a from scale-rotated-ellipse

Percentage Accurate: 2.8% → 24.5%

Time: 30.6s

Alternatives: 8

Speedup: 6.6×

• 69.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 2.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 24.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| \cdot b\\ t_1 := \left|y-scale\right| \cdot \left|x-scale\right|\\ t_2 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_3 := \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ t_4 := \mathsf{fma}\left(0.5 - t\_3 \cdot 0.5, \left|a\right| \cdot \left|a\right|, \left(b \cdot b\right) \cdot \mathsf{fma}\left(t\_3, 0.5, 0.5\right)\right)\\ \mathbf{if}\;\left|x-scale\right| \leq 1.02 \cdot 10^{-166}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left({t\_0}^{4} \cdot \left(\left|t\_4\right| + t\_4\right)\right) \cdot 8}}{\left|t\_1\right| \cdot \left|x-scale\right|}}{t\_0 \cdot 4}}{t\_0} \cdot t\_1\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\left|x-scale\right| \cdot \left|x-scale\right|\right) \cdot \frac{\left|a\right| \cdot \left(\left|y-scale\right| \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_2\right)}^{2}} + t\_2\right)\right)}\right)}{\left|\left|x-scale\right| \cdot \left|y-scale\right|\right|}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs a) b))
       (t_1 (* (fabs y-scale) (fabs x-scale)))
       (t_2 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
       (t_3 (cos (* (* PI angle) 0.011111111111111112)))
       (t_4
        (fma
         (- 0.5 (* t_3 0.5))
         (* (fabs a) (fabs a))
         (* (* b b) (fma t_3 0.5 0.5)))))
  (if (<= (fabs x-scale) 1.02e-166)
    (*
     (*
      (/
       (/
        (/
         (sqrt (* (* (pow t_0 4.0) (+ (fabs t_4) t_4)) 8.0))
         (* (fabs t_1) (fabs x-scale)))
        (* t_0 4.0))
       t_0)
      t_1)
     t_1)
    (*
     0.25
     (*
      (* (fabs x-scale) (fabs x-scale))
      (/
       (*
        (fabs a)
        (*
         (fabs y-scale)
         (sqrt (* 8.0 (+ 0.5 (+ (sqrt (pow (+ 0.5 t_2) 2.0)) t_2))))))
       (fabs (* (fabs x-scale) (fabs y-scale)))))))))

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(y_45_scale) * fabs(x_45_scale);
	double t_2 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_3 = cos(((((double) M_PI) * angle) * 0.011111111111111112));
	double t_4 = fma((0.5 - (t_3 * 0.5)), (fabs(a) * fabs(a)), ((b * b) * fma(t_3, 0.5, 0.5)));
	double tmp;
	if (fabs(x_45_scale) <= 1.02e-166) {
		tmp = ((((sqrt(((pow(t_0, 4.0) * (fabs(t_4) + t_4)) * 8.0)) / (fabs(t_1) * fabs(x_45_scale))) / (t_0 * 4.0)) / t_0) * t_1) * t_1;
	} else {
		tmp = 0.25 * ((fabs(x_45_scale) * fabs(x_45_scale)) * ((fabs(a) * (fabs(y_45_scale) * sqrt((8.0 * (0.5 + (sqrt(pow((0.5 + t_2), 2.0)) + t_2)))))) / fabs((fabs(x_45_scale) * fabs(y_45_scale)))));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(a) * b)
	t_1 = Float64(abs(y_45_scale) * abs(x_45_scale))
	t_2 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	t_3 = cos(Float64(Float64(pi * angle) * 0.011111111111111112))
	t_4 = fma(Float64(0.5 - Float64(t_3 * 0.5)), Float64(abs(a) * abs(a)), Float64(Float64(b * b) * fma(t_3, 0.5, 0.5)))
	tmp = 0.0
	if (abs(x_45_scale) <= 1.02e-166)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64((t_0 ^ 4.0) * Float64(abs(t_4) + t_4)) * 8.0)) / Float64(abs(t_1) * abs(x_45_scale))) / Float64(t_0 * 4.0)) / t_0) * t_1) * t_1);
	else
		tmp = Float64(0.25 * Float64(Float64(abs(x_45_scale) * abs(x_45_scale)) * Float64(Float64(abs(a) * Float64(abs(y_45_scale) * sqrt(Float64(8.0 * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_2) ^ 2.0)) + t_2)))))) / abs(Float64(abs(x_45_scale) * abs(y_45_scale))))));
	end
	return 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[y$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.5 - N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(t$95$3 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 1.02e-166], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[Power[t$95$0, 4.0], $MachinePrecision] * N[(N[Abs[t$95$4], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[t$95$1], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * 4.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision], N[(0.25 * N[(N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[a], $MachinePrecision] * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$2), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.0200000000000001e-166

   1. Initial program 2.8%

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

   3. Applied rewrites 6.2%

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

   4. Taylor expanded in x-scale around 0

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

   6. Applied rewrites 1.0%

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

   8. Applied rewrites 5.7%

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

   if 1.0200000000000001e-166 < x-scale

   1. Initial program 2.8%

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

   2. Taylor expanded in b around 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 1.2%

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

   6. Applied rewrites 2.4%

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

   7. Taylor expanded in a around 0

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left(\left(\sqrt{4 \cdot \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}\right)}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \]

   9. Applied rewrites 6.5%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left(\left(\sqrt{\mathsf{fma}\left(4, \frac{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(0.5 \cdot \frac{1}{{x-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}\right)} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}\right)}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \]

   10. Taylor expanded in y-scale around 0

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

       1. lower-*.f64 N/A

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

   12. Applied rewrites 11.3%

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

Alternative 2: 21.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ 0.25 \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\left|a\right| \cdot \left(\left|y-scale\right| \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_0\right)}^{2}} + t\_0\right)\right)}\right)}{\left|x-scale \cdot \left|y-scale\right|\right|}\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))
  (*
   0.25
   (*
    (* x-scale x-scale)
    (/
     (*
      (fabs a)
      (*
       (fabs y-scale)
       (sqrt (* 8.0 (+ 0.5 (+ (sqrt (pow (+ 0.5 t_0) 2.0)) t_0))))))
     (fabs (* x-scale (fabs y-scale))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	return 0.25 * ((x_45_scale * x_45_scale) * ((fabs(a) * (fabs(y_45_scale) * sqrt((8.0 * (0.5 + (sqrt(pow((0.5 + t_0), 2.0)) + t_0)))))) / fabs((x_45_scale * fabs(y_45_scale)))));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)));
	return 0.25 * ((x_45_scale * x_45_scale) * ((Math.abs(a) * (Math.abs(y_45_scale) * Math.sqrt((8.0 * (0.5 + (Math.sqrt(Math.pow((0.5 + t_0), 2.0)) + t_0)))))) / Math.abs((x_45_scale * Math.abs(y_45_scale)))));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))
	return 0.25 * ((x_45_scale * x_45_scale) * ((math.fabs(a) * (math.fabs(y_45_scale) * math.sqrt((8.0 * (0.5 + (math.sqrt(math.pow((0.5 + t_0), 2.0)) + t_0)))))) / math.fabs((x_45_scale * math.fabs(y_45_scale)))))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	return Float64(0.25 * Float64(Float64(x_45_scale * x_45_scale) * Float64(Float64(abs(a) * Float64(abs(y_45_scale) * sqrt(Float64(8.0 * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_0) ^ 2.0)) + t_0)))))) / abs(Float64(x_45_scale * abs(y_45_scale))))))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.5 * cos((0.011111111111111112 * (angle * pi)));
	tmp = 0.25 * ((x_45_scale * x_45_scale) * ((abs(a) * (abs(y_45_scale) * sqrt((8.0 * (0.5 + (sqrt(((0.5 + t_0) ^ 2.0)) + t_0)))))) / abs((x_45_scale * abs(y_45_scale)))));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(0.25 * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[(N[Abs[a], $MachinePrecision] * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$0), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 2.8%

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

2. Taylor expanded in b around 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 1.2%

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

6. Applied rewrites 2.4%

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

7. Taylor expanded in a around 0

   \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left(\left(\sqrt{4 \cdot \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}\right)}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \]

9. Applied rewrites 6.5%

   \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left(\left(\sqrt{\mathsf{fma}\left(4, \frac{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(0.5 \cdot \frac{1}{{x-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}\right)} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}\right)}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \]

10. Taylor expanded in y-scale around 0

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

    1. lower-*.f64 N/A

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

12. Applied rewrites 11.3%

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

Alternative 3: 5.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|x-scale\right|\right)}^{2}\\ \mathbf{if}\;\left|x-scale\right| \leq 2.6 \cdot 10^{+153}:\\ \;\;\;\;0.25 \cdot \frac{t\_0 \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{a}^{4}} + {a}^{2}\right)}{t\_0}}}{{a}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\left|x-scale\right| \cdot \left|x-scale\right|\right) \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\sqrt{\frac{{a}^{4}}{{y-scale}^{4}}} + \frac{{a}^{2}}{{y-scale}^{2}}\right) \cdot {a}^{4}\right)}}{\left|\left|x-scale\right| \cdot y-scale\right|} \cdot \left(y-scale \cdot y-scale\right)}{a \cdot a}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (pow (fabs x-scale) 2.0)))
  (if (<= (fabs x-scale) 2.6e+153)
    (*
     0.25
     (/
      (*
       t_0
       (sqrt
        (*
         8.0
         (/ (* (pow a 4.0) (+ (sqrt (pow a 4.0)) (pow a 2.0))) t_0))))
      (pow a 2.0)))
    (*
     0.25
     (*
      (* (fabs x-scale) (fabs x-scale))
      (/
       (*
        (/
         (sqrt
          (*
           8.0
           (*
            (+
             (sqrt (/ (pow a 4.0) (pow y-scale 4.0)))
             (/ (pow a 2.0) (pow y-scale 2.0)))
            (pow a 4.0))))
         (fabs (* (fabs x-scale) y-scale)))
        (* y-scale y-scale))
       (* a a)))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(x_45_scale), 2.0);
	double tmp;
	if (fabs(x_45_scale) <= 2.6e+153) {
		tmp = 0.25 * ((t_0 * sqrt((8.0 * ((pow(a, 4.0) * (sqrt(pow(a, 4.0)) + pow(a, 2.0))) / t_0)))) / pow(a, 2.0));
	} else {
		tmp = 0.25 * ((fabs(x_45_scale) * fabs(x_45_scale)) * (((sqrt((8.0 * ((sqrt((pow(a, 4.0) / pow(y_45_scale, 4.0))) + (pow(a, 2.0) / pow(y_45_scale, 2.0))) * pow(a, 4.0)))) / fabs((fabs(x_45_scale) * y_45_scale))) * (y_45_scale * y_45_scale)) / (a * a)));
	}
	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) :: tmp
    t_0 = abs(x_45scale) ** 2.0d0
    if (abs(x_45scale) <= 2.6d+153) then
        tmp = 0.25d0 * ((t_0 * sqrt((8.0d0 * (((a ** 4.0d0) * (sqrt((a ** 4.0d0)) + (a ** 2.0d0))) / t_0)))) / (a ** 2.0d0))
    else
        tmp = 0.25d0 * ((abs(x_45scale) * abs(x_45scale)) * (((sqrt((8.0d0 * ((sqrt(((a ** 4.0d0) / (y_45scale ** 4.0d0))) + ((a ** 2.0d0) / (y_45scale ** 2.0d0))) * (a ** 4.0d0)))) / abs((abs(x_45scale) * y_45scale))) * (y_45scale * y_45scale)) / (a * a)))
    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 = Math.pow(Math.abs(x_45_scale), 2.0);
	double tmp;
	if (Math.abs(x_45_scale) <= 2.6e+153) {
		tmp = 0.25 * ((t_0 * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt(Math.pow(a, 4.0)) + Math.pow(a, 2.0))) / t_0)))) / Math.pow(a, 2.0));
	} else {
		tmp = 0.25 * ((Math.abs(x_45_scale) * Math.abs(x_45_scale)) * (((Math.sqrt((8.0 * ((Math.sqrt((Math.pow(a, 4.0) / Math.pow(y_45_scale, 4.0))) + (Math.pow(a, 2.0) / Math.pow(y_45_scale, 2.0))) * Math.pow(a, 4.0)))) / Math.abs((Math.abs(x_45_scale) * y_45_scale))) * (y_45_scale * y_45_scale)) / (a * a)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pow(math.fabs(x_45_scale), 2.0)
	tmp = 0
	if math.fabs(x_45_scale) <= 2.6e+153:
		tmp = 0.25 * ((t_0 * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.sqrt(math.pow(a, 4.0)) + math.pow(a, 2.0))) / t_0)))) / math.pow(a, 2.0))
	else:
		tmp = 0.25 * ((math.fabs(x_45_scale) * math.fabs(x_45_scale)) * (((math.sqrt((8.0 * ((math.sqrt((math.pow(a, 4.0) / math.pow(y_45_scale, 4.0))) + (math.pow(a, 2.0) / math.pow(y_45_scale, 2.0))) * math.pow(a, 4.0)))) / math.fabs((math.fabs(x_45_scale) * y_45_scale))) * (y_45_scale * y_45_scale)) / (a * a)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(x_45_scale) ^ 2.0
	tmp = 0.0
	if (abs(x_45_scale) <= 2.6e+153)
		tmp = Float64(0.25 * Float64(Float64(t_0 * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(sqrt((a ^ 4.0)) + (a ^ 2.0))) / t_0)))) / (a ^ 2.0)));
	else
		tmp = Float64(0.25 * Float64(Float64(abs(x_45_scale) * abs(x_45_scale)) * Float64(Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64((a ^ 4.0) / (y_45_scale ^ 4.0))) + Float64((a ^ 2.0) / (y_45_scale ^ 2.0))) * (a ^ 4.0)))) / abs(Float64(abs(x_45_scale) * y_45_scale))) * Float64(y_45_scale * y_45_scale)) / Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(x_45_scale) ^ 2.0;
	tmp = 0.0;
	if (abs(x_45_scale) <= 2.6e+153)
		tmp = 0.25 * ((t_0 * sqrt((8.0 * (((a ^ 4.0) * (sqrt((a ^ 4.0)) + (a ^ 2.0))) / t_0)))) / (a ^ 2.0));
	else
		tmp = 0.25 * ((abs(x_45_scale) * abs(x_45_scale)) * (((sqrt((8.0 * ((sqrt(((a ^ 4.0) / (y_45_scale ^ 4.0))) + ((a ^ 2.0) / (y_45_scale ^ 2.0))) * (a ^ 4.0)))) / abs((abs(x_45_scale) * y_45_scale))) * (y_45_scale * y_45_scale)) / (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[x$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 2.6e+153], N[(0.25 * N[(N[(t$95$0 * N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[y$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(N[Abs[x$45$scale], $MachinePrecision] * y$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.5999999999999999e153

   1. Initial program 2.8%

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

   2. Taylor expanded in b around 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 1.2%

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

   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 4.2%

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

   8. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 4.2%

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

   10. Applied rewrites 4.2%

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

   if 2.5999999999999999e153 < x-scale

   1. Initial program 2.8%

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

   2. Taylor expanded in b around 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 1.2%

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

   6. Applied rewrites 2.4%

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

   7. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

   9. Applied rewrites 2.7%

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

Alternative 4: 4.9% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 2.8%

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

2. Taylor expanded in b around 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 1.2%

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

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 4.2%

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

8. Taylor expanded in angle around 0

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

   1. lower-+.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 4.2%

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

10. Applied rewrites 4.2%

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

Alternative 5: 4.2% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.8%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 4.3%

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

6. Applied rewrites 2.0%

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

8. Applied rewrites 3.8%

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

Alternative 6: 3.8% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.8%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 4.3%

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

6. Applied rewrites 2.0%

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

8. Applied rewrites 2.1%

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

10. Applied rewrites 4.9%

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

Alternative 7: 3.7% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.8%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 4.3%

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

6. Applied rewrites 2.0%

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

8. Applied rewrites 2.1%

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

10. Applied rewrites 3.7%

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

Alternative 8: 3.7% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.8%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 4.3%

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

6. Applied rewrites 2.0%

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

8. Applied rewrites 2.1%

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

10. Applied rewrites 3.7%

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

Reproduce

herbie shell --seed 2025214 
(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))))