a from scale-rotated-ellipse

Percentage Accurate: 2.7% → 27.6%

Time: 35.6s

Alternatives: 6

Speedup: 13.3×

• 30.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 2.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 27.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_0 := \cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)\\ t_1 := \mathsf{fma}\left(t\_0, 0.5, 0.5\right)\\ t_2 := \mathsf{fma}\left(\left(0.5 - t\_0 \cdot 0.5\right) \cdot a\_m, a\_m, \left(t\_1 \cdot b\_m\right) \cdot b\_m\right)\\ \mathbf{if}\;a\_m \leq 1.1 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\left(\frac{\sqrt{\left(\left|t\_2\right| + t\_2\right) \cdot \left({\left(a\_m \cdot b\_m\right)}^{4} \cdot 8\right)}}{\left|y-scale\right|} \cdot y-scale\right) \cdot y-scale}{b\_m} \cdot 0.25}{b\_m \cdot \left(a\_m \cdot a\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{\frac{\left(b\_m \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \sqrt{\left(8 \cdot \left(\left(a\_m \cdot a\_m\right) \cdot \left(a\_m \cdot a\_m\right)\right)\right) \cdot t\_1}}{\left|y-scale\right|}}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (cos (* (* 0.011111111111111112 angle) PI)))
        (t_1 (fma t_0 0.5 0.5))
        (t_2 (fma (* (- 0.5 (* t_0 0.5)) a_m) a_m (* (* t_1 b_m) b_m))))
   (if (<= a_m 1.1e-21)
     (/
      (*
       (/
        (*
         (*
          (/
           (sqrt (* (+ (fabs t_2) t_2) (* (pow (* a_m b_m) 4.0) 8.0)))
           (fabs y-scale))
          y-scale)
         y-scale)
        b_m)
       0.25)
      (* b_m (* a_m a_m)))
     (*
      (/ 0.25 a_m)
      (/
       (/
        (*
         (* b_m (* y-scale y-scale))
         (sqrt (* (* 8.0 (* (* a_m a_m) (* a_m a_m))) t_1)))
        (fabs y-scale))
       a_m)))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = cos(((0.011111111111111112 * angle) * ((double) M_PI)));
	double t_1 = fma(t_0, 0.5, 0.5);
	double t_2 = fma(((0.5 - (t_0 * 0.5)) * a_m), a_m, ((t_1 * b_m) * b_m));
	double tmp;
	if (a_m <= 1.1e-21) {
		tmp = (((((sqrt(((fabs(t_2) + t_2) * (pow((a_m * b_m), 4.0) * 8.0))) / fabs(y_45_scale)) * y_45_scale) * y_45_scale) / b_m) * 0.25) / (b_m * (a_m * a_m));
	} else {
		tmp = (0.25 / a_m) * ((((b_m * (y_45_scale * y_45_scale)) * sqrt(((8.0 * ((a_m * a_m) * (a_m * a_m))) * t_1))) / fabs(y_45_scale)) / a_m);
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	t_0 = cos(Float64(Float64(0.011111111111111112 * angle) * pi))
	t_1 = fma(t_0, 0.5, 0.5)
	t_2 = fma(Float64(Float64(0.5 - Float64(t_0 * 0.5)) * a_m), a_m, Float64(Float64(t_1 * b_m) * b_m))
	tmp = 0.0
	if (a_m <= 1.1e-21)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(abs(t_2) + t_2) * Float64((Float64(a_m * b_m) ^ 4.0) * 8.0))) / abs(y_45_scale)) * y_45_scale) * y_45_scale) / b_m) * 0.25) / Float64(b_m * Float64(a_m * a_m)));
	else
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(Float64(b_m * Float64(y_45_scale * y_45_scale)) * sqrt(Float64(Float64(8.0 * Float64(Float64(a_m * a_m) * Float64(a_m * a_m))) * t_1))) / abs(y_45_scale)) / a_m));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Cos[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 0.5 + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * a$95$m), $MachinePrecision] * a$95$m + N[(N[(t$95$1 * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 1.1e-21], N[(N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[Abs[t$95$2], $MachinePrecision] + t$95$2), $MachinePrecision] * N[(N[Power[N[(a$95$m * b$95$m), $MachinePrecision], 4.0], $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] / b$95$m), $MachinePrecision] * 0.25), $MachinePrecision] / N[(b$95$m * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(N[(b$95$m * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(8.0 * N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.1e-21

   1. Initial program 2.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 1.3%

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

   6. Applied rewrites 5.3%

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

   8. Applied rewrites 11.5%

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

   10. Applied rewrites 13.3%

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

   if 1.1e-21 < a

   1. Initial program 2.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 1.3%

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

   6. Applied rewrites 5.3%

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

   7. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 9.8%

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

   11. Applied rewrites 22.9%

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

Alternative 2: 26.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 - 1 \cdot 0.5, a\_m \cdot a\_m, \left(\mathsf{fma}\left(0.5, 1, 0.5\right) \cdot b\_m\right) \cdot b\_m\right)\\ \mathbf{if}\;a\_m \leq 1.05 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(8 \cdot {\left(b\_m \cdot a\_m\right)}^{4}\right) \cdot \left(\left|t\_0\right| + t\_0\right)}}{\left|y-scale\right|} \cdot \left(y-scale \cdot y-scale\right)}{b\_m} \cdot 0.25}{b\_m \cdot \left(a\_m \cdot a\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{\frac{\left(b\_m \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \sqrt{\left(8 \cdot \left(\left(a\_m \cdot a\_m\right) \cdot \left(a\_m \cdot a\_m\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right), 0.5, 0.5\right)}}{\left|y-scale\right|}}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (fma
          (- 0.5 (* 1.0 0.5))
          (* a_m a_m)
          (* (* (fma 0.5 1.0 0.5) b_m) b_m))))
   (if (<= a_m 1.05e-21)
     (/
      (*
       (/
        (*
         (/
          (sqrt (* (* 8.0 (pow (* b_m a_m) 4.0)) (+ (fabs t_0) t_0)))
          (fabs y-scale))
         (* y-scale y-scale))
        b_m)
       0.25)
      (* b_m (* a_m a_m)))
     (*
      (/ 0.25 a_m)
      (/
       (/
        (*
         (* b_m (* y-scale y-scale))
         (sqrt
          (*
           (* 8.0 (* (* a_m a_m) (* a_m a_m)))
           (fma (cos (* (* 0.011111111111111112 angle) PI)) 0.5 0.5))))
        (fabs y-scale))
       a_m)))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fma((0.5 - (1.0 * 0.5)), (a_m * a_m), ((fma(0.5, 1.0, 0.5) * b_m) * b_m));
	double tmp;
	if (a_m <= 1.05e-21) {
		tmp = ((((sqrt(((8.0 * pow((b_m * a_m), 4.0)) * (fabs(t_0) + t_0))) / fabs(y_45_scale)) * (y_45_scale * y_45_scale)) / b_m) * 0.25) / (b_m * (a_m * a_m));
	} else {
		tmp = (0.25 / a_m) * ((((b_m * (y_45_scale * y_45_scale)) * sqrt(((8.0 * ((a_m * a_m) * (a_m * a_m))) * fma(cos(((0.011111111111111112 * angle) * ((double) M_PI))), 0.5, 0.5)))) / fabs(y_45_scale)) / a_m);
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	t_0 = fma(Float64(0.5 - Float64(1.0 * 0.5)), Float64(a_m * a_m), Float64(Float64(fma(0.5, 1.0, 0.5) * b_m) * b_m))
	tmp = 0.0
	if (a_m <= 1.05e-21)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(8.0 * (Float64(b_m * a_m) ^ 4.0)) * Float64(abs(t_0) + t_0))) / abs(y_45_scale)) * Float64(y_45_scale * y_45_scale)) / b_m) * 0.25) / Float64(b_m * Float64(a_m * a_m)));
	else
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(Float64(b_m * Float64(y_45_scale * y_45_scale)) * sqrt(Float64(Float64(8.0 * Float64(Float64(a_m * a_m) * Float64(a_m * a_m))) * fma(cos(Float64(Float64(0.011111111111111112 * angle) * pi)), 0.5, 0.5)))) / abs(y_45_scale)) / a_m));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(0.5 - N[(1.0 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(a$95$m * a$95$m), $MachinePrecision] + N[(N[(N[(0.5 * 1.0 + 0.5), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 1.05e-21], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(8.0 * N[Power[N[(b$95$m * a$95$m), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[t$95$0], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] / b$95$m), $MachinePrecision] * 0.25), $MachinePrecision] / N[(b$95$m * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(N[(b$95$m * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(8.0 * N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.05000000000000006e-21

   1. Initial program 2.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 1.3%

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

   6. Applied rewrites 5.3%

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

   8. Applied rewrites 11.5%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 11.5%

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

   12. Taylor expanded in angle around 0

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

   14. Applied rewrites 11.4%

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

   15. Taylor expanded in angle around 0

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

   17. Applied rewrites 11.1%

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

   18. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{\frac{\sqrt{\left(8 \cdot {\left(b \cdot a\right)}^{4}\right) \cdot \left(\left|\mathsf{fma}\left(\frac{1}{2} - 1 \cdot \frac{1}{2}, a \cdot a, \left(\mathsf{fma}\left(\frac{1}{2}, 1, \frac{1}{2}\right) \cdot b\right) \cdot b\right)\right| + \mathsf{fma}\left(\frac{1}{2} - 1 \cdot \frac{1}{2}, a \cdot a, \left(\mathsf{fma}\left(\frac{1}{2}, 1, \frac{1}{2}\right) \cdot b\right) \cdot b\right)\right)}}{\left|y-scale\right|} \cdot \left(y-scale \cdot y-scale\right)}{b} \cdot \frac{1}{4}}{b \cdot \left(a \cdot a\right)} \]
   19. Step-by-step derivation

   20. Applied rewrites 11.1%

       \[\leadsto \frac{\frac{\frac{\sqrt{\left(8 \cdot {\left(b \cdot a\right)}^{4}\right) \cdot \left(\left|\mathsf{fma}\left(0.5 - 1 \cdot 0.5, a \cdot a, \left(\mathsf{fma}\left(0.5, 1, 0.5\right) \cdot b\right) \cdot b\right)\right| + \mathsf{fma}\left(0.5 - 1 \cdot 0.5, a \cdot a, \left(\mathsf{fma}\left(0.5, 1, 0.5\right) \cdot b\right) \cdot b\right)\right)}}{\left|y-scale\right|} \cdot \left(y-scale \cdot y-scale\right)}{b} \cdot 0.25}{b \cdot \left(a \cdot a\right)} \]

   if 1.05000000000000006e-21 < a

   1. Initial program 2.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 1.3%

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

   6. Applied rewrites 5.3%

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

   7. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 9.8%

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

   11. Applied rewrites 22.9%

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

Alternative 3: 24.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{0.25}{a\_m} \cdot \frac{\frac{\left(b\_m \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \sqrt{\left(8 \cdot \left(\left(a\_m \cdot a\_m\right) \cdot \left(a\_m \cdot a\_m\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right), 0.5, 0.5\right)}}{\left|y-scale\right|}}{a\_m}\\ \mathbf{if}\;a\_m \leq 2 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a\_m \leq 8 \cdot 10^{+143}:\\ \;\;\;\;\frac{0.25}{a\_m \cdot a\_m} \cdot \frac{b\_m \cdot \left({a\_m}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)\right)}{\left|y-scale\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (/ 0.25 a_m)
          (/
           (/
            (*
             (* b_m (* y-scale y-scale))
             (sqrt
              (*
               (* 8.0 (* (* a_m a_m) (* a_m a_m)))
               (fma (cos (* (* 0.011111111111111112 angle) PI)) 0.5 0.5))))
            (fabs y-scale))
           a_m))))
   (if (<= a_m 2e-156)
     t_0
     (if (<= a_m 8e+143)
       (*
        (/ 0.25 (* a_m a_m))
        (/
         (*
          b_m
          (*
           (pow a_m 2.0)
           (*
            (pow y-scale 2.0)
            (sqrt
             (*
              8.0
              (+ 0.5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))))))
         (fabs y-scale)))
       t_0))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (0.25 / a_m) * ((((b_m * (y_45_scale * y_45_scale)) * sqrt(((8.0 * ((a_m * a_m) * (a_m * a_m))) * fma(cos(((0.011111111111111112 * angle) * ((double) M_PI))), 0.5, 0.5)))) / fabs(y_45_scale)) / a_m);
	double tmp;
	if (a_m <= 2e-156) {
		tmp = t_0;
	} else if (a_m <= 8e+143) {
		tmp = (0.25 / (a_m * a_m)) * ((b_m * (pow(a_m, 2.0) * (pow(y_45_scale, 2.0) * sqrt((8.0 * (0.5 + (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))))))))))) / fabs(y_45_scale));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(Float64(b_m * Float64(y_45_scale * y_45_scale)) * sqrt(Float64(Float64(8.0 * Float64(Float64(a_m * a_m) * Float64(a_m * a_m))) * fma(cos(Float64(Float64(0.011111111111111112 * angle) * pi)), 0.5, 0.5)))) / abs(y_45_scale)) / a_m))
	tmp = 0.0
	if (a_m <= 2e-156)
		tmp = t_0;
	elseif (a_m <= 8e+143)
		tmp = Float64(Float64(0.25 / Float64(a_m * a_m)) * Float64(Float64(b_m * Float64((a_m ^ 2.0) * Float64((y_45_scale ^ 2.0) * sqrt(Float64(8.0 * Float64(0.5 + Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi)))))))))) / abs(y_45_scale)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(N[(b$95$m * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(8.0 * N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 2e-156], t$95$0, If[LessEqual[a$95$m, 8e+143], N[(N[(0.25 / N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(b$95$m * N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.00000000000000008e-156 or 8.0000000000000002e143 < a

   1. Initial program 2.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 1.3%

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

   6. Applied rewrites 5.3%

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

   7. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 9.8%

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

   11. Applied rewrites 22.9%

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

   if 2.00000000000000008e-156 < a < 8.0000000000000002e143

   1. Initial program 2.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 1.3%

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

   6. Applied rewrites 5.3%

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

   7. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 9.8%

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

   10. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-cos.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-PI.f64 11.4

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

   12. Applied rewrites 11.4%

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

Alternative 4: 22.9% accurate, 9.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \frac{0.25}{a\_m} \cdot \frac{\frac{\left(b\_m \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \sqrt{\left(8 \cdot \left(\left(a\_m \cdot a\_m\right) \cdot \left(a\_m \cdot a\_m\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right), 0.5, 0.5\right)}}{\left|y-scale\right|}}{a\_m} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (*
  (/ 0.25 a_m)
  (/
   (/
    (*
     (* b_m (* y-scale y-scale))
     (sqrt
      (*
       (* 8.0 (* (* a_m a_m) (* a_m a_m)))
       (fma (cos (* (* 0.011111111111111112 angle) PI)) 0.5 0.5))))
    (fabs y-scale))
   a_m)))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 / a_m) * ((((b_m * (y_45_scale * y_45_scale)) * sqrt(((8.0 * ((a_m * a_m) * (a_m * a_m))) * fma(cos(((0.011111111111111112 * angle) * ((double) M_PI))), 0.5, 0.5)))) / fabs(y_45_scale)) / a_m);
}

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	return Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(Float64(b_m * Float64(y_45_scale * y_45_scale)) * sqrt(Float64(Float64(8.0 * Float64(Float64(a_m * a_m) * Float64(a_m * a_m))) * fma(cos(Float64(Float64(0.011111111111111112 * angle) * pi)), 0.5, 0.5)))) / abs(y_45_scale)) / a_m))
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(N[(b$95$m * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(8.0 * N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.7%

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

2. Taylor expanded in x-scale around 0

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

4. Applied rewrites 1.3%

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

6. Applied rewrites 5.3%

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

7. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

9. Applied rewrites 9.8%

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

11. Applied rewrites 22.9%

    \[\leadsto \frac{0.25}{a} \cdot \color{blue}{\frac{\frac{\left(b \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \sqrt{\left(8 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right), 0.5, 0.5\right)}}{\left|y-scale\right|}}{a}} \]
12. Add Preprocessing

Alternative 5: 9.9% accurate, 9.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \frac{0.25}{a\_m \cdot a\_m} \cdot \left(b\_m \cdot \frac{\sqrt{\left(8 \cdot \left(\left(a\_m \cdot a\_m\right) \cdot \left(a\_m \cdot a\_m\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right), 0.5, 0.5\right)} \cdot \left(y-scale \cdot y-scale\right)}{\left|y-scale\right|}\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (*
  (/ 0.25 (* a_m a_m))
  (*
   b_m
   (/
    (*
     (sqrt
      (*
       (* 8.0 (* (* a_m a_m) (* a_m a_m)))
       (fma (cos (* (* 0.011111111111111112 angle) PI)) 0.5 0.5)))
     (* y-scale y-scale))
    (fabs y-scale)))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 / (a_m * a_m)) * (b_m * ((sqrt(((8.0 * ((a_m * a_m) * (a_m * a_m))) * fma(cos(((0.011111111111111112 * angle) * ((double) M_PI))), 0.5, 0.5))) * (y_45_scale * y_45_scale)) / fabs(y_45_scale)));
}

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle, x_45_scale, y_45_scale)
	return Float64(Float64(0.25 / Float64(a_m * a_m)) * Float64(b_m * Float64(Float64(sqrt(Float64(Float64(8.0 * Float64(Float64(a_m * a_m) * Float64(a_m * a_m))) * fma(cos(Float64(Float64(0.011111111111111112 * angle) * pi)), 0.5, 0.5))) * Float64(y_45_scale * y_45_scale)) / abs(y_45_scale))))
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(N[(0.25 / N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m * N[(N[(N[Sqrt[N[(N[(8.0 * N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.7%

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

2. Taylor expanded in x-scale around 0

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

4. Applied rewrites 1.3%

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

6. Applied rewrites 5.3%

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

7. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

9. Applied rewrites 9.8%

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

11. Applied rewrites 9.9%

    \[\leadsto \frac{0.25}{a \cdot a} \cdot \left(b \cdot \frac{\sqrt{\left(8 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right), 0.5, 0.5\right)} \cdot \left(y-scale \cdot y-scale\right)}{\color{blue}{\left|y-scale\right|}}\right) \]
12. Add Preprocessing

Alternative 6: 9.9% accurate, 13.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \frac{0.25}{a\_m \cdot a\_m} \cdot \frac{b\_m \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot {a\_m}^{4}}\right)}{\left|y-scale\right|} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (*
  (/ 0.25 (* a_m a_m))
  (/
   (* b_m (* (pow y-scale 2.0) (sqrt (* 8.0 (pow a_m 4.0)))))
   (fabs y-scale))))

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(N[(0.25 / N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(b$95$m * N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[Power[a$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.7%

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

2. Taylor expanded in x-scale around 0

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

4. Applied rewrites 1.3%

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

6. Applied rewrites 5.3%

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

7. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

9. Applied rewrites 9.8%

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

10. Taylor expanded in angle around 0

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

    1. lower-*.f64 N/A

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

    2. lower-pow.f64 9.9

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

12. Applied rewrites 9.9%

    \[\leadsto \frac{0.25}{a \cdot a} \cdot \frac{b \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot {a}^{4}}\right)}{\left|y-scale\right|} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025148 
(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))))