A

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

   7. Applied rewrites 1.6%

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

   9. Applied rewrites 9.1%

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

   if 1.7000000000000001e-138 < y-scale < 3.75000000000000023e152

   1. Initial program 2.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in x-scale around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-pow.64 N/A

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

   10. Applied rewrites 8.0%

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

Alternative 3: 15.9% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  (/ 0.25 a)
  (/
   (*
    (* (fabs b) (* x-scale x-scale))
    (/
     (sqrt
      (*
       8.0
       (*
        (+
         (- 0.5 (* (cos (* (* angle (+ PI PI)) 0.005555555555555556)) 0.5))
         (sqrt (pow (sin (* (* angle PI) 0.005555555555555556)) 4.0)))
        (pow a 4.0))))
     (fabs x-scale)))
   a)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 / a) * (((fabs(b) * (x_45_scale * x_45_scale)) * (sqrt((8.0 * (((0.5 - (cos(((angle * (((double) M_PI) + ((double) M_PI))) * 0.005555555555555556)) * 0.5)) + sqrt(pow(sin(((angle * ((double) M_PI)) * 0.005555555555555556)), 4.0))) * pow(a, 4.0)))) / fabs(x_45_scale))) / a);
}

Java

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

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (0.25 / a) * (((math.fabs(b) * (x_45_scale * x_45_scale)) * (math.sqrt((8.0 * (((0.5 - (math.cos(((angle * (math.pi + math.pi)) * 0.005555555555555556)) * 0.5)) + math.sqrt(math.pow(math.sin(((angle * math.pi) * 0.005555555555555556)), 4.0))) * math.pow(a, 4.0)))) / math.fabs(x_45_scale))) / a)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(0.25 / a) * Float64(Float64(Float64(abs(b) * Float64(x_45_scale * x_45_scale)) * Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(0.5 - Float64(cos(Float64(Float64(angle * Float64(pi + pi)) * 0.005555555555555556)) * 0.5)) + sqrt((sin(Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 4.0))) * (a ^ 4.0)))) / abs(x_45_scale))) / a))
end

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(0.25 / a), $MachinePrecision] * N[(N[(N[(N[Abs[b], $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[(N[(0.5 - N[(N[Cos[N[(N[(angle * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.8%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 0.8%

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

5. Taylor expanded in y-scale around 0

   \[\leadsto 0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2}}}\right)}{{a}^{2}} \]
6. Step-by-step derivation

   1. lower-sqrt.64 N/A

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

7. Applied rewrites 1.6%

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

9. Applied rewrites 9.1%

   \[\leadsto \frac{0.25}{a} \cdot \color{blue}{\frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\left(\left(0.5 - \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) + \sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {a}^{4}\right)}}{\left|x-scale\right|}}{a}} \]
10. Add Preprocessing

Alternative 4: 14.7% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  0.25
  (*
   (/ (* (fabs b) (* x-scale x-scale)) a)
   (/
    (/
     (sqrt
      (*
       8.0
       (*
        (+
         (- 0.5 (* (cos (* (* angle (+ PI PI)) 0.005555555555555556)) 0.5))
         (sqrt (pow (sin (* (* angle PI) 0.005555555555555556)) 4.0)))
        (pow a 4.0))))
     (fabs x-scale))
    a))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * (((fabs(b) * (x_45_scale * x_45_scale)) / a) * ((sqrt((8.0 * (((0.5 - (cos(((angle * (((double) M_PI) + ((double) M_PI))) * 0.005555555555555556)) * 0.5)) + sqrt(pow(sin(((angle * ((double) M_PI)) * 0.005555555555555556)), 4.0))) * pow(a, 4.0)))) / fabs(x_45_scale)) / a));
}

Java

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

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 0.25 * (((math.fabs(b) * (x_45_scale * x_45_scale)) / a) * ((math.sqrt((8.0 * (((0.5 - (math.cos(((angle * (math.pi + math.pi)) * 0.005555555555555556)) * 0.5)) + math.sqrt(math.pow(math.sin(((angle * math.pi) * 0.005555555555555556)), 4.0))) * math.pow(a, 4.0)))) / math.fabs(x_45_scale)) / a))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(0.25 * Float64(Float64(Float64(abs(b) * Float64(x_45_scale * x_45_scale)) / a) * Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(0.5 - Float64(cos(Float64(Float64(angle * Float64(pi + pi)) * 0.005555555555555556)) * 0.5)) + sqrt((sin(Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 4.0))) * (a ^ 4.0)))) / abs(x_45_scale)) / a)))
end

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(0.25 * N[(N[(N[(N[Abs[b], $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * N[(N[(N[Sqrt[N[(8.0 * N[(N[(N[(0.5 - N[(N[Cos[N[(N[(angle * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.8%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 0.8%

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

5. Taylor expanded in y-scale around 0

   \[\leadsto 0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2}}}\right)}{{a}^{2}} \]
6. Step-by-step derivation

   1. lower-sqrt.64 N/A

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

7. Applied rewrites 1.6%

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

9. Applied rewrites 8.5%

   \[\leadsto 0.25 \cdot \left(\frac{b \cdot \left(x-scale \cdot x-scale\right)}{a} \cdot \color{blue}{\frac{\frac{\sqrt{8 \cdot \left(\left(\left(0.5 - \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) + \sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {a}^{4}\right)}}{\left|x-scale\right|}}{a}}\right) \]
10. Add Preprocessing

Alternative 5: 8.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\left(-\left|a\right|\right) \cdot \left|b\right|\right) \cdot \left|b\right|\right) \cdot \left|a\right|\\ t_1 := \left|a\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|a\right| \leq 2.3 \cdot 10^{-118}:\\ \;\;\;\;\left(\frac{\frac{\sqrt{\left(8 \cdot t\_0\right) \cdot \left(t\_0 \cdot \left(\sqrt{\frac{{\left(\left|b\right|\right)}^{4}}{{x-scale}^{4}}} + \frac{{\left(\left|b\right|\right)}^{2}}{{x-scale}^{2}}\right)\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(t\_1 \cdot 4\right) \cdot t\_1} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left|b\right| \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\left(0.5 - \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) + \sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {\left(\left|a\right|\right)}^{4}\right)}}{\left|x-scale\right|} \cdot \left(x-scale \cdot x-scale\right)}{\left|a\right| \cdot \left|a\right|}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (* (- (fabs a)) (fabs b)) (fabs b)) (fabs a)))
        (t_1 (* (fabs a) (fabs b))))
   (if (<= (fabs a) 2.3e-118)
     (*
      (*
       (/
        (/
         (sqrt
          (*
           (* 8.0 t_0)
           (*
            t_0
            (+
             (sqrt (/ (pow (fabs b) 4.0) (pow x-scale 4.0)))
             (/ (pow (fabs b) 2.0) (pow x-scale 2.0))))))
         (fabs (* y-scale x-scale)))
        (* (* t_1 4.0) t_1))
       (* y-scale x-scale))
      (* y-scale x-scale))
     (*
      0.25
      (*
       (fabs b)
       (/
        (*
         (/
          (sqrt
           (*
            8.0
            (*
             (+
              (-
               0.5
               (* (cos (* (* angle (+ PI PI)) 0.005555555555555556)) 0.5))
              (sqrt (pow (sin (* (* angle PI) 0.005555555555555556)) 4.0)))
             (pow (fabs a) 4.0))))
          (fabs x-scale))
         (* x-scale x-scale))
        (* (fabs a) (fabs a))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((-fabs(a) * fabs(b)) * fabs(b)) * fabs(a);
	double t_1 = fabs(a) * fabs(b);
	double tmp;
	if (fabs(a) <= 2.3e-118) {
		tmp = (((sqrt(((8.0 * t_0) * (t_0 * (sqrt((pow(fabs(b), 4.0) / pow(x_45_scale, 4.0))) + (pow(fabs(b), 2.0) / pow(x_45_scale, 2.0)))))) / fabs((y_45_scale * x_45_scale))) / ((t_1 * 4.0) * t_1)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
	} else {
		tmp = 0.25 * (fabs(b) * (((sqrt((8.0 * (((0.5 - (cos(((angle * (((double) M_PI) + ((double) M_PI))) * 0.005555555555555556)) * 0.5)) + sqrt(pow(sin(((angle * ((double) M_PI)) * 0.005555555555555556)), 4.0))) * pow(fabs(a), 4.0)))) / fabs(x_45_scale)) * (x_45_scale * x_45_scale)) / (fabs(a) * fabs(a))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((-Math.abs(a) * Math.abs(b)) * Math.abs(b)) * Math.abs(a);
	double t_1 = Math.abs(a) * Math.abs(b);
	double tmp;
	if (Math.abs(a) <= 2.3e-118) {
		tmp = (((Math.sqrt(((8.0 * t_0) * (t_0 * (Math.sqrt((Math.pow(Math.abs(b), 4.0) / Math.pow(x_45_scale, 4.0))) + (Math.pow(Math.abs(b), 2.0) / Math.pow(x_45_scale, 2.0)))))) / Math.abs((y_45_scale * x_45_scale))) / ((t_1 * 4.0) * t_1)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
	} else {
		tmp = 0.25 * (Math.abs(b) * (((Math.sqrt((8.0 * (((0.5 - (Math.cos(((angle * (Math.PI + Math.PI)) * 0.005555555555555556)) * 0.5)) + Math.sqrt(Math.pow(Math.sin(((angle * Math.PI) * 0.005555555555555556)), 4.0))) * Math.pow(Math.abs(a), 4.0)))) / Math.abs(x_45_scale)) * (x_45_scale * x_45_scale)) / (Math.abs(a) * Math.abs(a))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = ((-math.fabs(a) * math.fabs(b)) * math.fabs(b)) * math.fabs(a)
	t_1 = math.fabs(a) * math.fabs(b)
	tmp = 0
	if math.fabs(a) <= 2.3e-118:
		tmp = (((math.sqrt(((8.0 * t_0) * (t_0 * (math.sqrt((math.pow(math.fabs(b), 4.0) / math.pow(x_45_scale, 4.0))) + (math.pow(math.fabs(b), 2.0) / math.pow(x_45_scale, 2.0)))))) / math.fabs((y_45_scale * x_45_scale))) / ((t_1 * 4.0) * t_1)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale)
	else:
		tmp = 0.25 * (math.fabs(b) * (((math.sqrt((8.0 * (((0.5 - (math.cos(((angle * (math.pi + math.pi)) * 0.005555555555555556)) * 0.5)) + math.sqrt(math.pow(math.sin(((angle * math.pi) * 0.005555555555555556)), 4.0))) * math.pow(math.fabs(a), 4.0)))) / math.fabs(x_45_scale)) * (x_45_scale * x_45_scale)) / (math.fabs(a) * math.fabs(a))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(Float64(-abs(a)) * abs(b)) * abs(b)) * abs(a))
	t_1 = Float64(abs(a) * abs(b))
	tmp = 0.0
	if (abs(a) <= 2.3e-118)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(Float64(8.0 * t_0) * Float64(t_0 * Float64(sqrt(Float64((abs(b) ^ 4.0) / (x_45_scale ^ 4.0))) + Float64((abs(b) ^ 2.0) / (x_45_scale ^ 2.0)))))) / abs(Float64(y_45_scale * x_45_scale))) / Float64(Float64(t_1 * 4.0) * t_1)) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale));
	else
		tmp = Float64(0.25 * Float64(abs(b) * Float64(Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(0.5 - Float64(cos(Float64(Float64(angle * Float64(pi + pi)) * 0.005555555555555556)) * 0.5)) + sqrt((sin(Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 4.0))) * (abs(a) ^ 4.0)))) / abs(x_45_scale)) * Float64(x_45_scale * x_45_scale)) / Float64(abs(a) * abs(a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = ((-abs(a) * abs(b)) * abs(b)) * abs(a);
	t_1 = abs(a) * abs(b);
	tmp = 0.0;
	if (abs(a) <= 2.3e-118)
		tmp = (((sqrt(((8.0 * t_0) * (t_0 * (sqrt(((abs(b) ^ 4.0) / (x_45_scale ^ 4.0))) + ((abs(b) ^ 2.0) / (x_45_scale ^ 2.0)))))) / abs((y_45_scale * x_45_scale))) / ((t_1 * 4.0) * t_1)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
	else
		tmp = 0.25 * (abs(b) * (((sqrt((8.0 * (((0.5 - (cos(((angle * (pi + pi)) * 0.005555555555555556)) * 0.5)) + sqrt((sin(((angle * pi) * 0.005555555555555556)) ^ 4.0))) * (abs(a) ^ 4.0)))) / abs(x_45_scale)) * (x_45_scale * x_45_scale)) / (abs(a) * abs(a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[((-N[Abs[a], $MachinePrecision]) * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[a], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 2.3e-118], N[(N[(N[(N[(N[Sqrt[N[(N[(8.0 * t$95$0), $MachinePrecision] * N[(t$95$0 * N[(N[Sqrt[N[(N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(y$45$scale * x$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * 4.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[Abs[b], $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[(N[(0.5 - N[(N[Cos[N[(N[(angle * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.30000000000000021e-118

   1. Initial program 2.8%

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

   3. Applied rewrites 6.2%

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

   4. Taylor expanded in y-scale around inf

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} + \frac{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} + \frac{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 5.1%

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

   7. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.64 N/A

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

      5. lower-pow.64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.64 N/A

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

      8. lower-pow.64 4.1%

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

   9. Applied rewrites 4.1%

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

   if 2.30000000000000021e-118 < a

   1. Initial program 2.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in y-scale around 0

      \[\leadsto 0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2}}}\right)}{{a}^{2}} \]
   6. Step-by-step derivation

      1. lower-sqrt.64 N/A

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

   7. Applied rewrites 1.6%

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

   9. Applied rewrites 3.6%

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

Alternative 6: 5.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\left(-\left|a\right|\right) \cdot \left|b\right|\right) \cdot \left|b\right|\right) \cdot \left|a\right|\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \left|a\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|a\right| \leq 6.5 \cdot 10^{-72}:\\ \;\;\;\;\left(\frac{\frac{\sqrt{\left(8 \cdot t\_0\right) \cdot \left(t\_0 \cdot \left(\sqrt{\frac{{\left(\left|b\right|\right)}^{4}}{{x-scale}^{4}}} + \frac{{\left(\left|b\right|\right)}^{2}}{{x-scale}^{2}}\right)\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(t\_2 \cdot 4\right) \cdot t\_2} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left|b\right| \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\left(\left|a\right|\right)}^{4} \cdot \left(\sqrt{{t\_1}^{4}} + {t\_1}^{2}\right)}{{x-scale}^{2}}}\right)}{{\left(\left|a\right|\right)}^{2}}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (* (- (fabs a)) (fabs b)) (fabs b)) (fabs a)))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (* (fabs a) (fabs b))))
   (if (<= (fabs a) 6.5e-72)
     (*
      (*
       (/
        (/
         (sqrt
          (*
           (* 8.0 t_0)
           (*
            t_0
            (+
             (sqrt (/ (pow (fabs b) 4.0) (pow x-scale 4.0)))
             (/ (pow (fabs b) 2.0) (pow x-scale 2.0))))))
         (fabs (* y-scale x-scale)))
        (* (* t_2 4.0) t_2))
       (* y-scale x-scale))
      (* y-scale x-scale))
     (*
      0.25
      (/
       (*
        (fabs b)
        (*
         (pow x-scale 2.0)
         (sqrt
          (*
           8.0
           (/
            (* (pow (fabs a) 4.0) (+ (sqrt (pow t_1 4.0)) (pow t_1 2.0)))
            (pow x-scale 2.0))))))
       (pow (fabs a) 2.0))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((-fabs(a) * fabs(b)) * fabs(b)) * fabs(a);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = fabs(a) * fabs(b);
	double tmp;
	if (fabs(a) <= 6.5e-72) {
		tmp = (((sqrt(((8.0 * t_0) * (t_0 * (sqrt((pow(fabs(b), 4.0) / pow(x_45_scale, 4.0))) + (pow(fabs(b), 2.0) / pow(x_45_scale, 2.0)))))) / fabs((y_45_scale * x_45_scale))) / ((t_2 * 4.0) * t_2)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
	} else {
		tmp = 0.25 * ((fabs(b) * (pow(x_45_scale, 2.0) * sqrt((8.0 * ((pow(fabs(a), 4.0) * (sqrt(pow(t_1, 4.0)) + pow(t_1, 2.0))) / pow(x_45_scale, 2.0)))))) / pow(fabs(a), 2.0));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((-Math.abs(a) * Math.abs(b)) * Math.abs(b)) * Math.abs(a);
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = Math.abs(a) * Math.abs(b);
	double tmp;
	if (Math.abs(a) <= 6.5e-72) {
		tmp = (((Math.sqrt(((8.0 * t_0) * (t_0 * (Math.sqrt((Math.pow(Math.abs(b), 4.0) / Math.pow(x_45_scale, 4.0))) + (Math.pow(Math.abs(b), 2.0) / Math.pow(x_45_scale, 2.0)))))) / Math.abs((y_45_scale * x_45_scale))) / ((t_2 * 4.0) * t_2)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
	} else {
		tmp = 0.25 * ((Math.abs(b) * (Math.pow(x_45_scale, 2.0) * Math.sqrt((8.0 * ((Math.pow(Math.abs(a), 4.0) * (Math.sqrt(Math.pow(t_1, 4.0)) + Math.pow(t_1, 2.0))) / Math.pow(x_45_scale, 2.0)))))) / Math.pow(Math.abs(a), 2.0));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = ((-math.fabs(a) * math.fabs(b)) * math.fabs(b)) * math.fabs(a)
	t_1 = 0.005555555555555556 * (angle * math.pi)
	t_2 = math.fabs(a) * math.fabs(b)
	tmp = 0
	if math.fabs(a) <= 6.5e-72:
		tmp = (((math.sqrt(((8.0 * t_0) * (t_0 * (math.sqrt((math.pow(math.fabs(b), 4.0) / math.pow(x_45_scale, 4.0))) + (math.pow(math.fabs(b), 2.0) / math.pow(x_45_scale, 2.0)))))) / math.fabs((y_45_scale * x_45_scale))) / ((t_2 * 4.0) * t_2)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale)
	else:
		tmp = 0.25 * ((math.fabs(b) * (math.pow(x_45_scale, 2.0) * math.sqrt((8.0 * ((math.pow(math.fabs(a), 4.0) * (math.sqrt(math.pow(t_1, 4.0)) + math.pow(t_1, 2.0))) / math.pow(x_45_scale, 2.0)))))) / math.pow(math.fabs(a), 2.0))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(Float64(-abs(a)) * abs(b)) * abs(b)) * abs(a))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = Float64(abs(a) * abs(b))
	tmp = 0.0
	if (abs(a) <= 6.5e-72)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(Float64(8.0 * t_0) * Float64(t_0 * Float64(sqrt(Float64((abs(b) ^ 4.0) / (x_45_scale ^ 4.0))) + Float64((abs(b) ^ 2.0) / (x_45_scale ^ 2.0)))))) / abs(Float64(y_45_scale * x_45_scale))) / Float64(Float64(t_2 * 4.0) * t_2)) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale));
	else
		tmp = Float64(0.25 * Float64(Float64(abs(b) * Float64((x_45_scale ^ 2.0) * sqrt(Float64(8.0 * Float64(Float64((abs(a) ^ 4.0) * Float64(sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0))) / (x_45_scale ^ 2.0)))))) / (abs(a) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = ((-abs(a) * abs(b)) * abs(b)) * abs(a);
	t_1 = 0.005555555555555556 * (angle * pi);
	t_2 = abs(a) * abs(b);
	tmp = 0.0;
	if (abs(a) <= 6.5e-72)
		tmp = (((sqrt(((8.0 * t_0) * (t_0 * (sqrt(((abs(b) ^ 4.0) / (x_45_scale ^ 4.0))) + ((abs(b) ^ 2.0) / (x_45_scale ^ 2.0)))))) / abs((y_45_scale * x_45_scale))) / ((t_2 * 4.0) * t_2)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
	else
		tmp = 0.25 * ((abs(b) * ((x_45_scale ^ 2.0) * sqrt((8.0 * (((abs(a) ^ 4.0) * (sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0))) / (x_45_scale ^ 2.0)))))) / (abs(a) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[((-N[Abs[a], $MachinePrecision]) * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[a], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 6.5e-72], N[(N[(N[(N[(N[Sqrt[N[(N[(8.0 * t$95$0), $MachinePrecision] * N[(t$95$0 * N[(N[Sqrt[N[(N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(y$45$scale * x$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[Abs[b], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.4999999999999997e-72

   1. Initial program 2.8%

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

   3. Applied rewrites 6.2%

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

   4. Taylor expanded in y-scale around inf

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} + \frac{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} + \frac{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 5.1%

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

   7. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.64 N/A

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

      5. lower-pow.64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.64 N/A

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

      8. lower-pow.64 4.1%

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

   9. Applied rewrites 4.1%

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

   if 6.4999999999999997e-72 < a

   1. Initial program 2.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in y-scale around 0

      \[\leadsto 0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2}}}\right)}{{a}^{2}} \]
   6. Step-by-step derivation

      1. lower-sqrt.64 N/A

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2}}}\right)}{{a}^{2}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 1.6%

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

   10. Applied rewrites 1.6%

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

   11. Taylor expanded in angle around 0

       \[\leadsto 0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2}}}\right)}{{a}^{2}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 1.6%

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

   13. Applied rewrites 1.6%

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

Alternative 7: 5.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|y-scale\right|\right)}^{2}\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;\left|y-scale\right| \leq 9 \cdot 10^{+144}:\\ \;\;\;\;0.25 \cdot \left(\left|b\right| \cdot \left({x-scale}^{2} \cdot \left(t\_0 \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}}{{x-scale}^{2} \cdot t\_0}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left|b\right| \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{t\_1}^{4}} + {t\_1}^{2}\right)}{{x-scale}^{2}}}\right)}{{a}^{2}}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs y-scale) 2.0))
        (t_1 (* 0.005555555555555556 (* angle PI))))
   (if (<= (fabs y-scale) 9e+144)
     (*
      0.25
      (*
       (fabs b)
       (*
        (pow x-scale 2.0)
        (*
         t_0
         (sqrt
          (*
           8.0
           (/
            (+ (sqrt (/ 1.0 (pow x-scale 4.0))) (/ 1.0 (pow x-scale 2.0)))
            (* (pow x-scale 2.0) t_0))))))))
     (*
      0.25
      (/
       (*
        (fabs b)
        (*
         (pow x-scale 2.0)
         (sqrt
          (*
           8.0
           (/
            (* (pow a 4.0) (+ (sqrt (pow t_1 4.0)) (pow t_1 2.0)))
            (pow x-scale 2.0))))))
       (pow a 2.0))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(y_45_scale), 2.0);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (fabs(y_45_scale) <= 9e+144) {
		tmp = 0.25 * (fabs(b) * (pow(x_45_scale, 2.0) * (t_0 * sqrt((8.0 * ((sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0))) / (pow(x_45_scale, 2.0) * t_0)))))));
	} else {
		tmp = 0.25 * ((fabs(b) * (pow(x_45_scale, 2.0) * sqrt((8.0 * ((pow(a, 4.0) * (sqrt(pow(t_1, 4.0)) + pow(t_1, 2.0))) / pow(x_45_scale, 2.0)))))) / pow(a, 2.0));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.pow(Math.abs(y_45_scale), 2.0);
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (Math.abs(y_45_scale) <= 9e+144) {
		tmp = 0.25 * (Math.abs(b) * (Math.pow(x_45_scale, 2.0) * (t_0 * Math.sqrt((8.0 * ((Math.sqrt((1.0 / Math.pow(x_45_scale, 4.0))) + (1.0 / Math.pow(x_45_scale, 2.0))) / (Math.pow(x_45_scale, 2.0) * t_0)))))));
	} else {
		tmp = 0.25 * ((Math.abs(b) * (Math.pow(x_45_scale, 2.0) * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt(Math.pow(t_1, 4.0)) + Math.pow(t_1, 2.0))) / Math.pow(x_45_scale, 2.0)))))) / Math.pow(a, 2.0));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pow(math.fabs(y_45_scale), 2.0)
	t_1 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if math.fabs(y_45_scale) <= 9e+144:
		tmp = 0.25 * (math.fabs(b) * (math.pow(x_45_scale, 2.0) * (t_0 * math.sqrt((8.0 * ((math.sqrt((1.0 / math.pow(x_45_scale, 4.0))) + (1.0 / math.pow(x_45_scale, 2.0))) / (math.pow(x_45_scale, 2.0) * t_0)))))))
	else:
		tmp = 0.25 * ((math.fabs(b) * (math.pow(x_45_scale, 2.0) * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.sqrt(math.pow(t_1, 4.0)) + math.pow(t_1, 2.0))) / math.pow(x_45_scale, 2.0)))))) / math.pow(a, 2.0))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (abs(y_45_scale) <= 9e+144)
		tmp = Float64(0.25 * Float64(abs(b) * Float64((x_45_scale ^ 2.0) * Float64(t_0 * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0))) / Float64((x_45_scale ^ 2.0) * t_0))))))));
	else
		tmp = Float64(0.25 * Float64(Float64(abs(b) * Float64((x_45_scale ^ 2.0) * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0))) / (x_45_scale ^ 2.0)))))) / (a ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0;
	t_1 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (abs(y_45_scale) <= 9e+144)
		tmp = 0.25 * (abs(b) * ((x_45_scale ^ 2.0) * (t_0 * sqrt((8.0 * ((sqrt((1.0 / (x_45_scale ^ 4.0))) + (1.0 / (x_45_scale ^ 2.0))) / ((x_45_scale ^ 2.0) * t_0)))))));
	else
		tmp = 0.25 * ((abs(b) * ((x_45_scale ^ 2.0) * sqrt((8.0 * (((a ^ 4.0) * (sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0))) / (x_45_scale ^ 2.0)))))) / (a ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 9e+144], N[(0.25 * N[(N[Abs[b], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[Abs[b], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 8.99999999999999935e144

   1. Initial program 2.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-sqrt.64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.64 2.2%

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

   10. Applied rewrites 2.2%

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

   if 8.99999999999999935e144 < y-scale

   1. Initial program 2.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in y-scale around 0

      \[\leadsto 0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2}}}\right)}{{a}^{2}} \]
   6. Step-by-step derivation

      1. lower-sqrt.64 N/A

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2}}}\right)}{{a}^{2}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 1.6%

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

   10. Applied rewrites 1.6%

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

   11. Taylor expanded in angle around 0

       \[\leadsto 0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2}}}\right)}{{a}^{2}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 1.6%

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

   13. Applied rewrites 1.6%

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

Alternative 8: 4.0% accurate, 5.8× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  0.25
  (*
   (fabs b)
   (*
    (pow x-scale 2.0)
    (*
     (pow y-scale 2.0)
     (sqrt
      (*
       8.0
       (/
        (+ (sqrt (/ 1.0 (pow x-scale 4.0))) (/ 1.0 (pow x-scale 2.0)))
        (* (pow x-scale 2.0) (pow y-scale 2.0))))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * (fabs(b) * (pow(x_45_scale, 2.0) * (pow(y_45_scale, 2.0) * sqrt((8.0 * ((sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0))) / (pow(x_45_scale, 2.0) * pow(y_45_scale, 2.0))))))));
}

Fortran

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, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = 0.25d0 * (abs(b) * ((x_45scale ** 2.0d0) * ((y_45scale ** 2.0d0) * sqrt((8.0d0 * ((sqrt((1.0d0 / (x_45scale ** 4.0d0))) + (1.0d0 / (x_45scale ** 2.0d0))) / ((x_45scale ** 2.0d0) * (y_45scale ** 2.0d0))))))))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * (Math.abs(b) * (Math.pow(x_45_scale, 2.0) * (Math.pow(y_45_scale, 2.0) * Math.sqrt((8.0 * ((Math.sqrt((1.0 / Math.pow(x_45_scale, 4.0))) + (1.0 / Math.pow(x_45_scale, 2.0))) / (Math.pow(x_45_scale, 2.0) * Math.pow(y_45_scale, 2.0))))))));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 0.25 * (math.fabs(b) * (math.pow(x_45_scale, 2.0) * (math.pow(y_45_scale, 2.0) * math.sqrt((8.0 * ((math.sqrt((1.0 / math.pow(x_45_scale, 4.0))) + (1.0 / math.pow(x_45_scale, 2.0))) / (math.pow(x_45_scale, 2.0) * math.pow(y_45_scale, 2.0))))))))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(0.25 * Float64(abs(b) * Float64((x_45_scale ^ 2.0) * Float64((y_45_scale ^ 2.0) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0))) / Float64((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0)))))))))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.25 * (abs(b) * ((x_45_scale ^ 2.0) * ((y_45_scale ^ 2.0) * sqrt((8.0 * ((sqrt((1.0 / (x_45_scale ^ 4.0))) + (1.0 / (x_45_scale ^ 2.0))) / ((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0))))))));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(0.25 * N[(N[Abs[b], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.8%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 0.8%

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

5. Taylor expanded in a around 0

   \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

7. Applied rewrites 2.7%

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

8. Taylor expanded in angle around 0

   \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)\right) \]
9. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-sqrt.64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.64 2.2%

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

10. Applied rewrites 2.2%

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

Alternative 9: 3.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{b}{x-scale \cdot x-scale}\\ t_1 := \frac{a}{y-scale \cdot y-scale}\\ t_2 := \left(-a\right) \cdot b\\ \frac{\frac{-\sqrt{\left(\left(\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 4\right) \cdot 2\right) \cdot \left(\left(t\_2 \cdot b\right) \cdot a\right)\right) \cdot \mathsf{fma}\left(b, t\_0, \mathsf{fma}\left(a, t\_1, \left|a \cdot t\_1 - b \cdot t\_0\right|\right)\right)}}{4 \cdot \left(a \cdot b\right)}}{t\_2} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* x-scale x-scale)))
        (t_1 (/ a (* y-scale y-scale)))
        (t_2 (* (- a) b)))
   (*
    (/
     (/
      (-
       (sqrt
        (*
         (*
          (*
           (*
            (*
             (* (* a b) b)
             (/ (- a) (* (* (* y-scale y-scale) x-scale) x-scale)))
            4.0)
           2.0)
          (* (* t_2 b) a))
         (fma b t_0 (fma a t_1 (fabs (- (* a t_1) (* b t_0))))))))
      (* 4.0 (* a b)))
     t_2)
    (* (* y-scale y-scale) (* x-scale x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (x_45_scale * x_45_scale);
	double t_1 = a / (y_45_scale * y_45_scale);
	double t_2 = -a * b;
	return ((-sqrt((((((((a * b) * b) * (-a / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * 4.0) * 2.0) * ((t_2 * b) * a)) * fma(b, t_0, fma(a, t_1, fabs(((a * t_1) - (b * t_0))))))) / (4.0 * (a * b))) / t_2) * ((y_45_scale * y_45_scale) * (x_45_scale * x_45_scale));
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(x_45_scale * x_45_scale))
	t_1 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_2 = Float64(Float64(-a) * b)
	return Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * b) * b) * Float64(Float64(-a) / Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * 4.0) * 2.0) * Float64(Float64(t_2 * b) * a)) * fma(b, t_0, fma(a, t_1, abs(Float64(Float64(a * t_1) - Float64(b * t_0)))))))) / Float64(4.0 * Float64(a * b))) / t_2) * Float64(Float64(y_45_scale * y_45_scale) * Float64(x_45_scale * x_45_scale)))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-a) * b), $MachinePrecision]}, N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(N[(N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision] * N[((-a) / N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(t$95$2 * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[(b * t$95$0 + N[(a * t$95$1 + N[Abs[N[(N[(a * t$95$1), $MachinePrecision] - N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 2.8%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 4.3%

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

6. Applied rewrites 1.9%

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

8. Applied rewrites 3.4%

   \[\leadsto \color{blue}{\frac{\frac{-\sqrt{\left(\left(\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 4\right) \cdot 2\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{x-scale \cdot x-scale}, \mathsf{fma}\left(a, \frac{a}{y-scale \cdot y-scale}, \left|a \cdot \frac{a}{y-scale \cdot y-scale} - b \cdot \frac{b}{x-scale \cdot x-scale}\right|\right)\right)}}{4 \cdot \left(a \cdot b\right)}}{\left(-a\right) \cdot b}} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]
9. Add Preprocessing

Alternative 10: 3.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} t_0 := \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot x-scale\\ \frac{\sqrt{\left(\left(\left(\left(\frac{-a}{t\_0} \cdot \left(\left(a \cdot b\right) \cdot b\right)\right) \cdot 8\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \mathsf{fma}\left(\frac{b}{x-scale \cdot x-scale}, b, \mathsf{fma}\left(\frac{a}{y-scale \cdot y-scale}, a, \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right)}}{\left(\left(4 \cdot \left(a \cdot b\right)\right) \cdot b\right) \cdot a} \cdot t\_0 \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* y-scale (* x-scale y-scale)) x-scale)))
   (*
    (/
     (sqrt
      (*
       (* (* (* (* (/ (- a) t_0) (* (* a b) b)) 8.0) (* (* (- a) b) b)) a)
       (fma
        (/ b (* x-scale x-scale))
        b
        (fma
         (/ a (* y-scale y-scale))
         a
         (fabs
          (-
           (/ (* b b) (* x-scale x-scale))
           (/ (* a a) (* y-scale y-scale))))))))
     (* (* (* 4.0 (* a b)) b) a))
    t_0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (y_45_scale * (x_45_scale * y_45_scale)) * x_45_scale;
	return (sqrt(((((((-a / t_0) * ((a * b) * b)) * 8.0) * ((-a * b) * b)) * a) * fma((b / (x_45_scale * x_45_scale)), b, fma((a / (y_45_scale * y_45_scale)), a, fabs((((b * b) / (x_45_scale * x_45_scale)) - ((a * a) / (y_45_scale * y_45_scale)))))))) / (((4.0 * (a * b)) * b) * a)) * t_0;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(y_45_scale * Float64(x_45_scale * y_45_scale)) * x_45_scale)
	return Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-a) / t_0) * Float64(Float64(a * b) * b)) * 8.0) * Float64(Float64(Float64(-a) * b) * b)) * a) * fma(Float64(b / Float64(x_45_scale * x_45_scale)), b, fma(Float64(a / Float64(y_45_scale * y_45_scale)), a, abs(Float64(Float64(Float64(b * b) / Float64(x_45_scale * x_45_scale)) - Float64(Float64(a * a) / Float64(y_45_scale * y_45_scale)))))))) / Float64(Float64(Float64(4.0 * Float64(a * b)) * b) * a)) * t_0)
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]}, N[(N[(N[Sqrt[N[(N[(N[(N[(N[(N[((-a) / t$95$0), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision] * N[(N[((-a) * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * N[(N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * b + N[(N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a + N[Abs[N[(N[(N[(b * b), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 2.8%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 4.3%

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

6. Applied rewrites 1.9%

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

8. Applied rewrites 2.0%

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

10. Applied rewrites 3.2%

    \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\left(\left(\frac{-a}{\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot x-scale} \cdot \left(\left(a \cdot b\right) \cdot b\right)\right) \cdot 8\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \mathsf{fma}\left(\frac{b}{x-scale \cdot x-scale}, b, \mathsf{fma}\left(\frac{a}{y-scale \cdot y-scale}, a, \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right)}}{\left(\left(4 \cdot \left(a \cdot b\right)\right) \cdot b\right) \cdot a} \cdot \left(\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot x-scale\right)} \]
11. Add Preprocessing

Alternative 11: 3.0% accurate, 6.6× speedup

Math

\[\left(\left(\frac{\sqrt{\left(\left(\left(\left(\frac{-a}{\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot x-scale} \cdot \left(\left(a \cdot b\right) \cdot b\right)\right) \cdot 8\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \mathsf{fma}\left(\frac{b}{x-scale \cdot x-scale}, b, \mathsf{fma}\left(\frac{a}{y-scale \cdot y-scale}, a, \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right)}}{\left(\left(4 \cdot \left(a \cdot b\right)\right) \cdot b\right) \cdot a} \cdot y-scale\right) \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right) \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  (*
   (*
    (/
     (sqrt
      (*
       (*
        (*
         (*
          (*
           (/ (- a) (* (* y-scale (* x-scale y-scale)) x-scale))
           (* (* a b) b))
          8.0)
         (* (* (- a) b) b))
        a)
       (fma
        (/ b (* x-scale x-scale))
        b
        (fma
         (/ a (* y-scale y-scale))
         a
         (fabs
          (-
           (/ (* b b) (* x-scale x-scale))
           (/ (* a a) (* y-scale y-scale))))))))
     (* (* (* 4.0 (* a b)) b) a))
    y-scale)
   y-scale)
  (* x-scale x-scale)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (((sqrt(((((((-a / ((y_45_scale * (x_45_scale * y_45_scale)) * x_45_scale)) * ((a * b) * b)) * 8.0) * ((-a * b) * b)) * a) * fma((b / (x_45_scale * x_45_scale)), b, fma((a / (y_45_scale * y_45_scale)), a, fabs((((b * b) / (x_45_scale * x_45_scale)) - ((a * a) / (y_45_scale * y_45_scale)))))))) / (((4.0 * (a * b)) * b) * a)) * y_45_scale) * y_45_scale) * (x_45_scale * x_45_scale);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-a) / Float64(Float64(y_45_scale * Float64(x_45_scale * y_45_scale)) * x_45_scale)) * Float64(Float64(a * b) * b)) * 8.0) * Float64(Float64(Float64(-a) * b) * b)) * a) * fma(Float64(b / Float64(x_45_scale * x_45_scale)), b, fma(Float64(a / Float64(y_45_scale * y_45_scale)), a, abs(Float64(Float64(Float64(b * b) / Float64(x_45_scale * x_45_scale)) - Float64(Float64(a * a) / Float64(y_45_scale * y_45_scale)))))))) / Float64(Float64(Float64(4.0 * Float64(a * b)) * b) * a)) * y_45_scale) * y_45_scale) * Float64(x_45_scale * x_45_scale))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[(N[(N[((-a) / N[(N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision] * N[(N[((-a) * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * N[(N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * b + N[(N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a + N[Abs[N[(N[(N[(b * b), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.8%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 4.3%

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

6. Applied rewrites 1.9%

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

8. Applied rewrites 2.0%

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

10. Applied rewrites 3.0%

    \[\leadsto \color{blue}{\left(\left(\frac{\sqrt{\left(\left(\left(\left(\frac{-a}{\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot x-scale} \cdot \left(\left(a \cdot b\right) \cdot b\right)\right) \cdot 8\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \mathsf{fma}\left(\frac{b}{x-scale \cdot x-scale}, b, \mathsf{fma}\left(\frac{a}{y-scale \cdot y-scale}, a, \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right)}}{\left(\left(4 \cdot \left(a \cdot b\right)\right) \cdot b\right) \cdot a} \cdot y-scale\right) \cdot y-scale\right)} \cdot \left(x-scale \cdot x-scale\right) \]
11. Add Preprocessing

Alternative 12: 1.8% accurate, 6.6× speedup

Math

\[\left(\frac{\sqrt{\left(\left(\left(\left(\frac{-a}{\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot x-scale} \cdot \left(\left(a \cdot b\right) \cdot b\right)\right) \cdot 8\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \mathsf{fma}\left(\frac{b}{x-scale \cdot x-scale}, b, \mathsf{fma}\left(\frac{a}{y-scale \cdot y-scale}, a, \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right)}}{\left(\left(4 \cdot \left(a \cdot b\right)\right) \cdot b\right) \cdot a} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right) \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  (*
   (/
    (sqrt
     (*
      (*
       (*
        (*
         (*
          (/ (- a) (* (* y-scale (* x-scale y-scale)) x-scale))
          (* (* a b) b))
         8.0)
        (* (* (- a) b) b))
       a)
      (fma
       (/ b (* x-scale x-scale))
       b
       (fma
        (/ a (* y-scale y-scale))
        a
        (fabs
         (-
          (/ (* b b) (* x-scale x-scale))
          (/ (* a a) (* y-scale y-scale))))))))
    (* (* (* 4.0 (* a b)) b) a))
   (* y-scale y-scale))
  (* x-scale x-scale)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((sqrt(((((((-a / ((y_45_scale * (x_45_scale * y_45_scale)) * x_45_scale)) * ((a * b) * b)) * 8.0) * ((-a * b) * b)) * a) * fma((b / (x_45_scale * x_45_scale)), b, fma((a / (y_45_scale * y_45_scale)), a, fabs((((b * b) / (x_45_scale * x_45_scale)) - ((a * a) / (y_45_scale * y_45_scale)))))))) / (((4.0 * (a * b)) * b) * a)) * (y_45_scale * y_45_scale)) * (x_45_scale * x_45_scale);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-a) / Float64(Float64(y_45_scale * Float64(x_45_scale * y_45_scale)) * x_45_scale)) * Float64(Float64(a * b) * b)) * 8.0) * Float64(Float64(Float64(-a) * b) * b)) * a) * fma(Float64(b / Float64(x_45_scale * x_45_scale)), b, fma(Float64(a / Float64(y_45_scale * y_45_scale)), a, abs(Float64(Float64(Float64(b * b) / Float64(x_45_scale * x_45_scale)) - Float64(Float64(a * a) / Float64(y_45_scale * y_45_scale)))))))) / Float64(Float64(Float64(4.0 * Float64(a * b)) * b) * a)) * Float64(y_45_scale * y_45_scale)) * Float64(x_45_scale * x_45_scale))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[Sqrt[N[(N[(N[(N[(N[(N[((-a) / N[(N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision] * N[(N[((-a) * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * N[(N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * b + N[(N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a + N[Abs[N[(N[(N[(b * b), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.8%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 4.3%

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

6. Applied rewrites 1.9%

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

8. Applied rewrites 2.0%

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

10. Applied rewrites 1.8%

    \[\leadsto \color{blue}{\left(\frac{\sqrt{\left(\left(\left(\left(\frac{-a}{\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot x-scale} \cdot \left(\left(a \cdot b\right) \cdot b\right)\right) \cdot 8\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \mathsf{fma}\left(\frac{b}{x-scale \cdot x-scale}, b, \mathsf{fma}\left(\frac{a}{y-scale \cdot y-scale}, a, \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right)}}{\left(\left(4 \cdot \left(a \cdot b\right)\right) \cdot b\right) \cdot a} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(FPCore (a b angle x-scale y-scale)
  :name "a 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))))