a from scale-rotated-ellipse

Percentage Accurate: 2.6% → 24.3%

Time: 29.7s

Alternatives: 10

Speedup: 12.2×

• 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: 2.6% 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: 24.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\sqrt{\left(8 \cdot \left(\sqrt{{\left(\cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right)}^{4}} + \left(\left(0.5 - \cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot -0.5\right) \cdot a\right) \cdot a\right)\right) \cdot {a}^{4}}}{\left|\left|x-scale\right|\right|}\\ \mathbf{if}\;\left|x-scale\right| \leq 3 \cdot 10^{-164}:\\ \;\;\;\;\left(\left|x-scale\right| \cdot \left(\left|x-scale\right| \cdot \frac{t\_0}{a \cdot a}\right)\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|x-scale\right| \cdot \left|x-scale\right|\right) \cdot \frac{\frac{t\_0}{a}}{a}\right) \cdot 0.25\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0
        (/
         (sqrt
          (*
           (*
            8.0
            (+
             (sqrt
              (pow
               (* (cos (* (* PI 0.005555555555555556) angle)) a)
               4.0))
             (*
              (*
               (-
                0.5
                (* (cos (* (* 0.011111111111111112 angle) PI)) -0.5))
               a)
              a)))
           (pow a 4.0)))
         (fabs (fabs x-scale)))))
  (if (<= (fabs x-scale) 3e-164)
    (* (* (fabs x-scale) (* (fabs x-scale) (/ t_0 (* a a)))) 0.25)
    (* (* (* (fabs x-scale) (fabs x-scale)) (/ (/ t_0 a) a)) 0.25))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sqrt(((8.0 * (sqrt(pow((cos(((((double) M_PI) * 0.005555555555555556) * angle)) * a), 4.0)) + (((0.5 - (cos(((0.011111111111111112 * angle) * ((double) M_PI))) * -0.5)) * a) * a))) * pow(a, 4.0))) / fabs(fabs(x_45_scale));
	double tmp;
	if (fabs(x_45_scale) <= 3e-164) {
		tmp = (fabs(x_45_scale) * (fabs(x_45_scale) * (t_0 / (a * a)))) * 0.25;
	} else {
		tmp = ((fabs(x_45_scale) * fabs(x_45_scale)) * ((t_0 / a) / a)) * 0.25;
	}
	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.sqrt(((8.0 * (Math.sqrt(Math.pow((Math.cos(((Math.PI * 0.005555555555555556) * angle)) * a), 4.0)) + (((0.5 - (Math.cos(((0.011111111111111112 * angle) * Math.PI)) * -0.5)) * a) * a))) * Math.pow(a, 4.0))) / Math.abs(Math.abs(x_45_scale));
	double tmp;
	if (Math.abs(x_45_scale) <= 3e-164) {
		tmp = (Math.abs(x_45_scale) * (Math.abs(x_45_scale) * (t_0 / (a * a)))) * 0.25;
	} else {
		tmp = ((Math.abs(x_45_scale) * Math.abs(x_45_scale)) * ((t_0 / a) / a)) * 0.25;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.sqrt(((8.0 * (math.sqrt(math.pow((math.cos(((math.pi * 0.005555555555555556) * angle)) * a), 4.0)) + (((0.5 - (math.cos(((0.011111111111111112 * angle) * math.pi)) * -0.5)) * a) * a))) * math.pow(a, 4.0))) / math.fabs(math.fabs(x_45_scale))
	tmp = 0
	if math.fabs(x_45_scale) <= 3e-164:
		tmp = (math.fabs(x_45_scale) * (math.fabs(x_45_scale) * (t_0 / (a * a)))) * 0.25
	else:
		tmp = ((math.fabs(x_45_scale) * math.fabs(x_45_scale)) * ((t_0 / a) / a)) * 0.25
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(sqrt(Float64(Float64(8.0 * Float64(sqrt((Float64(cos(Float64(Float64(pi * 0.005555555555555556) * angle)) * a) ^ 4.0)) + Float64(Float64(Float64(0.5 - Float64(cos(Float64(Float64(0.011111111111111112 * angle) * pi)) * -0.5)) * a) * a))) * (a ^ 4.0))) / abs(abs(x_45_scale)))
	tmp = 0.0
	if (abs(x_45_scale) <= 3e-164)
		tmp = Float64(Float64(abs(x_45_scale) * Float64(abs(x_45_scale) * Float64(t_0 / Float64(a * a)))) * 0.25);
	else
		tmp = Float64(Float64(Float64(abs(x_45_scale) * abs(x_45_scale)) * Float64(Float64(t_0 / a) / a)) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sqrt(((8.0 * (sqrt(((cos(((pi * 0.005555555555555556) * angle)) * a) ^ 4.0)) + (((0.5 - (cos(((0.011111111111111112 * angle) * pi)) * -0.5)) * a) * a))) * (a ^ 4.0))) / abs(abs(x_45_scale));
	tmp = 0.0;
	if (abs(x_45_scale) <= 3e-164)
		tmp = (abs(x_45_scale) * (abs(x_45_scale) * (t_0 / (a * a)))) * 0.25;
	else
		tmp = ((abs(x_45_scale) * abs(x_45_scale)) * ((t_0 / a) / a)) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(8.0 * N[(N[Sqrt[N[Power[N[(N[Cos[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + N[(N[(N[(0.5 - N[(N[Cos[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[Abs[x$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 3e-164], N[(N[(N[Abs[x$45$scale], $MachinePrecision] * N[(N[Abs[x$45$scale], $MachinePrecision] * N[(t$95$0 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / a), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 3.0000000000000001e-164

   1. Initial program 2.6%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 1.1%

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

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 4.1%

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

   9. Applied rewrites 9.7%

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

   11. Applied rewrites 11.2%

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

   if 3.0000000000000001e-164 < x-scale

   1. Initial program 2.6%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 1.1%

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

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 4.1%

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

   9. Applied rewrites 9.7%

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

   11. Applied rewrites 23.6%

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

Alternative 2: 14.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} t_0 := \left(b \cdot \left|a\right|\right) \cdot 4\\ t_1 := \left(-\left|a\right|\right) \cdot b\\ t_2 := b \cdot \frac{b}{x-scale \cdot x-scale}\\ t_3 := \left|a\right| \cdot \frac{\left|a\right|}{y-scale \cdot y-scale}\\ t_4 := \left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale\\ \mathbf{if}\;\left|a\right| \leq 1.6 \cdot 10^{+153}:\\ \;\;\;\;\left(x-scale \cdot \left(x-scale \cdot \frac{\frac{\sqrt{\left(8 \cdot \left(\sqrt{{\left(\cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \left|a\right|\right)}^{4}} + \left(\left(0.5 - \cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot -0.5\right) \cdot \left|a\right|\right) \cdot \left|a\right|\right)\right) \cdot {\left(\left|a\right|\right)}^{4}}}{\left|x-scale\right|}}{\left|a\right| \cdot \left|a\right|}\right)\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(\left|t\_3 - t\_2\right| + \left(t\_2 + t\_3\right)\right) \cdot \left(\left(t\_0 \cdot \frac{t\_1}{t\_4}\right) \cdot \left(2 \cdot \left(\left(t\_1 \cdot b\right) \cdot \left|a\right|\right)\right)\right)}}{t\_0}}{t\_1} \cdot t\_4\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (* b (fabs a)) 4.0))
       (t_1 (* (- (fabs a)) b))
       (t_2 (* b (/ b (* x-scale x-scale))))
       (t_3 (* (fabs a) (/ (fabs a) (* y-scale y-scale))))
       (t_4 (* (* (* x-scale y-scale) x-scale) y-scale)))
  (if (<= (fabs a) 1.6e+153)
    (*
     (*
      x-scale
      (*
       x-scale
       (/
        (/
         (sqrt
          (*
           (*
            8.0
            (+
             (sqrt
              (pow
               (*
                (cos (* (* PI 0.005555555555555556) angle))
                (fabs a))
               4.0))
             (*
              (*
               (-
                0.5
                (* (cos (* (* 0.011111111111111112 angle) PI)) -0.5))
               (fabs a))
              (fabs a))))
           (pow (fabs a) 4.0)))
         (fabs x-scale))
        (* (fabs a) (fabs a)))))
     0.25)
    (*
     (/
      (/
       (-
        (sqrt
         (*
          (+ (fabs (- t_3 t_2)) (+ t_2 t_3))
          (* (* t_0 (/ t_1 t_4)) (* 2.0 (* (* t_1 b) (fabs a)))))))
       t_0)
      t_1)
     t_4))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * fabs(a)) * 4.0;
	double t_1 = -fabs(a) * b;
	double t_2 = b * (b / (x_45_scale * x_45_scale));
	double t_3 = fabs(a) * (fabs(a) / (y_45_scale * y_45_scale));
	double t_4 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	double tmp;
	if (fabs(a) <= 1.6e+153) {
		tmp = (x_45_scale * (x_45_scale * ((sqrt(((8.0 * (sqrt(pow((cos(((((double) M_PI) * 0.005555555555555556) * angle)) * fabs(a)), 4.0)) + (((0.5 - (cos(((0.011111111111111112 * angle) * ((double) M_PI))) * -0.5)) * fabs(a)) * fabs(a)))) * pow(fabs(a), 4.0))) / fabs(x_45_scale)) / (fabs(a) * fabs(a))))) * 0.25;
	} else {
		tmp = ((-sqrt(((fabs((t_3 - t_2)) + (t_2 + t_3)) * ((t_0 * (t_1 / t_4)) * (2.0 * ((t_1 * b) * fabs(a)))))) / t_0) / t_1) * t_4;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * Math.abs(a)) * 4.0;
	double t_1 = -Math.abs(a) * b;
	double t_2 = b * (b / (x_45_scale * x_45_scale));
	double t_3 = Math.abs(a) * (Math.abs(a) / (y_45_scale * y_45_scale));
	double t_4 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	double tmp;
	if (Math.abs(a) <= 1.6e+153) {
		tmp = (x_45_scale * (x_45_scale * ((Math.sqrt(((8.0 * (Math.sqrt(Math.pow((Math.cos(((Math.PI * 0.005555555555555556) * angle)) * Math.abs(a)), 4.0)) + (((0.5 - (Math.cos(((0.011111111111111112 * angle) * Math.PI)) * -0.5)) * Math.abs(a)) * Math.abs(a)))) * Math.pow(Math.abs(a), 4.0))) / Math.abs(x_45_scale)) / (Math.abs(a) * Math.abs(a))))) * 0.25;
	} else {
		tmp = ((-Math.sqrt(((Math.abs((t_3 - t_2)) + (t_2 + t_3)) * ((t_0 * (t_1 / t_4)) * (2.0 * ((t_1 * b) * Math.abs(a)))))) / t_0) / t_1) * t_4;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b * math.fabs(a)) * 4.0
	t_1 = -math.fabs(a) * b
	t_2 = b * (b / (x_45_scale * x_45_scale))
	t_3 = math.fabs(a) * (math.fabs(a) / (y_45_scale * y_45_scale))
	t_4 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale
	tmp = 0
	if math.fabs(a) <= 1.6e+153:
		tmp = (x_45_scale * (x_45_scale * ((math.sqrt(((8.0 * (math.sqrt(math.pow((math.cos(((math.pi * 0.005555555555555556) * angle)) * math.fabs(a)), 4.0)) + (((0.5 - (math.cos(((0.011111111111111112 * angle) * math.pi)) * -0.5)) * math.fabs(a)) * math.fabs(a)))) * math.pow(math.fabs(a), 4.0))) / math.fabs(x_45_scale)) / (math.fabs(a) * math.fabs(a))))) * 0.25
	else:
		tmp = ((-math.sqrt(((math.fabs((t_3 - t_2)) + (t_2 + t_3)) * ((t_0 * (t_1 / t_4)) * (2.0 * ((t_1 * b) * math.fabs(a)))))) / t_0) / t_1) * t_4
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b * abs(a)) * 4.0)
	t_1 = Float64(Float64(-abs(a)) * b)
	t_2 = Float64(b * Float64(b / Float64(x_45_scale * x_45_scale)))
	t_3 = Float64(abs(a) * Float64(abs(a) / Float64(y_45_scale * y_45_scale)))
	t_4 = Float64(Float64(Float64(x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)
	tmp = 0.0
	if (abs(a) <= 1.6e+153)
		tmp = Float64(Float64(x_45_scale * Float64(x_45_scale * Float64(Float64(sqrt(Float64(Float64(8.0 * Float64(sqrt((Float64(cos(Float64(Float64(pi * 0.005555555555555556) * angle)) * abs(a)) ^ 4.0)) + Float64(Float64(Float64(0.5 - Float64(cos(Float64(Float64(0.011111111111111112 * angle) * pi)) * -0.5)) * abs(a)) * abs(a)))) * (abs(a) ^ 4.0))) / abs(x_45_scale)) / Float64(abs(a) * abs(a))))) * 0.25);
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(abs(Float64(t_3 - t_2)) + Float64(t_2 + t_3)) * Float64(Float64(t_0 * Float64(t_1 / t_4)) * Float64(2.0 * Float64(Float64(t_1 * b) * abs(a))))))) / t_0) / t_1) * t_4);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b * abs(a)) * 4.0;
	t_1 = -abs(a) * b;
	t_2 = b * (b / (x_45_scale * x_45_scale));
	t_3 = abs(a) * (abs(a) / (y_45_scale * y_45_scale));
	t_4 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	tmp = 0.0;
	if (abs(a) <= 1.6e+153)
		tmp = (x_45_scale * (x_45_scale * ((sqrt(((8.0 * (sqrt(((cos(((pi * 0.005555555555555556) * angle)) * abs(a)) ^ 4.0)) + (((0.5 - (cos(((0.011111111111111112 * angle) * pi)) * -0.5)) * abs(a)) * abs(a)))) * (abs(a) ^ 4.0))) / abs(x_45_scale)) / (abs(a) * abs(a))))) * 0.25;
	else
		tmp = ((-sqrt(((abs((t_3 - t_2)) + (t_2 + t_3)) * ((t_0 * (t_1 / t_4)) * (2.0 * ((t_1 * b) * abs(a)))))) / t_0) / t_1) * t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Abs[a], $MachinePrecision]) * b), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[a], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1.6e+153], N[(N[(x$45$scale * N[(x$45$scale * N[(N[(N[Sqrt[N[(N[(8.0 * N[(N[Sqrt[N[Power[N[(N[Cos[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + N[(N[(N[(0.5 - N[(N[Cos[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(N[((-N[Sqrt[N[(N[(N[Abs[N[(t$95$3 - t$95$2), $MachinePrecision]], $MachinePrecision] + N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[(t$95$1 / t$95$4), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(t$95$1 * b), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$4), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.6000000000000001e153

   1. Initial program 2.6%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 1.1%

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

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 4.1%

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

   9. Applied rewrites 9.7%

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

   11. Applied rewrites 11.2%

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

   if 1.6000000000000001e153 < a

   1. Initial program 2.6%

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

   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 4.6%

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

   8. Applied rewrites 8.4%

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

Alternative 3: 12.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} t_0 := \left(-\left|a\right|\right) \cdot b\\ t_1 := \left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale\\ t_2 := \left|a\right| \cdot \frac{\left|a\right|}{y-scale \cdot y-scale}\\ t_3 := b \cdot \frac{b}{x-scale \cdot x-scale}\\ t_4 := \left(b \cdot \left|a\right|\right) \cdot 4\\ \mathbf{if}\;\left|a\right| \leq 1.7 \cdot 10^{+139}:\\ \;\;\;\;\left(\left(x-scale \cdot x-scale\right) \cdot \frac{\frac{{\left(\left|a\right|\right)}^{3} \cdot \sqrt{8 \cdot \left(\left(0.5 + \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right) - -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}}{\left|x-scale\right|}}{\left|a\right| \cdot \left|a\right|}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(\left|t\_2 - t\_3\right| + \left(t\_3 + t\_2\right)\right) \cdot \left(\left(t\_4 \cdot \frac{t\_0}{t\_1}\right) \cdot \left(2 \cdot \left(\left(t\_0 \cdot b\right) \cdot \left|a\right|\right)\right)\right)}}{t\_4}}{t\_0} \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (- (fabs a)) b))
       (t_1 (* (* (* x-scale y-scale) x-scale) y-scale))
       (t_2 (* (fabs a) (/ (fabs a) (* y-scale y-scale))))
       (t_3 (* b (/ b (* x-scale x-scale))))
       (t_4 (* (* b (fabs a)) 4.0)))
  (if (<= (fabs a) 1.7e+139)
    (*
     (*
      (* x-scale x-scale)
      (/
       (/
        (*
         (pow (fabs a) 3.0)
         (sqrt
          (*
           8.0
           (-
            (+
             0.5
             (sqrt
              (pow (cos (* 0.005555555555555556 (* angle PI))) 4.0)))
            (* -0.5 (cos (* 0.011111111111111112 (* angle PI))))))))
        (fabs x-scale))
       (* (fabs a) (fabs a))))
     0.25)
    (*
     (/
      (/
       (-
        (sqrt
         (*
          (+ (fabs (- t_2 t_3)) (+ t_3 t_2))
          (* (* t_4 (/ t_0 t_1)) (* 2.0 (* (* t_0 b) (fabs a)))))))
       t_4)
      t_0)
     t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = -fabs(a) * b;
	double t_1 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	double t_2 = fabs(a) * (fabs(a) / (y_45_scale * y_45_scale));
	double t_3 = b * (b / (x_45_scale * x_45_scale));
	double t_4 = (b * fabs(a)) * 4.0;
	double tmp;
	if (fabs(a) <= 1.7e+139) {
		tmp = ((x_45_scale * x_45_scale) * (((pow(fabs(a), 3.0) * sqrt((8.0 * ((0.5 + sqrt(pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 4.0))) - (-0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))))))))) / fabs(x_45_scale)) / (fabs(a) * fabs(a)))) * 0.25;
	} else {
		tmp = ((-sqrt(((fabs((t_2 - t_3)) + (t_3 + t_2)) * ((t_4 * (t_0 / t_1)) * (2.0 * ((t_0 * b) * fabs(a)))))) / t_4) / t_0) * 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.abs(a) * b;
	double t_1 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	double t_2 = Math.abs(a) * (Math.abs(a) / (y_45_scale * y_45_scale));
	double t_3 = b * (b / (x_45_scale * x_45_scale));
	double t_4 = (b * Math.abs(a)) * 4.0;
	double tmp;
	if (Math.abs(a) <= 1.7e+139) {
		tmp = ((x_45_scale * x_45_scale) * (((Math.pow(Math.abs(a), 3.0) * Math.sqrt((8.0 * ((0.5 + Math.sqrt(Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 4.0))) - (-0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)))))))) / Math.abs(x_45_scale)) / (Math.abs(a) * Math.abs(a)))) * 0.25;
	} else {
		tmp = ((-Math.sqrt(((Math.abs((t_2 - t_3)) + (t_3 + t_2)) * ((t_4 * (t_0 / t_1)) * (2.0 * ((t_0 * b) * Math.abs(a)))))) / t_4) / t_0) * t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = -math.fabs(a) * b
	t_1 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale
	t_2 = math.fabs(a) * (math.fabs(a) / (y_45_scale * y_45_scale))
	t_3 = b * (b / (x_45_scale * x_45_scale))
	t_4 = (b * math.fabs(a)) * 4.0
	tmp = 0
	if math.fabs(a) <= 1.7e+139:
		tmp = ((x_45_scale * x_45_scale) * (((math.pow(math.fabs(a), 3.0) * math.sqrt((8.0 * ((0.5 + math.sqrt(math.pow(math.cos((0.005555555555555556 * (angle * math.pi))), 4.0))) - (-0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))))))) / math.fabs(x_45_scale)) / (math.fabs(a) * math.fabs(a)))) * 0.25
	else:
		tmp = ((-math.sqrt(((math.fabs((t_2 - t_3)) + (t_3 + t_2)) * ((t_4 * (t_0 / t_1)) * (2.0 * ((t_0 * b) * math.fabs(a)))))) / t_4) / t_0) * t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(-abs(a)) * b)
	t_1 = Float64(Float64(Float64(x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)
	t_2 = Float64(abs(a) * Float64(abs(a) / Float64(y_45_scale * y_45_scale)))
	t_3 = Float64(b * Float64(b / Float64(x_45_scale * x_45_scale)))
	t_4 = Float64(Float64(b * abs(a)) * 4.0)
	tmp = 0.0
	if (abs(a) <= 1.7e+139)
		tmp = Float64(Float64(Float64(x_45_scale * x_45_scale) * Float64(Float64(Float64((abs(a) ^ 3.0) * sqrt(Float64(8.0 * Float64(Float64(0.5 + sqrt((cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 4.0))) - Float64(-0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi)))))))) / abs(x_45_scale)) / Float64(abs(a) * abs(a)))) * 0.25);
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(abs(Float64(t_2 - t_3)) + Float64(t_3 + t_2)) * Float64(Float64(t_4 * Float64(t_0 / t_1)) * Float64(2.0 * Float64(Float64(t_0 * b) * abs(a))))))) / t_4) / t_0) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = -abs(a) * b;
	t_1 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	t_2 = abs(a) * (abs(a) / (y_45_scale * y_45_scale));
	t_3 = b * (b / (x_45_scale * x_45_scale));
	t_4 = (b * abs(a)) * 4.0;
	tmp = 0.0;
	if (abs(a) <= 1.7e+139)
		tmp = ((x_45_scale * x_45_scale) * ((((abs(a) ^ 3.0) * sqrt((8.0 * ((0.5 + sqrt((cos((0.005555555555555556 * (angle * pi))) ^ 4.0))) - (-0.5 * cos((0.011111111111111112 * (angle * pi)))))))) / abs(x_45_scale)) / (abs(a) * abs(a)))) * 0.25;
	else
		tmp = ((-sqrt(((abs((t_2 - t_3)) + (t_3 + t_2)) * ((t_4 * (t_0 / t_1)) * (2.0 * ((t_0 * b) * abs(a)))))) / t_4) / t_0) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[((-N[Abs[a], $MachinePrecision]) * b), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[a], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(b * N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1.7e+139], N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[(N[(N[Power[N[Abs[a], $MachinePrecision], 3.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(0.5 + N[Sqrt[N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(N[((-N[Sqrt[N[(N[(N[Abs[N[(t$95$2 - t$95$3), $MachinePrecision]], $MachinePrecision] + N[(t$95$3 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$4 * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(t$95$0 * b), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.7000000000000001e139

   1. Initial program 2.6%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 1.1%

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

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 4.1%

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

   9. Applied rewrites 9.7%

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

   10. Taylor expanded in a around 0

       \[\leadsto \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\frac{{a}^{3} \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right) - \frac{-1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}}{\left|x-scale\right|}}{a \cdot a}\right) \cdot 0.25 \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\frac{{a}^{3} \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right) - \frac{-1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}{\left|x-scale\right|}}{a \cdot a}\right) \cdot \frac{1}{4} \]

       2. lower-pow.f64 N/A

          \[\leadsto \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\frac{{a}^{3} \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right) - \frac{-1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}{\left|x-scale\right|}}{a \cdot a}\right) \cdot \frac{1}{4} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\frac{{a}^{3} \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right) - \frac{-1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}{\left|x-scale\right|}}{a \cdot a}\right) \cdot \frac{1}{4} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\frac{{a}^{3} \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right) - \frac{-1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}{\left|x-scale\right|}}{a \cdot a}\right) \cdot \frac{1}{4} \]

       5. lower--.f64 N/A

          \[\leadsto \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\frac{{a}^{3} \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right) - \frac{-1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}{\left|x-scale\right|}}{a \cdot a}\right) \cdot \frac{1}{4} \]

   12. Applied rewrites 6.1%

       \[\leadsto \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\frac{{a}^{3} \cdot \sqrt{8 \cdot \left(\left(0.5 + \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right) - -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}}{\left|x-scale\right|}}{a \cdot a}\right) \cdot 0.25 \]

   if 1.7000000000000001e139 < a

   1. Initial program 2.6%

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

   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 4.6%

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

   8. Applied rewrites 8.4%

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

Alternative 4: 12.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} t_0 := \left(-\left|a\right|\right) \cdot b\\ t_1 := \left|a\right| \cdot \left|a\right|\\ t_2 := \left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale\\ t_3 := \left|a\right| \cdot \frac{\left|a\right|}{y-scale \cdot y-scale}\\ t_4 := b \cdot \frac{b}{x-scale \cdot x-scale}\\ t_5 := \left(b \cdot \left|a\right|\right) \cdot 4\\ \mathbf{if}\;\left|a\right| \leq 1.7 \cdot 10^{+139}:\\ \;\;\;\;\left(\left(x-scale \cdot x-scale\right) \cdot \frac{\left(t\_1 \cdot \left|a\right|\right) \cdot \frac{\sqrt{8 \cdot \left(\left(0.5 - \cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot -0.5\right) + \sqrt{{\cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}\right)}}{\left|x-scale\right|}}{t\_1}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(\left|t\_3 - t\_4\right| + \left(t\_4 + t\_3\right)\right) \cdot \left(\left(t\_5 \cdot \frac{t\_0}{t\_2}\right) \cdot \left(2 \cdot \left(\left(t\_0 \cdot b\right) \cdot \left|a\right|\right)\right)\right)}}{t\_5}}{t\_0} \cdot t\_2\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (- (fabs a)) b))
       (t_1 (* (fabs a) (fabs a)))
       (t_2 (* (* (* x-scale y-scale) x-scale) y-scale))
       (t_3 (* (fabs a) (/ (fabs a) (* y-scale y-scale))))
       (t_4 (* b (/ b (* x-scale x-scale))))
       (t_5 (* (* b (fabs a)) 4.0)))
  (if (<= (fabs a) 1.7e+139)
    (*
     (*
      (* x-scale x-scale)
      (/
       (*
        (* t_1 (fabs a))
        (/
         (sqrt
          (*
           8.0
           (+
            (-
             0.5
             (* (cos (* (* 0.011111111111111112 angle) PI)) -0.5))
            (sqrt
             (pow (cos (* (* PI 0.005555555555555556) angle)) 4.0)))))
         (fabs x-scale)))
       t_1))
     0.25)
    (*
     (/
      (/
       (-
        (sqrt
         (*
          (+ (fabs (- t_3 t_4)) (+ t_4 t_3))
          (* (* t_5 (/ t_0 t_2)) (* 2.0 (* (* t_0 b) (fabs a)))))))
       t_5)
      t_0)
     t_2))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = -fabs(a) * b;
	double t_1 = fabs(a) * fabs(a);
	double t_2 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	double t_3 = fabs(a) * (fabs(a) / (y_45_scale * y_45_scale));
	double t_4 = b * (b / (x_45_scale * x_45_scale));
	double t_5 = (b * fabs(a)) * 4.0;
	double tmp;
	if (fabs(a) <= 1.7e+139) {
		tmp = ((x_45_scale * x_45_scale) * (((t_1 * fabs(a)) * (sqrt((8.0 * ((0.5 - (cos(((0.011111111111111112 * angle) * ((double) M_PI))) * -0.5)) + sqrt(pow(cos(((((double) M_PI) * 0.005555555555555556) * angle)), 4.0))))) / fabs(x_45_scale))) / t_1)) * 0.25;
	} else {
		tmp = ((-sqrt(((fabs((t_3 - t_4)) + (t_4 + t_3)) * ((t_5 * (t_0 / t_2)) * (2.0 * ((t_0 * b) * fabs(a)))))) / t_5) / t_0) * t_2;
	}
	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) * b;
	double t_1 = Math.abs(a) * Math.abs(a);
	double t_2 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	double t_3 = Math.abs(a) * (Math.abs(a) / (y_45_scale * y_45_scale));
	double t_4 = b * (b / (x_45_scale * x_45_scale));
	double t_5 = (b * Math.abs(a)) * 4.0;
	double tmp;
	if (Math.abs(a) <= 1.7e+139) {
		tmp = ((x_45_scale * x_45_scale) * (((t_1 * Math.abs(a)) * (Math.sqrt((8.0 * ((0.5 - (Math.cos(((0.011111111111111112 * angle) * Math.PI)) * -0.5)) + Math.sqrt(Math.pow(Math.cos(((Math.PI * 0.005555555555555556) * angle)), 4.0))))) / Math.abs(x_45_scale))) / t_1)) * 0.25;
	} else {
		tmp = ((-Math.sqrt(((Math.abs((t_3 - t_4)) + (t_4 + t_3)) * ((t_5 * (t_0 / t_2)) * (2.0 * ((t_0 * b) * Math.abs(a)))))) / t_5) / t_0) * t_2;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = -math.fabs(a) * b
	t_1 = math.fabs(a) * math.fabs(a)
	t_2 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale
	t_3 = math.fabs(a) * (math.fabs(a) / (y_45_scale * y_45_scale))
	t_4 = b * (b / (x_45_scale * x_45_scale))
	t_5 = (b * math.fabs(a)) * 4.0
	tmp = 0
	if math.fabs(a) <= 1.7e+139:
		tmp = ((x_45_scale * x_45_scale) * (((t_1 * math.fabs(a)) * (math.sqrt((8.0 * ((0.5 - (math.cos(((0.011111111111111112 * angle) * math.pi)) * -0.5)) + math.sqrt(math.pow(math.cos(((math.pi * 0.005555555555555556) * angle)), 4.0))))) / math.fabs(x_45_scale))) / t_1)) * 0.25
	else:
		tmp = ((-math.sqrt(((math.fabs((t_3 - t_4)) + (t_4 + t_3)) * ((t_5 * (t_0 / t_2)) * (2.0 * ((t_0 * b) * math.fabs(a)))))) / t_5) / t_0) * t_2
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(-abs(a)) * b)
	t_1 = Float64(abs(a) * abs(a))
	t_2 = Float64(Float64(Float64(x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)
	t_3 = Float64(abs(a) * Float64(abs(a) / Float64(y_45_scale * y_45_scale)))
	t_4 = Float64(b * Float64(b / Float64(x_45_scale * x_45_scale)))
	t_5 = Float64(Float64(b * abs(a)) * 4.0)
	tmp = 0.0
	if (abs(a) <= 1.7e+139)
		tmp = Float64(Float64(Float64(x_45_scale * x_45_scale) * Float64(Float64(Float64(t_1 * abs(a)) * Float64(sqrt(Float64(8.0 * Float64(Float64(0.5 - Float64(cos(Float64(Float64(0.011111111111111112 * angle) * pi)) * -0.5)) + sqrt((cos(Float64(Float64(pi * 0.005555555555555556) * angle)) ^ 4.0))))) / abs(x_45_scale))) / t_1)) * 0.25);
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(abs(Float64(t_3 - t_4)) + Float64(t_4 + t_3)) * Float64(Float64(t_5 * Float64(t_0 / t_2)) * Float64(2.0 * Float64(Float64(t_0 * b) * abs(a))))))) / t_5) / t_0) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = -abs(a) * b;
	t_1 = abs(a) * abs(a);
	t_2 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	t_3 = abs(a) * (abs(a) / (y_45_scale * y_45_scale));
	t_4 = b * (b / (x_45_scale * x_45_scale));
	t_5 = (b * abs(a)) * 4.0;
	tmp = 0.0;
	if (abs(a) <= 1.7e+139)
		tmp = ((x_45_scale * x_45_scale) * (((t_1 * abs(a)) * (sqrt((8.0 * ((0.5 - (cos(((0.011111111111111112 * angle) * pi)) * -0.5)) + sqrt((cos(((pi * 0.005555555555555556) * angle)) ^ 4.0))))) / abs(x_45_scale))) / t_1)) * 0.25;
	else
		tmp = ((-sqrt(((abs((t_3 - t_4)) + (t_4 + t_3)) * ((t_5 * (t_0 / t_2)) * (2.0 * ((t_0 * b) * abs(a)))))) / t_5) / t_0) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[((-N[Abs[a], $MachinePrecision]) * b), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[a], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(b * N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1.7e+139], N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[(N[(t$95$1 * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[(0.5 - N[(N[Cos[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Cos[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(N[((-N[Sqrt[N[(N[(N[Abs[N[(t$95$3 - t$95$4), $MachinePrecision]], $MachinePrecision] + N[(t$95$4 + t$95$3), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$5 * N[(t$95$0 / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(t$95$0 * b), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$5), $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.7000000000000001e139

   1. Initial program 2.6%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 1.1%

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

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 4.1%

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

   8. Taylor expanded in a around 0

      \[\leadsto 0.25 \cdot \frac{{x-scale}^{2} \cdot \left({a}^{3} \cdot \sqrt{8 \cdot \frac{\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}\right)}{{a}^{2}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({a}^{3} \cdot \sqrt{8 \cdot \frac{\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}\right)}{{a}^{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({a}^{3} \cdot \sqrt{8 \cdot \frac{\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}\right)}{{a}^{2}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({a}^{3} \cdot \sqrt{8 \cdot \frac{\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}\right)}{{a}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({a}^{3} \cdot \sqrt{8 \cdot \frac{\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}\right)}{{a}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({a}^{3} \cdot \sqrt{8 \cdot \frac{\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}\right)}{{a}^{2}} \]

   10. Applied rewrites 2.9%

       \[\leadsto 0.25 \cdot \frac{{x-scale}^{2} \cdot \left({a}^{3} \cdot \sqrt{8 \cdot \frac{\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}\right)}{{a}^{2}} \]

   12. Applied rewrites 6.1%

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

   if 1.7000000000000001e139 < a

   1. Initial program 2.6%

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

   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 4.6%

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

   8. Applied rewrites 8.4%

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

Alternative 5: 11.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| \cdot \frac{\left|a\right|}{y-scale \cdot y-scale}\\ t_1 := b \cdot \frac{b}{x-scale \cdot x-scale}\\ t_2 := \left|a\right| \cdot \left|a\right|\\ t_3 := \left(b \cdot \left|a\right|\right) \cdot 4\\ t_4 := \left(-\left|a\right|\right) \cdot b\\ t_5 := \left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale\\ \mathbf{if}\;\left|a\right| \leq 1.7 \cdot 10^{+139}:\\ \;\;\;\;\left(\left(x-scale \cdot x-scale\right) \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\left(0.5 - -0.5\right) \cdot t\_2 + \sqrt{{\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left|a\right|\right)}^{4}}\right) \cdot {\left(\left|a\right|\right)}^{4}\right)}}{\left|x-scale\right|}}{t\_2}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(\left|t\_0 - t\_1\right| + \left(t\_1 + t\_0\right)\right) \cdot \left(\left(t\_3 \cdot \frac{t\_4}{t\_5}\right) \cdot \left(2 \cdot \left(\left(t\_4 \cdot b\right) \cdot \left|a\right|\right)\right)\right)}}{t\_3}}{t\_4} \cdot t\_5\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs a) (/ (fabs a) (* y-scale y-scale))))
       (t_1 (* b (/ b (* x-scale x-scale))))
       (t_2 (* (fabs a) (fabs a)))
       (t_3 (* (* b (fabs a)) 4.0))
       (t_4 (* (- (fabs a)) b))
       (t_5 (* (* (* x-scale y-scale) x-scale) y-scale)))
  (if (<= (fabs a) 1.7e+139)
    (*
     (*
      (* x-scale x-scale)
      (/
       (/
        (sqrt
         (*
          8.0
          (*
           (+
            (* (- 0.5 -0.5) t_2)
            (sqrt
             (pow
              (* (cos (* (* PI angle) 0.005555555555555556)) (fabs a))
              4.0)))
           (pow (fabs a) 4.0))))
        (fabs x-scale))
       t_2))
     0.25)
    (*
     (/
      (/
       (-
        (sqrt
         (*
          (+ (fabs (- t_0 t_1)) (+ t_1 t_0))
          (* (* t_3 (/ t_4 t_5)) (* 2.0 (* (* t_4 b) (fabs a)))))))
       t_3)
      t_4)
     t_5))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(a) * (fabs(a) / (y_45_scale * y_45_scale));
	double t_1 = b * (b / (x_45_scale * x_45_scale));
	double t_2 = fabs(a) * fabs(a);
	double t_3 = (b * fabs(a)) * 4.0;
	double t_4 = -fabs(a) * b;
	double t_5 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	double tmp;
	if (fabs(a) <= 1.7e+139) {
		tmp = ((x_45_scale * x_45_scale) * ((sqrt((8.0 * ((((0.5 - -0.5) * t_2) + sqrt(pow((cos(((((double) M_PI) * angle) * 0.005555555555555556)) * fabs(a)), 4.0))) * pow(fabs(a), 4.0)))) / fabs(x_45_scale)) / t_2)) * 0.25;
	} else {
		tmp = ((-sqrt(((fabs((t_0 - t_1)) + (t_1 + t_0)) * ((t_3 * (t_4 / t_5)) * (2.0 * ((t_4 * b) * fabs(a)))))) / t_3) / t_4) * t_5;
	}
	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(a) / (y_45_scale * y_45_scale));
	double t_1 = b * (b / (x_45_scale * x_45_scale));
	double t_2 = Math.abs(a) * Math.abs(a);
	double t_3 = (b * Math.abs(a)) * 4.0;
	double t_4 = -Math.abs(a) * b;
	double t_5 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	double tmp;
	if (Math.abs(a) <= 1.7e+139) {
		tmp = ((x_45_scale * x_45_scale) * ((Math.sqrt((8.0 * ((((0.5 - -0.5) * t_2) + Math.sqrt(Math.pow((Math.cos(((Math.PI * angle) * 0.005555555555555556)) * Math.abs(a)), 4.0))) * Math.pow(Math.abs(a), 4.0)))) / Math.abs(x_45_scale)) / t_2)) * 0.25;
	} else {
		tmp = ((-Math.sqrt(((Math.abs((t_0 - t_1)) + (t_1 + t_0)) * ((t_3 * (t_4 / t_5)) * (2.0 * ((t_4 * b) * Math.abs(a)))))) / t_3) / t_4) * t_5;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(a) * (math.fabs(a) / (y_45_scale * y_45_scale))
	t_1 = b * (b / (x_45_scale * x_45_scale))
	t_2 = math.fabs(a) * math.fabs(a)
	t_3 = (b * math.fabs(a)) * 4.0
	t_4 = -math.fabs(a) * b
	t_5 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale
	tmp = 0
	if math.fabs(a) <= 1.7e+139:
		tmp = ((x_45_scale * x_45_scale) * ((math.sqrt((8.0 * ((((0.5 - -0.5) * t_2) + math.sqrt(math.pow((math.cos(((math.pi * angle) * 0.005555555555555556)) * math.fabs(a)), 4.0))) * math.pow(math.fabs(a), 4.0)))) / math.fabs(x_45_scale)) / t_2)) * 0.25
	else:
		tmp = ((-math.sqrt(((math.fabs((t_0 - t_1)) + (t_1 + t_0)) * ((t_3 * (t_4 / t_5)) * (2.0 * ((t_4 * b) * math.fabs(a)))))) / t_3) / t_4) * t_5
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(a) * Float64(abs(a) / Float64(y_45_scale * y_45_scale)))
	t_1 = Float64(b * Float64(b / Float64(x_45_scale * x_45_scale)))
	t_2 = Float64(abs(a) * abs(a))
	t_3 = Float64(Float64(b * abs(a)) * 4.0)
	t_4 = Float64(Float64(-abs(a)) * b)
	t_5 = Float64(Float64(Float64(x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)
	tmp = 0.0
	if (abs(a) <= 1.7e+139)
		tmp = Float64(Float64(Float64(x_45_scale * x_45_scale) * Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(Float64(0.5 - -0.5) * t_2) + sqrt((Float64(cos(Float64(Float64(pi * angle) * 0.005555555555555556)) * abs(a)) ^ 4.0))) * (abs(a) ^ 4.0)))) / abs(x_45_scale)) / t_2)) * 0.25);
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(abs(Float64(t_0 - t_1)) + Float64(t_1 + t_0)) * Float64(Float64(t_3 * Float64(t_4 / t_5)) * Float64(2.0 * Float64(Float64(t_4 * b) * abs(a))))))) / t_3) / t_4) * t_5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(a) * (abs(a) / (y_45_scale * y_45_scale));
	t_1 = b * (b / (x_45_scale * x_45_scale));
	t_2 = abs(a) * abs(a);
	t_3 = (b * abs(a)) * 4.0;
	t_4 = -abs(a) * b;
	t_5 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	tmp = 0.0;
	if (abs(a) <= 1.7e+139)
		tmp = ((x_45_scale * x_45_scale) * ((sqrt((8.0 * ((((0.5 - -0.5) * t_2) + sqrt(((cos(((pi * angle) * 0.005555555555555556)) * abs(a)) ^ 4.0))) * (abs(a) ^ 4.0)))) / abs(x_45_scale)) / t_2)) * 0.25;
	else
		tmp = ((-sqrt(((abs((t_0 - t_1)) + (t_1 + t_0)) * ((t_3 * (t_4 / t_5)) * (2.0 * ((t_4 * b) * abs(a)))))) / t_3) / t_4) * t_5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Abs[a], $MachinePrecision]) * b), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1.7e+139], N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[(N[Sqrt[N[(8.0 * N[(N[(N[(N[(0.5 - -0.5), $MachinePrecision] * t$95$2), $MachinePrecision] + N[Sqrt[N[Power[N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[Abs[a], $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] / t$95$2), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(N[((-N[Sqrt[N[(N[(N[Abs[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 * N[(t$95$4 / t$95$5), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(t$95$4 * b), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$5), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.7000000000000001e139

   1. Initial program 2.6%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 1.1%

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

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 4.1%

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

   9. Applied rewrites 9.7%

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

   10. Taylor expanded in angle around 0

       \[\leadsto \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\left(0.5 - \frac{-1}{2}\right) \cdot \left(a \cdot a\right) + \sqrt{{\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{4}}\right) \cdot {a}^{4}\right)}}{\left|x-scale\right|}}{a \cdot a}\right) \cdot 0.25 \]
   11. Step-by-step derivation

   12. Applied rewrites 9.7%

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

   if 1.7000000000000001e139 < a

   1. Initial program 2.6%

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

   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 4.6%

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

   8. Applied rewrites 8.4%

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

Alternative 6: 11.9% accurate, 7.3× speedup

Math

\[\begin{array}{l} t_0 := \left(b \cdot \left|a\right|\right) \cdot 4\\ t_1 := \left(-\left|a\right|\right) \cdot b\\ t_2 := \left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale\\ t_3 := b \cdot \frac{b}{x-scale \cdot x-scale}\\ t_4 := {\left(\left|a\right|\right)}^{4}\\ t_5 := \left|a\right| \cdot \frac{\left|a\right|}{y-scale \cdot y-scale}\\ \mathbf{if}\;\left|a\right| \leq 1.7 \cdot 10^{+139}:\\ \;\;\;\;\left(\left(x-scale \cdot x-scale\right) \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\sqrt{t\_4} + {\left(\left|a\right|\right)}^{2}\right) \cdot t\_4\right)}}{\left|x-scale\right|}}{\left|a\right| \cdot \left|a\right|}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(\left|t\_5 - t\_3\right| + \left(t\_3 + t\_5\right)\right) \cdot \left(\left(t\_0 \cdot \frac{t\_1}{t\_2}\right) \cdot \left(2 \cdot \left(\left(t\_1 \cdot b\right) \cdot \left|a\right|\right)\right)\right)}}{t\_0}}{t\_1} \cdot t\_2\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (* b (fabs a)) 4.0))
       (t_1 (* (- (fabs a)) b))
       (t_2 (* (* (* x-scale y-scale) x-scale) y-scale))
       (t_3 (* b (/ b (* x-scale x-scale))))
       (t_4 (pow (fabs a) 4.0))
       (t_5 (* (fabs a) (/ (fabs a) (* y-scale y-scale)))))
  (if (<= (fabs a) 1.7e+139)
    (*
     (*
      (* x-scale x-scale)
      (/
       (/
        (sqrt (* 8.0 (* (+ (sqrt t_4) (pow (fabs a) 2.0)) t_4)))
        (fabs x-scale))
       (* (fabs a) (fabs a))))
     0.25)
    (*
     (/
      (/
       (-
        (sqrt
         (*
          (+ (fabs (- t_5 t_3)) (+ t_3 t_5))
          (* (* t_0 (/ t_1 t_2)) (* 2.0 (* (* t_1 b) (fabs a)))))))
       t_0)
      t_1)
     t_2))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * fabs(a)) * 4.0;
	double t_1 = -fabs(a) * b;
	double t_2 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	double t_3 = b * (b / (x_45_scale * x_45_scale));
	double t_4 = pow(fabs(a), 4.0);
	double t_5 = fabs(a) * (fabs(a) / (y_45_scale * y_45_scale));
	double tmp;
	if (fabs(a) <= 1.7e+139) {
		tmp = ((x_45_scale * x_45_scale) * ((sqrt((8.0 * ((sqrt(t_4) + pow(fabs(a), 2.0)) * t_4))) / fabs(x_45_scale)) / (fabs(a) * fabs(a)))) * 0.25;
	} else {
		tmp = ((-sqrt(((fabs((t_5 - t_3)) + (t_3 + t_5)) * ((t_0 * (t_1 / t_2)) * (2.0 * ((t_1 * b) * fabs(a)))))) / t_0) / t_1) * t_2;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (b * abs(a)) * 4.0d0
    t_1 = -abs(a) * b
    t_2 = ((x_45scale * y_45scale) * x_45scale) * y_45scale
    t_3 = b * (b / (x_45scale * x_45scale))
    t_4 = abs(a) ** 4.0d0
    t_5 = abs(a) * (abs(a) / (y_45scale * y_45scale))
    if (abs(a) <= 1.7d+139) then
        tmp = ((x_45scale * x_45scale) * ((sqrt((8.0d0 * ((sqrt(t_4) + (abs(a) ** 2.0d0)) * t_4))) / abs(x_45scale)) / (abs(a) * abs(a)))) * 0.25d0
    else
        tmp = ((-sqrt(((abs((t_5 - t_3)) + (t_3 + t_5)) * ((t_0 * (t_1 / t_2)) * (2.0d0 * ((t_1 * b) * abs(a)))))) / t_0) / t_1) * t_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * Math.abs(a)) * 4.0;
	double t_1 = -Math.abs(a) * b;
	double t_2 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	double t_3 = b * (b / (x_45_scale * x_45_scale));
	double t_4 = Math.pow(Math.abs(a), 4.0);
	double t_5 = Math.abs(a) * (Math.abs(a) / (y_45_scale * y_45_scale));
	double tmp;
	if (Math.abs(a) <= 1.7e+139) {
		tmp = ((x_45_scale * x_45_scale) * ((Math.sqrt((8.0 * ((Math.sqrt(t_4) + Math.pow(Math.abs(a), 2.0)) * t_4))) / Math.abs(x_45_scale)) / (Math.abs(a) * Math.abs(a)))) * 0.25;
	} else {
		tmp = ((-Math.sqrt(((Math.abs((t_5 - t_3)) + (t_3 + t_5)) * ((t_0 * (t_1 / t_2)) * (2.0 * ((t_1 * b) * Math.abs(a)))))) / t_0) / t_1) * t_2;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b * math.fabs(a)) * 4.0
	t_1 = -math.fabs(a) * b
	t_2 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale
	t_3 = b * (b / (x_45_scale * x_45_scale))
	t_4 = math.pow(math.fabs(a), 4.0)
	t_5 = math.fabs(a) * (math.fabs(a) / (y_45_scale * y_45_scale))
	tmp = 0
	if math.fabs(a) <= 1.7e+139:
		tmp = ((x_45_scale * x_45_scale) * ((math.sqrt((8.0 * ((math.sqrt(t_4) + math.pow(math.fabs(a), 2.0)) * t_4))) / math.fabs(x_45_scale)) / (math.fabs(a) * math.fabs(a)))) * 0.25
	else:
		tmp = ((-math.sqrt(((math.fabs((t_5 - t_3)) + (t_3 + t_5)) * ((t_0 * (t_1 / t_2)) * (2.0 * ((t_1 * b) * math.fabs(a)))))) / t_0) / t_1) * t_2
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b * abs(a)) * 4.0)
	t_1 = Float64(Float64(-abs(a)) * b)
	t_2 = Float64(Float64(Float64(x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)
	t_3 = Float64(b * Float64(b / Float64(x_45_scale * x_45_scale)))
	t_4 = abs(a) ^ 4.0
	t_5 = Float64(abs(a) * Float64(abs(a) / Float64(y_45_scale * y_45_scale)))
	tmp = 0.0
	if (abs(a) <= 1.7e+139)
		tmp = Float64(Float64(Float64(x_45_scale * x_45_scale) * Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(sqrt(t_4) + (abs(a) ^ 2.0)) * t_4))) / abs(x_45_scale)) / Float64(abs(a) * abs(a)))) * 0.25);
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(abs(Float64(t_5 - t_3)) + Float64(t_3 + t_5)) * Float64(Float64(t_0 * Float64(t_1 / t_2)) * Float64(2.0 * Float64(Float64(t_1 * b) * abs(a))))))) / t_0) / t_1) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b * abs(a)) * 4.0;
	t_1 = -abs(a) * b;
	t_2 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	t_3 = b * (b / (x_45_scale * x_45_scale));
	t_4 = abs(a) ^ 4.0;
	t_5 = abs(a) * (abs(a) / (y_45_scale * y_45_scale));
	tmp = 0.0;
	if (abs(a) <= 1.7e+139)
		tmp = ((x_45_scale * x_45_scale) * ((sqrt((8.0 * ((sqrt(t_4) + (abs(a) ^ 2.0)) * t_4))) / abs(x_45_scale)) / (abs(a) * abs(a)))) * 0.25;
	else
		tmp = ((-sqrt(((abs((t_5 - t_3)) + (t_3 + t_5)) * ((t_0 * (t_1 / t_2)) * (2.0 * ((t_1 * b) * abs(a)))))) / t_0) / t_1) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Abs[a], $MachinePrecision]) * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]}, Block[{t$95$3 = N[(b * N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[a], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1.7e+139], N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[(N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[t$95$4], $MachinePrecision] + N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(N[((-N[Sqrt[N[(N[(N[Abs[N[(t$95$5 - t$95$3), $MachinePrecision]], $MachinePrecision] + N[(t$95$3 + t$95$5), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(t$95$1 * b), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.7000000000000001e139

   1. Initial program 2.6%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 1.1%

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

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 4.1%

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

   9. Applied rewrites 9.7%

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

   10. Taylor expanded in angle around 0

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

       1. lower-+.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-pow.f64 9.7%

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

   12. Applied rewrites 9.7%

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

   if 1.7000000000000001e139 < a

   1. Initial program 2.6%

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

   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 4.6%

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

   8. Applied rewrites 8.4%

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

Alternative 7: 8.4% accurate, 11.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale\\ t_1 := b \cdot \frac{b}{x-scale \cdot x-scale}\\ t_2 := \left(b \cdot a\right) \cdot 4\\ t_3 := a \cdot \frac{a}{y-scale \cdot y-scale}\\ t_4 := \left(-a\right) \cdot b\\ \frac{\frac{-\sqrt{\left(\left|t\_3 - t\_1\right| + \left(t\_1 + t\_3\right)\right) \cdot \left(\left(t\_2 \cdot \frac{t\_4}{t\_0}\right) \cdot \left(2 \cdot \left(\left(t\_4 \cdot b\right) \cdot a\right)\right)\right)}}{t\_2}}{t\_4} \cdot t\_0 \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (* (* x-scale y-scale) x-scale) y-scale))
       (t_1 (* b (/ b (* x-scale x-scale))))
       (t_2 (* (* b a) 4.0))
       (t_3 (* a (/ a (* y-scale y-scale))))
       (t_4 (* (- a) b)))
  (*
   (/
    (/
     (-
      (sqrt
       (*
        (+ (fabs (- t_3 t_1)) (+ t_1 t_3))
        (* (* t_2 (/ t_4 t_0)) (* 2.0 (* (* t_4 b) a))))))
     t_2)
    t_4)
   t_0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	double t_1 = b * (b / (x_45_scale * x_45_scale));
	double t_2 = (b * a) * 4.0;
	double t_3 = a * (a / (y_45_scale * y_45_scale));
	double t_4 = -a * b;
	return ((-sqrt(((fabs((t_3 - t_1)) + (t_1 + t_3)) * ((t_2 * (t_4 / t_0)) * (2.0 * ((t_4 * b) * a))))) / t_2) / t_4) * t_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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    t_0 = ((x_45scale * y_45scale) * x_45scale) * y_45scale
    t_1 = b * (b / (x_45scale * x_45scale))
    t_2 = (b * a) * 4.0d0
    t_3 = a * (a / (y_45scale * y_45scale))
    t_4 = -a * b
    code = ((-sqrt(((abs((t_3 - t_1)) + (t_1 + t_3)) * ((t_2 * (t_4 / t_0)) * (2.0d0 * ((t_4 * b) * a))))) / t_2) / t_4) * t_0
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	double t_1 = b * (b / (x_45_scale * x_45_scale));
	double t_2 = (b * a) * 4.0;
	double t_3 = a * (a / (y_45_scale * y_45_scale));
	double t_4 = -a * b;
	return ((-Math.sqrt(((Math.abs((t_3 - t_1)) + (t_1 + t_3)) * ((t_2 * (t_4 / t_0)) * (2.0 * ((t_4 * b) * a))))) / t_2) / t_4) * t_0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale
	t_1 = b * (b / (x_45_scale * x_45_scale))
	t_2 = (b * a) * 4.0
	t_3 = a * (a / (y_45_scale * y_45_scale))
	t_4 = -a * b
	return ((-math.sqrt(((math.fabs((t_3 - t_1)) + (t_1 + t_3)) * ((t_2 * (t_4 / t_0)) * (2.0 * ((t_4 * b) * a))))) / t_2) / t_4) * t_0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)
	t_1 = Float64(b * Float64(b / Float64(x_45_scale * x_45_scale)))
	t_2 = Float64(Float64(b * a) * 4.0)
	t_3 = Float64(a * Float64(a / Float64(y_45_scale * y_45_scale)))
	t_4 = Float64(Float64(-a) * b)
	return Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(abs(Float64(t_3 - t_1)) + Float64(t_1 + t_3)) * Float64(Float64(t_2 * Float64(t_4 / t_0)) * Float64(2.0 * Float64(Float64(t_4 * b) * a)))))) / t_2) / t_4) * t_0)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	t_1 = b * (b / (x_45_scale * x_45_scale));
	t_2 = (b * a) * 4.0;
	t_3 = a * (a / (y_45_scale * y_45_scale));
	t_4 = -a * b;
	tmp = ((-sqrt(((abs((t_3 - t_1)) + (t_1 + t_3)) * ((t_2 * (t_4 / t_0)) * (2.0 * ((t_4 * b) * a))))) / t_2) / t_4) * t_0;
end

Wolfram

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

Derivation

1. Initial program 2.6%

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

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 4.6%

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

8. Applied rewrites 8.4%

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

Alternative 8: 7.3% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = -a * b
    t_1 = b * (b / (x_45scale * x_45scale))
    t_2 = a * (a / (y_45scale * y_45scale))
    code = ((-sqrt(((abs((t_2 - t_1)) + (t_1 + t_2)) * (((((a * b) * (4.0d0 / (y_45scale * x_45scale))) * t_0) / (y_45scale * x_45scale)) * (2.0d0 * ((t_0 * b) * a))))) / ((b * a) * (4.0d0 * t_0))) * ((x_45scale * y_45scale) * x_45scale)) * y_45scale
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = -a * b;
	double t_1 = b * (b / (x_45_scale * x_45_scale));
	double t_2 = a * (a / (y_45_scale * y_45_scale));
	return ((-Math.sqrt(((Math.abs((t_2 - t_1)) + (t_1 + t_2)) * (((((a * b) * (4.0 / (y_45_scale * x_45_scale))) * t_0) / (y_45_scale * x_45_scale)) * (2.0 * ((t_0 * b) * a))))) / ((b * a) * (4.0 * t_0))) * ((x_45_scale * y_45_scale) * x_45_scale)) * y_45_scale;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = -a * b
	t_1 = b * (b / (x_45_scale * x_45_scale))
	t_2 = a * (a / (y_45_scale * y_45_scale))
	return ((-math.sqrt(((math.fabs((t_2 - t_1)) + (t_1 + t_2)) * (((((a * b) * (4.0 / (y_45_scale * x_45_scale))) * t_0) / (y_45_scale * x_45_scale)) * (2.0 * ((t_0 * b) * a))))) / ((b * a) * (4.0 * t_0))) * ((x_45_scale * y_45_scale) * x_45_scale)) * y_45_scale

Julia

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

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = -a * b;
	t_1 = b * (b / (x_45_scale * x_45_scale));
	t_2 = a * (a / (y_45_scale * y_45_scale));
	tmp = ((-sqrt(((abs((t_2 - t_1)) + (t_1 + t_2)) * (((((a * b) * (4.0 / (y_45_scale * x_45_scale))) * t_0) / (y_45_scale * x_45_scale)) * (2.0 * ((t_0 * b) * a))))) / ((b * a) * (4.0 * t_0))) * ((x_45_scale * y_45_scale) * x_45_scale)) * y_45_scale;
end

Wolfram

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

Derivation

1. Initial program 2.6%

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

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 4.6%

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

8. Applied rewrites 5.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. frac-times N/A

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

   16. lift-/.f64 N/A

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

   17. associate-*r/ N/A

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

   18. lower-/.f64 N/A

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

10. Applied rewrites 7.3%

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

Alternative 9: 5.6% accurate, 12.1× speedup

Math

\[\begin{array}{l} t_0 := \left(x-scale \cdot y-scale\right) \cdot x-scale\\ t_1 := b \cdot \frac{b}{x-scale \cdot x-scale}\\ t_2 := a \cdot \frac{a}{y-scale \cdot y-scale}\\ t_3 := \left(-a\right) \cdot b\\ \left(\frac{-\sqrt{\left(\left|t\_2 - t\_1\right| + \left(t\_1 + t\_2\right)\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot 4\right) \cdot \frac{t\_3}{t\_0 \cdot y-scale}\right) \cdot \left(2 \cdot \left(t\_3 \cdot \left(a \cdot b\right)\right)\right)\right)}}{\left(b \cdot a\right) \cdot \left(4 \cdot t\_3\right)} \cdot t\_0\right) \cdot y-scale \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (* x-scale y-scale) x-scale))
       (t_1 (* b (/ b (* x-scale x-scale))))
       (t_2 (* a (/ a (* y-scale y-scale))))
       (t_3 (* (- a) b)))
  (*
   (*
    (/
     (-
      (sqrt
       (*
        (+ (fabs (- t_2 t_1)) (+ t_1 t_2))
        (*
         (* (* (* b a) 4.0) (/ t_3 (* t_0 y-scale)))
         (* 2.0 (* t_3 (* a b)))))))
     (* (* b a) (* 4.0 t_3)))
    t_0)
   y-scale)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (x_45_scale * y_45_scale) * x_45_scale;
	double t_1 = b * (b / (x_45_scale * x_45_scale));
	double t_2 = a * (a / (y_45_scale * y_45_scale));
	double t_3 = -a * b;
	return ((-sqrt(((fabs((t_2 - t_1)) + (t_1 + t_2)) * ((((b * a) * 4.0) * (t_3 / (t_0 * y_45_scale))) * (2.0 * (t_3 * (a * b)))))) / ((b * a) * (4.0 * t_3))) * t_0) * y_45_scale;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = (x_45scale * y_45scale) * x_45scale
    t_1 = b * (b / (x_45scale * x_45scale))
    t_2 = a * (a / (y_45scale * y_45scale))
    t_3 = -a * b
    code = ((-sqrt(((abs((t_2 - t_1)) + (t_1 + t_2)) * ((((b * a) * 4.0d0) * (t_3 / (t_0 * y_45scale))) * (2.0d0 * (t_3 * (a * b)))))) / ((b * a) * (4.0d0 * t_3))) * t_0) * y_45scale
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (x_45_scale * y_45_scale) * x_45_scale;
	double t_1 = b * (b / (x_45_scale * x_45_scale));
	double t_2 = a * (a / (y_45_scale * y_45_scale));
	double t_3 = -a * b;
	return ((-Math.sqrt(((Math.abs((t_2 - t_1)) + (t_1 + t_2)) * ((((b * a) * 4.0) * (t_3 / (t_0 * y_45_scale))) * (2.0 * (t_3 * (a * b)))))) / ((b * a) * (4.0 * t_3))) * t_0) * y_45_scale;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (x_45_scale * y_45_scale) * x_45_scale
	t_1 = b * (b / (x_45_scale * x_45_scale))
	t_2 = a * (a / (y_45_scale * y_45_scale))
	t_3 = -a * b
	return ((-math.sqrt(((math.fabs((t_2 - t_1)) + (t_1 + t_2)) * ((((b * a) * 4.0) * (t_3 / (t_0 * y_45_scale))) * (2.0 * (t_3 * (a * b)))))) / ((b * a) * (4.0 * t_3))) * t_0) * y_45_scale

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(x_45_scale * y_45_scale) * x_45_scale)
	t_1 = Float64(b * Float64(b / Float64(x_45_scale * x_45_scale)))
	t_2 = Float64(a * Float64(a / Float64(y_45_scale * y_45_scale)))
	t_3 = Float64(Float64(-a) * b)
	return Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(abs(Float64(t_2 - t_1)) + Float64(t_1 + t_2)) * Float64(Float64(Float64(Float64(b * a) * 4.0) * Float64(t_3 / Float64(t_0 * y_45_scale))) * Float64(2.0 * Float64(t_3 * Float64(a * b))))))) / Float64(Float64(b * a) * Float64(4.0 * t_3))) * t_0) * y_45_scale)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (x_45_scale * y_45_scale) * x_45_scale;
	t_1 = b * (b / (x_45_scale * x_45_scale));
	t_2 = a * (a / (y_45_scale * y_45_scale));
	t_3 = -a * b;
	tmp = ((-sqrt(((abs((t_2 - t_1)) + (t_1 + t_2)) * ((((b * a) * 4.0) * (t_3 / (t_0 * y_45_scale))) * (2.0 * (t_3 * (a * b)))))) / ((b * a) * (4.0 * t_3))) * t_0) * y_45_scale;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-a) * b), $MachinePrecision]}, N[(N[(N[((-N[Sqrt[N[(N[(N[Abs[N[(t$95$2 - t$95$1), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(b * a), $MachinePrecision] * 4.0), $MachinePrecision] * N[(t$95$3 / N[(t$95$0 * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$3 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(b * a), $MachinePrecision] * N[(4.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * y$45$scale), $MachinePrecision]]]]]

Derivation

1. Initial program 2.6%

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

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 4.6%

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

8. Applied rewrites 5.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lower-*.f64 5.6%

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 5.6%

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

10. Applied rewrites 5.6%

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

Alternative 10: 5.5% accurate, 12.2× speedup

Math

\[\begin{array}{l} t_0 := \left(x-scale \cdot y-scale\right) \cdot x-scale\\ t_1 := b \cdot \frac{b}{x-scale \cdot x-scale}\\ t_2 := a \cdot \frac{a}{y-scale \cdot y-scale}\\ t_3 := \left(-a\right) \cdot b\\ \left(\frac{-\sqrt{\left(\left|t\_2 - t\_1\right| + \left(t\_1 + t\_2\right)\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot 4\right) \cdot \frac{t\_3}{t\_0 \cdot y-scale}\right) \cdot \left(2 \cdot \left(\left(t\_3 \cdot b\right) \cdot a\right)\right)\right)}}{\left(b \cdot a\right) \cdot \left(-4 \cdot \left(a \cdot b\right)\right)} \cdot t\_0\right) \cdot y-scale \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (* x-scale y-scale) x-scale))
       (t_1 (* b (/ b (* x-scale x-scale))))
       (t_2 (* a (/ a (* y-scale y-scale))))
       (t_3 (* (- a) b)))
  (*
   (*
    (/
     (-
      (sqrt
       (*
        (+ (fabs (- t_2 t_1)) (+ t_1 t_2))
        (*
         (* (* (* b a) 4.0) (/ t_3 (* t_0 y-scale)))
         (* 2.0 (* (* t_3 b) a))))))
     (* (* b a) (* -4.0 (* a b))))
    t_0)
   y-scale)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (x_45_scale * y_45_scale) * x_45_scale;
	double t_1 = b * (b / (x_45_scale * x_45_scale));
	double t_2 = a * (a / (y_45_scale * y_45_scale));
	double t_3 = -a * b;
	return ((-sqrt(((fabs((t_2 - t_1)) + (t_1 + t_2)) * ((((b * a) * 4.0) * (t_3 / (t_0 * y_45_scale))) * (2.0 * ((t_3 * b) * a))))) / ((b * a) * (-4.0 * (a * b)))) * t_0) * y_45_scale;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = (x_45scale * y_45scale) * x_45scale
    t_1 = b * (b / (x_45scale * x_45scale))
    t_2 = a * (a / (y_45scale * y_45scale))
    t_3 = -a * b
    code = ((-sqrt(((abs((t_2 - t_1)) + (t_1 + t_2)) * ((((b * a) * 4.0d0) * (t_3 / (t_0 * y_45scale))) * (2.0d0 * ((t_3 * b) * a))))) / ((b * a) * ((-4.0d0) * (a * b)))) * t_0) * y_45scale
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (x_45_scale * y_45_scale) * x_45_scale;
	double t_1 = b * (b / (x_45_scale * x_45_scale));
	double t_2 = a * (a / (y_45_scale * y_45_scale));
	double t_3 = -a * b;
	return ((-Math.sqrt(((Math.abs((t_2 - t_1)) + (t_1 + t_2)) * ((((b * a) * 4.0) * (t_3 / (t_0 * y_45_scale))) * (2.0 * ((t_3 * b) * a))))) / ((b * a) * (-4.0 * (a * b)))) * t_0) * y_45_scale;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (x_45_scale * y_45_scale) * x_45_scale
	t_1 = b * (b / (x_45_scale * x_45_scale))
	t_2 = a * (a / (y_45_scale * y_45_scale))
	t_3 = -a * b
	return ((-math.sqrt(((math.fabs((t_2 - t_1)) + (t_1 + t_2)) * ((((b * a) * 4.0) * (t_3 / (t_0 * y_45_scale))) * (2.0 * ((t_3 * b) * a))))) / ((b * a) * (-4.0 * (a * b)))) * t_0) * y_45_scale

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(x_45_scale * y_45_scale) * x_45_scale)
	t_1 = Float64(b * Float64(b / Float64(x_45_scale * x_45_scale)))
	t_2 = Float64(a * Float64(a / Float64(y_45_scale * y_45_scale)))
	t_3 = Float64(Float64(-a) * b)
	return Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(abs(Float64(t_2 - t_1)) + Float64(t_1 + t_2)) * Float64(Float64(Float64(Float64(b * a) * 4.0) * Float64(t_3 / Float64(t_0 * y_45_scale))) * Float64(2.0 * Float64(Float64(t_3 * b) * a)))))) / Float64(Float64(b * a) * Float64(-4.0 * Float64(a * b)))) * t_0) * y_45_scale)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (x_45_scale * y_45_scale) * x_45_scale;
	t_1 = b * (b / (x_45_scale * x_45_scale));
	t_2 = a * (a / (y_45_scale * y_45_scale));
	t_3 = -a * b;
	tmp = ((-sqrt(((abs((t_2 - t_1)) + (t_1 + t_2)) * ((((b * a) * 4.0) * (t_3 / (t_0 * y_45_scale))) * (2.0 * ((t_3 * b) * a))))) / ((b * a) * (-4.0 * (a * b)))) * t_0) * y_45_scale;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-a) * b), $MachinePrecision]}, N[(N[(N[((-N[Sqrt[N[(N[(N[Abs[N[(t$95$2 - t$95$1), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(b * a), $MachinePrecision] * 4.0), $MachinePrecision] * N[(t$95$3 / N[(t$95$0 * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(t$95$3 * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(b * a), $MachinePrecision] * N[(-4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * y$45$scale), $MachinePrecision]]]]]

Derivation

1. Initial program 2.6%

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

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 4.6%

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

8. Applied rewrites 5.5%

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

9. Taylor expanded in a around 0

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

    1. lower-*.f64 N/A

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

    2. lower-*.f64 5.5%

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

11. Applied rewrites 5.5%

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

Reproduce

herbie shell --seed 2025258 
(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))))