a from scale-rotated-ellipse

Percentage Accurate: 2.7% → 47.2%

Time: 27.6s

Alternatives: 8

Speedup: 484.7×

• 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: 47.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 7.4 \cdot 10^{-110}:\\ \;\;\;\;b\_m \cdot y-scale\_m\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot \left(0.25 \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a\_m \cdot \left(\sin \left(0.5 \cdot \pi\right) + 0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot \cos \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b\_m \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a_m 7.4e-110)
   (* b_m y-scale_m)
   (*
    x-scale_m
    (*
     0.25
     (sqrt
      (*
       8.0
       (fma
        2.0
        (pow
         (*
          a_m
          (+
           (sin (* 0.5 PI))
           (* 0.005555555555555556 (* angle (* PI (cos (* 0.5 PI)))))))
         2.0)
        (*
         2.0
         (pow (* b_m (sin (* 0.005555555555555556 (* angle PI)))) 2.0)))))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a_m <= 7.4e-110) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = x_45_scale_m * (0.25 * sqrt((8.0 * fma(2.0, pow((a_m * (sin((0.5 * ((double) M_PI))) + (0.005555555555555556 * (angle * (((double) M_PI) * cos((0.5 * ((double) M_PI)))))))), 2.0), (2.0 * pow((b_m * sin((0.005555555555555556 * (angle * ((double) M_PI))))), 2.0))))));
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a_m <= 7.4e-110)
		tmp = Float64(b_m * y_45_scale_m);
	else
		tmp = Float64(x_45_scale_m * Float64(0.25 * sqrt(Float64(8.0 * fma(2.0, (Float64(a_m * Float64(sin(Float64(0.5 * pi)) + Float64(0.005555555555555556 * Float64(angle * Float64(pi * cos(Float64(0.5 * pi))))))) ^ 2.0), Float64(2.0 * (Float64(b_m * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0)))))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a$95$m, 7.4e-110], N[(b$95$m * y$45$scale$95$m), $MachinePrecision], N[(x$45$scale$95$m * N[(0.25 * N[Sqrt[N[(8.0 * N[(2.0 * N[Power[N[(a$95$m * N[(N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] + N[(0.005555555555555556 * N[(angle * N[(Pi * N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(2.0 * N[Power[N[(b$95$m * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.40000000000000032e-110

   1. Initial program 2.0%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

   5. Taylor expanded in b around 0

      \[\leadsto b \cdot \color{blue}{y-scale} \]
   6. Step-by-step derivation

      1. lower-*.f64 44.1

         \[\leadsto b \cdot y-scale \]

   7. Applied rewrites 44.1%

      \[\leadsto b \cdot \color{blue}{y-scale} \]

   if 7.40000000000000032e-110 < a

   1. Initial program 3.1%

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

   2. Taylor expanded in x-scale around inf

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

   4. Applied rewrites 5.4%

      \[\leadsto \color{blue}{x-scale \cdot \mathsf{fma}\left(0.125, \frac{y-scale \cdot \left(\sqrt{8} \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right) \cdot \frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}, 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}{\frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}}, {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(2, \frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}, 2 \cdot \frac{{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}}, 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\mathsf{fma}\left(2, \frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}, 2 \cdot \frac{{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}\right)\right)} \]

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 48.4%

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

   8. Taylor expanded in angle around 0

      \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. sin-+PI/2-rev N/A

         \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

      4. pow-prod-down N/A

         \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot {\left(angle \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot {\left(angle \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot {\left(angle \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

      7. lift-PI.f64 48.8

         \[\leadsto x-scale \cdot \left(0.25 \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + -1.54320987654321 \cdot 10^{-5} \cdot {\left(angle \cdot \pi\right)}^{2}\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

   10. Applied rewrites 48.8%

       \[\leadsto x-scale \cdot \left(0.25 \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + -1.54320987654321 \cdot 10^{-5} \cdot {\left(angle \cdot \pi\right)}^{2}\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

   11. Taylor expanded in angle around 0

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

       1. lower-+.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-PI.f64 N/A

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

       9. lower-cos.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-PI.f64 48.6

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

   13. Applied rewrites 48.6%

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

Alternative 2: 47.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 7.4 \cdot 10^{-110}:\\ \;\;\;\;b\_m \cdot y-scale\_m\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot \left(0.25 \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a\_m \cdot \left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b\_m \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a_m 7.4e-110)
   (* b_m y-scale_m)
   (*
    x-scale_m
    (*
     0.25
     (sqrt
      (*
       8.0
       (fma
        2.0
        (pow
         (* a_m (+ 1.0 (* -1.54320987654321e-5 (* (* angle PI) (* angle PI)))))
         2.0)
        (*
         2.0
         (pow (* b_m (sin (* 0.005555555555555556 (* angle PI)))) 2.0)))))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a_m <= 7.4e-110) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = x_45_scale_m * (0.25 * sqrt((8.0 * fma(2.0, pow((a_m * (1.0 + (-1.54320987654321e-5 * ((angle * ((double) M_PI)) * (angle * ((double) M_PI)))))), 2.0), (2.0 * pow((b_m * sin((0.005555555555555556 * (angle * ((double) M_PI))))), 2.0))))));
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a_m <= 7.4e-110)
		tmp = Float64(b_m * y_45_scale_m);
	else
		tmp = Float64(x_45_scale_m * Float64(0.25 * sqrt(Float64(8.0 * fma(2.0, (Float64(a_m * Float64(1.0 + Float64(-1.54320987654321e-5 * Float64(Float64(angle * pi) * Float64(angle * pi))))) ^ 2.0), Float64(2.0 * (Float64(b_m * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0)))))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a$95$m, 7.4e-110], N[(b$95$m * y$45$scale$95$m), $MachinePrecision], N[(x$45$scale$95$m * N[(0.25 * N[Sqrt[N[(8.0 * N[(2.0 * N[Power[N[(a$95$m * N[(1.0 + N[(-1.54320987654321e-5 * N[(N[(angle * Pi), $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(2.0 * N[Power[N[(b$95$m * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.40000000000000032e-110

   1. Initial program 2.0%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

   5. Taylor expanded in b around 0

      \[\leadsto b \cdot \color{blue}{y-scale} \]
   6. Step-by-step derivation

      1. lower-*.f64 44.1

         \[\leadsto b \cdot y-scale \]

   7. Applied rewrites 44.1%

      \[\leadsto b \cdot \color{blue}{y-scale} \]

   if 7.40000000000000032e-110 < a

   1. Initial program 3.1%

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

   2. Taylor expanded in x-scale around inf

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

   4. Applied rewrites 5.4%

      \[\leadsto \color{blue}{x-scale \cdot \mathsf{fma}\left(0.125, \frac{y-scale \cdot \left(\sqrt{8} \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right) \cdot \frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}, 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}{\frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}}, {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(2, \frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}, 2 \cdot \frac{{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}}, 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\mathsf{fma}\left(2, \frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}, 2 \cdot \frac{{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}\right)\right)} \]

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 48.4%

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

   8. Taylor expanded in angle around 0

      \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. sin-+PI/2-rev N/A

         \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

      4. pow-prod-down N/A

         \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot {\left(angle \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot {\left(angle \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot {\left(angle \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

      7. lift-PI.f64 48.8

         \[\leadsto x-scale \cdot \left(0.25 \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + -1.54320987654321 \cdot 10^{-5} \cdot {\left(angle \cdot \pi\right)}^{2}\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

   10. Applied rewrites 48.8%

       \[\leadsto x-scale \cdot \left(0.25 \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + -1.54320987654321 \cdot 10^{-5} \cdot {\left(angle \cdot \pi\right)}^{2}\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]
   11. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot {\left(angle \cdot \pi\right)}^{2}\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

       2. lift-PI.f64 N/A

          \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot {\left(angle \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot {\left(angle \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

       7. lift-PI.f64 N/A

          \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

       9. lift-PI.f64 48.8

          \[\leadsto x-scale \cdot \left(0.25 \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]

   12. Applied rewrites 48.8%

       \[\leadsto x-scale \cdot \left(0.25 \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, {\left(a \cdot \left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 47.0% accurate, 10.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 7.4 \cdot 10^{-110}:\\ \;\;\;\;b\_m \cdot y-scale\_m\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \sqrt{\mathsf{fma}\left(2, a\_m \cdot a\_m, 2 \cdot {\left(b\_m \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a_m 7.4e-110)
   (* b_m y-scale_m)
   (*
    0.25
    (*
     (* x-scale_m (sqrt 8.0))
     (sqrt
      (fma
       2.0
       (* a_m a_m)
       (*
        2.0
        (pow (* b_m (sin (* 0.005555555555555556 (* angle PI)))) 2.0))))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a_m <= 7.4e-110) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * sqrt(fma(2.0, (a_m * a_m), (2.0 * pow((b_m * sin((0.005555555555555556 * (angle * ((double) M_PI))))), 2.0)))));
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a_m <= 7.4e-110)
		tmp = Float64(b_m * y_45_scale_m);
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * sqrt(fma(2.0, Float64(a_m * a_m), Float64(2.0 * (Float64(b_m * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0))))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a$95$m, 7.4e-110], N[(b$95$m * y$45$scale$95$m), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(a$95$m * a$95$m), $MachinePrecision] + N[(2.0 * N[Power[N[(b$95$m * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.40000000000000032e-110

   1. Initial program 2.0%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

   5. Taylor expanded in b around 0

      \[\leadsto b \cdot \color{blue}{y-scale} \]
   6. Step-by-step derivation

      1. lower-*.f64 44.1

         \[\leadsto b \cdot y-scale \]

   7. Applied rewrites 44.1%

      \[\leadsto b \cdot \color{blue}{y-scale} \]

   if 7.40000000000000032e-110 < a

   1. Initial program 3.1%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in angle around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   7. Applied rewrites 48.3%

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

Alternative 4: 47.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 7.4 \cdot 10^{-110}:\\ \;\;\;\;b\_m \cdot y-scale\_m\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot \left(0.25 \cdot \sqrt{8 \cdot \mathsf{fma}\left(2, a\_m \cdot a\_m, 2 \cdot {\left(b\_m \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a_m 7.4e-110)
   (* b_m y-scale_m)
   (*
    x-scale_m
    (*
     0.25
     (sqrt
      (*
       8.0
       (fma
        2.0
        (* a_m a_m)
        (*
         2.0
         (pow (* b_m (sin (* 0.005555555555555556 (* angle PI)))) 2.0)))))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a_m <= 7.4e-110) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = x_45_scale_m * (0.25 * sqrt((8.0 * fma(2.0, (a_m * a_m), (2.0 * pow((b_m * sin((0.005555555555555556 * (angle * ((double) M_PI))))), 2.0))))));
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a_m <= 7.4e-110)
		tmp = Float64(b_m * y_45_scale_m);
	else
		tmp = Float64(x_45_scale_m * Float64(0.25 * sqrt(Float64(8.0 * fma(2.0, Float64(a_m * a_m), Float64(2.0 * (Float64(b_m * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0)))))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a$95$m, 7.4e-110], N[(b$95$m * y$45$scale$95$m), $MachinePrecision], N[(x$45$scale$95$m * N[(0.25 * N[Sqrt[N[(8.0 * N[(2.0 * N[(a$95$m * a$95$m), $MachinePrecision] + N[(2.0 * N[Power[N[(b$95$m * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.40000000000000032e-110

   1. Initial program 2.0%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

   5. Taylor expanded in b around 0

      \[\leadsto b \cdot \color{blue}{y-scale} \]
   6. Step-by-step derivation

      1. lower-*.f64 44.1

         \[\leadsto b \cdot y-scale \]

   7. Applied rewrites 44.1%

      \[\leadsto b \cdot \color{blue}{y-scale} \]

   if 7.40000000000000032e-110 < a

   1. Initial program 3.1%

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

   2. Taylor expanded in x-scale around inf

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

   4. Applied rewrites 5.4%

      \[\leadsto \color{blue}{x-scale \cdot \mathsf{fma}\left(0.125, \frac{y-scale \cdot \left(\sqrt{8} \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right) \cdot \frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}, 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}{\frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}}, {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(2, \frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}, 2 \cdot \frac{{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}}, 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\mathsf{fma}\left(2, \frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}, 2 \cdot \frac{{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}\right)\right)} \]

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 48.4%

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

   8. Taylor expanded in angle around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.5

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

   10. Applied rewrites 48.5%

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

Alternative 5: 47.0% accurate, 15.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 4 \cdot 10^{-79}:\\ \;\;\;\;b\_m \cdot y-scale\_m\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(a\_m \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5}, \pi \cdot \pi, 3.969161205100849 \cdot 10^{-11} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a_m 4e-79)
   (* b_m y-scale_m)
   (*
    0.25
    (*
     (* x-scale_m (sqrt 8.0))
     (*
      a_m
      (*
       (+
        1.0
        (*
         (* angle angle)
         (fma
          -1.54320987654321e-5
          (* PI PI)
          (* 3.969161205100849e-11 (* (* angle angle) (pow PI 4.0))))))
       (sqrt 2.0)))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a_m <= 4e-79) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (a_m * ((1.0 + ((angle * angle) * fma(-1.54320987654321e-5, (((double) M_PI) * ((double) M_PI)), (3.969161205100849e-11 * ((angle * angle) * pow(((double) M_PI), 4.0)))))) * sqrt(2.0))));
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a_m <= 4e-79)
		tmp = Float64(b_m * y_45_scale_m);
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * Float64(a_m * Float64(Float64(1.0 + Float64(Float64(angle * angle) * fma(-1.54320987654321e-5, Float64(pi * pi), Float64(3.969161205100849e-11 * Float64(Float64(angle * angle) * (pi ^ 4.0)))))) * sqrt(2.0)))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a$95$m, 4e-79], N[(b$95$m * y$45$scale$95$m), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(a$95$m * N[(N[(1.0 + N[(N[(angle * angle), $MachinePrecision] * N[(-1.54320987654321e-5 * N[(Pi * Pi), $MachinePrecision] + N[(3.969161205100849e-11 * N[(N[(angle * angle), $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4e-79

   1. Initial program 2.1%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 44.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

   5. Taylor expanded in b around 0

      \[\leadsto b \cdot \color{blue}{y-scale} \]
   6. Step-by-step derivation

      1. lower-*.f64 44.0

         \[\leadsto b \cdot y-scale \]

   7. Applied rewrites 44.0%

      \[\leadsto b \cdot \color{blue}{y-scale} \]

   if 4e-79 < a

   1. Initial program 3.1%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-sqrt.f64 35.0

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

   7. Applied rewrites 35.0%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]
   9. Step-by-step derivation

      1. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(\frac{-1}{64800}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(\frac{-1}{64800}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(\frac{-1}{64800}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(\frac{-1}{64800}, \pi \cdot \mathsf{PI}\left(\right), \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(\frac{-1}{64800}, \pi \cdot \pi, \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(\frac{-1}{64800}, \pi \cdot \pi, \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(\frac{-1}{64800}, \pi \cdot \pi, \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(\frac{-1}{64800}, \pi \cdot \pi, \frac{1}{25194240000} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(\frac{-1}{64800}, \pi \cdot \pi, \frac{1}{25194240000} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(\frac{-1}{64800}, \pi \cdot \pi, \frac{1}{25194240000} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

      16. lift-PI.f64 48.7

          \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5}, \pi \cdot \pi, 3.969161205100849 \cdot 10^{-11} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]

   10. Applied rewrites 48.7%

       \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(\left(1 + \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5}, \pi \cdot \pi, 3.969161205100849 \cdot 10^{-11} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \sqrt{2}\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 43.3% accurate, 78.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 2.5 \cdot 10^{-78}:\\ \;\;\;\;b\_m \cdot y-scale\_m\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot \left(0.25 \cdot \sqrt{16 \cdot \left(a\_m \cdot a\_m\right)}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a_m 2.5e-78)
   (* b_m y-scale_m)
   (* x-scale_m (* 0.25 (sqrt (* 16.0 (* a_m a_m)))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a_m <= 2.5e-78) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = x_45_scale_m * (0.25 * sqrt((16.0 * (a_m * a_m))));
	}
	return tmp;
}

Fortran

a_m =     private
b_m =     private
x-scale_m =     private
y-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
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_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (a_m <= 2.5d-78) then
        tmp = b_m * y_45scale_m
    else
        tmp = x_45scale_m * (0.25d0 * sqrt((16.0d0 * (a_m * a_m))))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a_m <= 2.5e-78) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = x_45_scale_m * (0.25 * Math.sqrt((16.0 * (a_m * a_m))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a_m <= 2.5e-78:
		tmp = b_m * y_45_scale_m
	else:
		tmp = x_45_scale_m * (0.25 * math.sqrt((16.0 * (a_m * a_m))))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a_m <= 2.5e-78)
		tmp = Float64(b_m * y_45_scale_m);
	else
		tmp = Float64(x_45_scale_m * Float64(0.25 * sqrt(Float64(16.0 * Float64(a_m * a_m)))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a_m <= 2.5e-78)
		tmp = b_m * y_45_scale_m;
	else
		tmp = x_45_scale_m * (0.25 * sqrt((16.0 * (a_m * a_m))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a$95$m, 2.5e-78], N[(b$95$m * y$45$scale$95$m), $MachinePrecision], N[(x$45$scale$95$m * N[(0.25 * N[Sqrt[N[(16.0 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.4999999999999998e-78

   1. Initial program 2.1%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 44.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

   5. Taylor expanded in b around 0

      \[\leadsto b \cdot \color{blue}{y-scale} \]
   6. Step-by-step derivation

      1. lower-*.f64 44.0

         \[\leadsto b \cdot y-scale \]

   7. Applied rewrites 44.0%

      \[\leadsto b \cdot \color{blue}{y-scale} \]

   if 2.4999999999999998e-78 < a

   1. Initial program 3.1%

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

   2. Taylor expanded in x-scale around inf

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

   4. Applied rewrites 5.2%

      \[\leadsto \color{blue}{x-scale \cdot \mathsf{fma}\left(0.125, \frac{y-scale \cdot \left(\sqrt{8} \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right) \cdot \frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}, 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}{\frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}}, {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(2, \frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}, 2 \cdot \frac{{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}}, 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\mathsf{fma}\left(2, \frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}, 2 \cdot \frac{{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}\right)\right)} \]

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 49.1%

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

   8. Taylor expanded in angle around 0

      \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{16 \cdot {a}^{2}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{16 \cdot {a}^{2}}\right) \]

      2. pow2 N/A

         \[\leadsto x-scale \cdot \left(\frac{1}{4} \cdot \sqrt{16 \cdot \left(a \cdot a\right)}\right) \]

      3. lift-*.f64 42.9

         \[\leadsto x-scale \cdot \left(0.25 \cdot \sqrt{16 \cdot \left(a \cdot a\right)}\right) \]

   10. Applied rewrites 42.9%

       \[\leadsto x-scale \cdot \left(0.25 \cdot \sqrt{16 \cdot \left(a \cdot a\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 42.8% accurate, 132.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 2.5 \cdot 10^{-78}:\\ \;\;\;\;b\_m \cdot y-scale\_m\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(a\_m \cdot \left(x-scale\_m \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a_m 2.5e-78) (* b_m y-scale_m) (* 0.25 (* a_m (* x-scale_m 4.0)))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a_m <= 2.5e-78) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = 0.25 * (a_m * (x_45_scale_m * 4.0));
	}
	return tmp;
}

Fortran

a_m =     private
b_m =     private
x-scale_m =     private
y-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
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_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (a_m <= 2.5d-78) then
        tmp = b_m * y_45scale_m
    else
        tmp = 0.25d0 * (a_m * (x_45scale_m * 4.0d0))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a_m <= 2.5e-78) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = 0.25 * (a_m * (x_45_scale_m * 4.0));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a_m <= 2.5e-78:
		tmp = b_m * y_45_scale_m
	else:
		tmp = 0.25 * (a_m * (x_45_scale_m * 4.0))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a_m <= 2.5e-78)
		tmp = Float64(b_m * y_45_scale_m);
	else
		tmp = Float64(0.25 * Float64(a_m * Float64(x_45_scale_m * 4.0)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a_m <= 2.5e-78)
		tmp = b_m * y_45_scale_m;
	else
		tmp = 0.25 * (a_m * (x_45_scale_m * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a$95$m, 2.5e-78], N[(b$95$m * y$45$scale$95$m), $MachinePrecision], N[(0.25 * N[(a$95$m * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.4999999999999998e-78

   1. Initial program 2.1%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 44.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

   5. Taylor expanded in b around 0

      \[\leadsto b \cdot \color{blue}{y-scale} \]
   6. Step-by-step derivation

      1. lower-*.f64 44.0

         \[\leadsto b \cdot y-scale \]

   7. Applied rewrites 44.0%

      \[\leadsto b \cdot \color{blue}{y-scale} \]

   if 2.4999999999999998e-78 < a

   1. Initial program 3.1%

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

   2. Taylor expanded in x-scale around inf

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

   4. Applied rewrites 25.5%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \sqrt{16}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot 4\right)\right) \]

      5. lower-*.f64 42.1

         \[\leadsto 0.25 \cdot \left(a \cdot \left(x-scale \cdot 4\right)\right) \]

   7. Applied rewrites 42.1%

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

Alternative 8: 32.5% accurate, 484.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ b\_m \cdot y-scale\_m \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (* b_m y-scale_m))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return b_m * y_45_scale_m;
}

Fortran

a_m =     private
b_m =     private
x-scale_m =     private
y-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
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_m
    real(8), intent (in) :: y_45scale_m
    code = b_m * y_45scale_m
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return b_m * y_45_scale_m;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return b_m * y_45_scale_m

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(b_m * y_45_scale_m)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = b_m * y_45_scale_m;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(b$95$m * y$45$scale$95$m), $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 angle around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
3. Step-by-step derivation

4. Applied rewrites 32.5%

   \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

5. Taylor expanded in b around 0

   \[\leadsto b \cdot \color{blue}{y-scale} \]
6. Step-by-step derivation

   1. lower-*.f64 32.5

      \[\leadsto b \cdot y-scale \]

7. Applied rewrites 32.5%

   \[\leadsto b \cdot \color{blue}{y-scale} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025098 
(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))))