b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 44.0%

Time: 1.3min

Alternatives: 8

Speedup: 919.0×

• Could not converge on a ground truth

  1. reg0 = (bf #e5859071.11993806064128875732421875)
  2. reg1 = (bf "4.657185204154174409399676542607769299405e-295")
  3. reg2 = (bf "-2.235958058358246165593055494862762905467e213")
  4. reg3 = (bf "4.453327012456984610677469335610601650672e288")
  5. reg4 = (bf "1.610314102319441231010045947080057637315e211")

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: 0.1% 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: 44.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\\ \mathbf{if}\;y-scale\_m \leq 6.5 \cdot 10^{-87}:\\ \;\;\;\;0.25 \cdot \left(\left(b \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right) \cdot \sqrt{t\_0 - t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (pow (cos (* 0.005555555555555556 (* angle PI))) 2.0)))
   (if (<= y-scale_m 6.5e-87)
     (* 0.25 (* (* b (* y-scale_m (sqrt 8.0))) (sqrt (- t_0 t_0))))
     (* a_m x-scale_m))))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0);
	double tmp;
	if (y_45_scale_m <= 6.5e-87) {
		tmp = 0.25 * ((b * (y_45_scale_m * sqrt(8.0))) * sqrt((t_0 - t_0)));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 2.0);
	double tmp;
	if (y_45_scale_m <= 6.5e-87) {
		tmp = 0.25 * ((b * (y_45_scale_m * Math.sqrt(8.0))) * Math.sqrt((t_0 - t_0)));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.pow(math.cos((0.005555555555555556 * (angle * math.pi))), 2.0)
	tmp = 0
	if y_45_scale_m <= 6.5e-87:
		tmp = 0.25 * ((b * (y_45_scale_m * math.sqrt(8.0))) * math.sqrt((t_0 - t_0)))
	else:
		tmp = a_m * x_45_scale_m
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0
	tmp = 0.0
	if (y_45_scale_m <= 6.5e-87)
		tmp = Float64(0.25 * Float64(Float64(b * Float64(y_45_scale_m * sqrt(8.0))) * sqrt(Float64(t_0 - t_0))));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = cos((0.005555555555555556 * (angle * pi))) ^ 2.0;
	tmp = 0.0;
	if (y_45_scale_m <= 6.5e-87)
		tmp = 0.25 * ((b * (y_45_scale_m * sqrt(8.0))) * sqrt((t_0 - t_0)));
	else
		tmp = a_m * x_45_scale_m;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 6.5e-87], N[(0.25 * N[(N[(b * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 6.5000000000000003e-87

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0 3.5%

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

   4. Taylor expanded in x-scale around inf 0.4%

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

   6. Simplified 0.3%

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

   7. Taylor expanded in b around 0 38.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(b \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{-1 \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)} \]

   if 6.5000000000000003e-87 < y-scale

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 26.6%

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

      1. associate-*r* 26.6%

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

      2. *-commutative 26.6%

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

   5. Simplified 26.6%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)} \]
   6. Step-by-step derivation

      1. pow1 26.6%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 26.7%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)}^{1} \]

      3. metadata-eval 26.7%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \sqrt{\color{blue}{16}}\right)}^{1} \]

      4. metadata-eval 26.7%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{4}\right)}^{1} \]

   7. Applied egg-rr 26.7%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot 4\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 26.7%

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

   9. Simplified 26.7%

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

   10. Taylor expanded in a around 0 26.7%

       \[\leadsto \color{blue}{a \cdot x-scale} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 6.5 \cdot 10^{-87}:\\ \;\;\;\;0.25 \cdot \left(\left(b \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 35.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-144}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot \sqrt{8}\right) \cdot \sqrt{{b}^{2} - {b}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b 9.5e-144)
   (* 0.25 (* (* y-scale_m (sqrt 8.0)) (sqrt (- (pow b 2.0) (pow b 2.0)))))
   (* a_m x-scale_m)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 9.5e-144) {
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * sqrt((pow(b, 2.0) - pow(b, 2.0))));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Fortran

a_m = abs(a)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (b <= 9.5d-144) then
        tmp = 0.25d0 * ((y_45scale_m * sqrt(8.0d0)) * sqrt(((b ** 2.0d0) - (b ** 2.0d0))))
    else
        tmp = a_m * x_45scale_m
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 9.5e-144) {
		tmp = 0.25 * ((y_45_scale_m * Math.sqrt(8.0)) * Math.sqrt((Math.pow(b, 2.0) - Math.pow(b, 2.0))));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 9.5e-144:
		tmp = 0.25 * ((y_45_scale_m * math.sqrt(8.0)) * math.sqrt((math.pow(b, 2.0) - math.pow(b, 2.0))))
	else:
		tmp = a_m * x_45_scale_m
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 9.5e-144)
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * sqrt(8.0)) * sqrt(Float64((b ^ 2.0) - (b ^ 2.0)))));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b <= 9.5e-144)
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * sqrt(((b ^ 2.0) - (b ^ 2.0))));
	else
		tmp = a_m * x_45_scale_m;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 9.5e-144], N[(0.25 * N[(N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.49999999999999953e-144

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0 6.5%

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

   4. Taylor expanded in x-scale around inf 0.3%

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

   6. Simplified 0.3%

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

   7. Taylor expanded in angle around 0 33.3%

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

   if 9.49999999999999953e-144 < b

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 23.1%

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

      1. associate-*r* 23.1%

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

      2. *-commutative 23.1%

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

   5. Simplified 23.1%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)} \]
   6. Step-by-step derivation

      1. pow1 23.1%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 23.3%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)}^{1} \]

      3. metadata-eval 23.3%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \sqrt{\color{blue}{16}}\right)}^{1} \]

      4. metadata-eval 23.3%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{4}\right)}^{1} \]

   7. Applied egg-rr 23.3%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot 4\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 23.3%

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

   9. Simplified 23.3%

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

   10. Taylor expanded in a around 0 23.3%

       \[\leadsto \color{blue}{a \cdot x-scale} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-144}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{{b}^{2} - {b}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 34.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-178}:\\ \;\;\;\;\left(0.25 \cdot a\_m\right) \cdot \left(-1 + e^{\mathsf{log1p}\left(x-scale\_m \cdot 4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b 7.5e-178)
   (* (* 0.25 a_m) (+ -1.0 (exp (log1p (* x-scale_m 4.0)))))
   (* a_m x-scale_m)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 7.5e-178) {
		tmp = (0.25 * a_m) * (-1.0 + exp(log1p((x_45_scale_m * 4.0))));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 7.5e-178) {
		tmp = (0.25 * a_m) * (-1.0 + Math.exp(Math.log1p((x_45_scale_m * 4.0))));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 7.5e-178:
		tmp = (0.25 * a_m) * (-1.0 + math.exp(math.log1p((x_45_scale_m * 4.0))))
	else:
		tmp = a_m * x_45_scale_m
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 7.5e-178)
		tmp = Float64(Float64(0.25 * a_m) * Float64(-1.0 + exp(log1p(Float64(x_45_scale_m * 4.0)))));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 7.5e-178], N[(N[(0.25 * a$95$m), $MachinePrecision] * N[(-1.0 + N[Exp[N[Log[1 + N[(x$45$scale$95$m * 4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.50000000000000019e-178

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0 27.0%

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

      1. associate-*r* 27.0%

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

      2. *-commutative 27.0%

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

   5. Simplified 27.0%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 23.1%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      2. expm1-undefine 31.3%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)} - 1\right)} \]

      3. sqrt-unprod 31.3%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \left(e^{\mathsf{log1p}\left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)} - 1\right) \]

      4. metadata-eval 31.3%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \left(e^{\mathsf{log1p}\left(x-scale \cdot \sqrt{\color{blue}{16}}\right)} - 1\right) \]

      5. metadata-eval 31.3%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \left(e^{\mathsf{log1p}\left(x-scale \cdot \color{blue}{4}\right)} - 1\right) \]

   7. Applied egg-rr 31.3%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x-scale \cdot 4\right)} - 1\right)} \]

   if 7.50000000000000019e-178 < b

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 23.1%

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

      1. associate-*r* 23.1%

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

      2. *-commutative 23.1%

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

   5. Simplified 23.1%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)} \]
   6. Step-by-step derivation

      1. pow1 23.1%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 23.2%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)}^{1} \]

      3. metadata-eval 23.2%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \sqrt{\color{blue}{16}}\right)}^{1} \]

      4. metadata-eval 23.2%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{4}\right)}^{1} \]

   7. Applied egg-rr 23.2%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot 4\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 23.2%

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

   9. Simplified 23.2%

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

   10. Taylor expanded in a around 0 23.2%

       \[\leadsto \color{blue}{a \cdot x-scale} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-178}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(-1 + e^{\mathsf{log1p}\left(x-scale \cdot 4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 30.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-178}:\\ \;\;\;\;\left(0.25 \cdot a\_m\right) \cdot \log \left(1 + \mathsf{expm1}\left(x-scale\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b 6.5e-178)
   (* (* 0.25 a_m) (log (+ 1.0 (expm1 (* x-scale_m 4.0)))))
   (* a_m x-scale_m)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 6.5e-178) {
		tmp = (0.25 * a_m) * log((1.0 + expm1((x_45_scale_m * 4.0))));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 6.5e-178) {
		tmp = (0.25 * a_m) * Math.log((1.0 + Math.expm1((x_45_scale_m * 4.0))));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 6.5e-178:
		tmp = (0.25 * a_m) * math.log((1.0 + math.expm1((x_45_scale_m * 4.0))))
	else:
		tmp = a_m * x_45_scale_m
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 6.5e-178)
		tmp = Float64(Float64(0.25 * a_m) * log(Float64(1.0 + expm1(Float64(x_45_scale_m * 4.0)))));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 6.5e-178], N[(N[(0.25 * a$95$m), $MachinePrecision] * N[Log[N[(1.0 + N[(Exp[N[(x$45$scale$95$m * 4.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.5000000000000002e-178

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0 27.0%

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

      1. associate-*r* 27.0%

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

      2. *-commutative 27.0%

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

   5. Simplified 27.0%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 22.3%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      2. log1p-undefine 30.5%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      3. sqrt-unprod 30.5%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \log \left(1 + \mathsf{expm1}\left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      4. metadata-eval 30.5%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \log \left(1 + \mathsf{expm1}\left(x-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      5. metadata-eval 30.5%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \log \left(1 + \mathsf{expm1}\left(x-scale \cdot \color{blue}{4}\right)\right) \]

   7. Applied egg-rr 30.5%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(x-scale \cdot 4\right)\right)} \]

   if 6.5000000000000002e-178 < b

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 23.1%

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

      1. associate-*r* 23.1%

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

      2. *-commutative 23.1%

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

   5. Simplified 23.1%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)} \]
   6. Step-by-step derivation

      1. pow1 23.1%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 23.2%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)}^{1} \]

      3. metadata-eval 23.2%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \sqrt{\color{blue}{16}}\right)}^{1} \]

      4. metadata-eval 23.2%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{4}\right)}^{1} \]

   7. Applied egg-rr 23.2%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot 4\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 23.2%

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

   9. Simplified 23.2%

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

   10. Taylor expanded in a around 0 23.2%

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

Alternative 5: 36.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 1.8 \cdot 10^{-198}:\\ \;\;\;\;\left(x-scale\_m \cdot 4\right) \cdot \log \left(1 + \mathsf{expm1}\left(0.25 \cdot a\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 1.8e-198)
   (* (* x-scale_m 4.0) (log (+ 1.0 (expm1 (* 0.25 a_m)))))
   (* a_m x-scale_m)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 1.8e-198) {
		tmp = (x_45_scale_m * 4.0) * log((1.0 + expm1((0.25 * a_m))));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 1.8e-198) {
		tmp = (x_45_scale_m * 4.0) * Math.log((1.0 + Math.expm1((0.25 * a_m))));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 1.8e-198:
		tmp = (x_45_scale_m * 4.0) * math.log((1.0 + math.expm1((0.25 * a_m))))
	else:
		tmp = a_m * x_45_scale_m
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 1.8e-198)
		tmp = Float64(Float64(x_45_scale_m * 4.0) * log(Float64(1.0 + expm1(Float64(0.25 * a_m)))));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 1.8e-198], N[(N[(x$45$scale$95$m * 4.0), $MachinePrecision] * N[Log[N[(1.0 + N[(Exp[N[(0.25 * a$95$m), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.79999999999999999e-198

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0 24.4%

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

      1. associate-*r* 24.4%

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

      2. *-commutative 24.4%

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

   5. Simplified 24.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)} \]
   6. Step-by-step derivation

      1. pow1 24.4%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 24.6%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)}^{1} \]

      3. metadata-eval 24.6%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \sqrt{\color{blue}{16}}\right)}^{1} \]

      4. metadata-eval 24.6%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{4}\right)}^{1} \]

   7. Applied egg-rr 24.6%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot 4\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 24.6%

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

   9. Simplified 24.6%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{\left(x-scale \cdot 4\right)} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 19.9%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.25 \cdot a\right)\right)} \cdot \left(x-scale \cdot 4\right) \]

       2. log1p-undefine 25.4%

          \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.25 \cdot a\right)\right)} \cdot \left(x-scale \cdot 4\right) \]

   11. Applied egg-rr 25.4%

       \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.25 \cdot a\right)\right)} \cdot \left(x-scale \cdot 4\right) \]

   if 1.79999999999999999e-198 < y-scale

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 26.8%

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

      1. associate-*r* 26.8%

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

      2. *-commutative 26.8%

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

   5. Simplified 26.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)} \]
   6. Step-by-step derivation

      1. pow1 26.8%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 26.9%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)}^{1} \]

      3. metadata-eval 26.9%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \sqrt{\color{blue}{16}}\right)}^{1} \]

      4. metadata-eval 26.9%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{4}\right)}^{1} \]

   7. Applied egg-rr 26.9%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot 4\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 26.9%

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

   9. Simplified 26.9%

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

   10. Taylor expanded in a around 0 26.9%

       \[\leadsto \color{blue}{a \cdot x-scale} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.8 \cdot 10^{-198}:\\ \;\;\;\;\left(x-scale \cdot 4\right) \cdot \log \left(1 + \mathsf{expm1}\left(0.25 \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 1.8 \cdot 10^{-198}:\\ \;\;\;\;\left(x-scale\_m \cdot 4\right) \cdot \sqrt[3]{{\left(0.25 \cdot a\_m\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 1.8e-198)
   (* (* x-scale_m 4.0) (cbrt (pow (* 0.25 a_m) 3.0)))
   (* a_m x-scale_m)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 1.8e-198) {
		tmp = (x_45_scale_m * 4.0) * cbrt(pow((0.25 * a_m), 3.0));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 1.8e-198) {
		tmp = (x_45_scale_m * 4.0) * Math.cbrt(Math.pow((0.25 * a_m), 3.0));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 1.8e-198)
		tmp = Float64(Float64(x_45_scale_m * 4.0) * cbrt((Float64(0.25 * a_m) ^ 3.0)));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 1.8e-198], N[(N[(x$45$scale$95$m * 4.0), $MachinePrecision] * N[Power[N[Power[N[(0.25 * a$95$m), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.79999999999999999e-198

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0 24.4%

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

      1. associate-*r* 24.4%

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

      2. *-commutative 24.4%

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

   5. Simplified 24.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)} \]
   6. Step-by-step derivation

      1. pow1 24.4%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 24.6%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)}^{1} \]

      3. metadata-eval 24.6%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \sqrt{\color{blue}{16}}\right)}^{1} \]

      4. metadata-eval 24.6%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{4}\right)}^{1} \]

   7. Applied egg-rr 24.6%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot 4\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 24.6%

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

   9. Simplified 24.6%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{\left(x-scale \cdot 4\right)} \]
   10. Step-by-step derivation

       1. add-cbrt-cube 28.4%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.25 \cdot a\right) \cdot \left(0.25 \cdot a\right)\right) \cdot \left(0.25 \cdot a\right)}} \cdot \left(x-scale \cdot 4\right) \]

       2. pow3 28.3%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(0.25 \cdot a\right)}^{3}}} \cdot \left(x-scale \cdot 4\right) \]

   11. Applied egg-rr 28.3%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(0.25 \cdot a\right)}^{3}}} \cdot \left(x-scale \cdot 4\right) \]

   if 1.79999999999999999e-198 < y-scale

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 26.8%

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

      1. associate-*r* 26.8%

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

      2. *-commutative 26.8%

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

   5. Simplified 26.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)} \]
   6. Step-by-step derivation

      1. pow1 26.8%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 26.9%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)}^{1} \]

      3. metadata-eval 26.9%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \sqrt{\color{blue}{16}}\right)}^{1} \]

      4. metadata-eval 26.9%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{4}\right)}^{1} \]

   7. Applied egg-rr 26.9%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot 4\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 26.9%

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

   9. Simplified 26.9%

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

   10. Taylor expanded in a around 0 26.9%

       \[\leadsto \color{blue}{a \cdot x-scale} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.8 \cdot 10^{-198}:\\ \;\;\;\;\left(x-scale \cdot 4\right) \cdot \sqrt[3]{{\left(0.25 \cdot a\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.8% accurate, 25.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-199}:\\ \;\;\;\;1 + \mathsf{fma}\left(a\_m, x-scale\_m, -1\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b 1.35e-199) (+ 1.0 (fma a_m x-scale_m -1.0)) (* a_m x-scale_m)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 1.35e-199) {
		tmp = 1.0 + fma(a_m, x_45_scale_m, -1.0);
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 1.35e-199)
		tmp = Float64(1.0 + fma(a_m, x_45_scale_m, -1.0));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 1.35e-199], N[(1.0 + N[(a$95$m * x$45$scale$95$m + -1.0), $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.34999999999999995e-199

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0 27.4%

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

      1. associate-*r* 27.4%

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

      2. *-commutative 27.4%

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

   5. Simplified 27.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)} \]
   6. Step-by-step derivation

      1. pow1 27.4%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 27.6%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)}^{1} \]

      3. metadata-eval 27.6%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \sqrt{\color{blue}{16}}\right)}^{1} \]

      4. metadata-eval 27.6%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{4}\right)}^{1} \]

   7. Applied egg-rr 27.6%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot 4\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 27.6%

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

   9. Simplified 27.6%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{\left(x-scale \cdot 4\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 26.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot 4\right)\right)\right)} \]

       2. expm1-undefine 29.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot 4\right)\right)} - 1} \]

       3. associate-*l* 29.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.25 \cdot \left(a \cdot \left(x-scale \cdot 4\right)\right)}\right)} - 1 \]

   11. Applied egg-rr 29.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.25 \cdot \left(a \cdot \left(x-scale \cdot 4\right)\right)\right)} - 1} \]
   12. Step-by-step derivation

       1. associate-*r* 29.1%

          \[\leadsto e^{\mathsf{log1p}\left(0.25 \cdot \color{blue}{\left(\left(a \cdot x-scale\right) \cdot 4\right)}\right)} - 1 \]

       2. log1p-undefine 29.1%

          \[\leadsto e^{\color{blue}{\log \left(1 + 0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot 4\right)\right)}} - 1 \]

       3. associate-*r* 29.1%

          \[\leadsto e^{\log \left(1 + 0.25 \cdot \color{blue}{\left(a \cdot \left(x-scale \cdot 4\right)\right)}\right)} - 1 \]

       4. rem-exp-log 29.8%

          \[\leadsto \color{blue}{\left(1 + 0.25 \cdot \left(a \cdot \left(x-scale \cdot 4\right)\right)\right)} - 1 \]

       5. associate-+r- 29.8%

          \[\leadsto \color{blue}{1 + \left(0.25 \cdot \left(a \cdot \left(x-scale \cdot 4\right)\right) - 1\right)} \]

       6. associate-*r* 29.8%

          \[\leadsto 1 + \left(\color{blue}{\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot 4\right)} - 1\right) \]

       7. *-commutative 29.8%

          \[\leadsto 1 + \left(\color{blue}{\left(x-scale \cdot 4\right) \cdot \left(0.25 \cdot a\right)} - 1\right) \]

       8. associate-*l* 29.8%

          \[\leadsto 1 + \left(\color{blue}{x-scale \cdot \left(4 \cdot \left(0.25 \cdot a\right)\right)} - 1\right) \]

       9. fma-neg 29.8%

          \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(x-scale, 4 \cdot \left(0.25 \cdot a\right), -1\right)} \]

       10. associate-*r* 29.8%

           \[\leadsto 1 + \mathsf{fma}\left(x-scale, \color{blue}{\left(4 \cdot 0.25\right) \cdot a}, -1\right) \]

       11. metadata-eval 29.8%

           \[\leadsto 1 + \mathsf{fma}\left(x-scale, \color{blue}{1} \cdot a, -1\right) \]

       12. *-lft-identity 29.8%

           \[\leadsto 1 + \mathsf{fma}\left(x-scale, \color{blue}{a}, -1\right) \]

       13. fma-neg 29.8%

           \[\leadsto 1 + \color{blue}{\left(x-scale \cdot a - 1\right)} \]

       14. *-commutative 29.8%

           \[\leadsto 1 + \left(\color{blue}{a \cdot x-scale} - 1\right) \]

       15. fma-neg 29.8%

           \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(a, x-scale, -1\right)} \]

       16. metadata-eval 29.8%

           \[\leadsto 1 + \mathsf{fma}\left(a, x-scale, \color{blue}{-1}\right) \]

   13. Simplified 29.8%

       \[\leadsto \color{blue}{1 + \mathsf{fma}\left(a, x-scale, -1\right)} \]

   if 1.34999999999999995e-199 < b

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 22.4%

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

      1. associate-*r* 22.4%

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

      2. *-commutative 22.4%

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

   5. Simplified 22.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)} \]
   6. Step-by-step derivation

      1. pow1 22.4%

         \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 22.6%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)}^{1} \]

      3. metadata-eval 22.6%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \sqrt{\color{blue}{16}}\right)}^{1} \]

      4. metadata-eval 22.6%

         \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{4}\right)}^{1} \]

   7. Applied egg-rr 22.6%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot 4\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 22.6%

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

   9. Simplified 22.6%

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

   10. Taylor expanded in a around 0 22.6%

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

Alternative 8: 32.6% accurate, 919.0× speedup

Math

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

FPCore

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

C

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

Fortran

a_m = abs(a)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = a_m * x_45scale_m
end function

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	return a_m * x_45_scale_m;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.0%

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

3. Taylor expanded in angle around 0 25.5%

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

   1. associate-*r* 25.5%

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

   2. *-commutative 25.5%

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

5. Simplified 25.5%

   \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)} \]
6. Step-by-step derivation

   1. pow1 25.5%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}^{1}} \]

   2. sqrt-unprod 25.6%

      \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)}^{1} \]

   3. metadata-eval 25.6%

      \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \sqrt{\color{blue}{16}}\right)}^{1} \]

   4. metadata-eval 25.6%

      \[\leadsto \left(0.25 \cdot a\right) \cdot {\left(x-scale \cdot \color{blue}{4}\right)}^{1} \]

7. Applied egg-rr 25.6%

   \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{{\left(x-scale \cdot 4\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 25.6%

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

9. Simplified 25.6%

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

10. Taylor expanded in a around 0 25.6%

    \[\leadsto \color{blue}{a \cdot x-scale} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024148 
(FPCore (a b angle x-scale y-scale)
  :name "b 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))))