b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 43.3%

Time: 1.4min

Alternatives: 7

Speedup: 919.0×

• Could not converge on a ground truth

  1. a = 2.9502036831713314e-34
  2. b = 3.646004943625498e+199
  3. angle = -1.1351786212857545e-19
  4. x-scale = 3.281535657384979e-9
  5. y-scale = 82247418659596530.0

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: 43.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 1.9 \cdot 10^{-51}:\\ \;\;\;\;-0.25 \cdot \left(a\_m \cdot \left(\left(\left(y-scale\_m \cdot angle\right) \cdot \sqrt{8}\right) \cdot \sqrt{0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

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 1.9e-51)
   (* -0.25 (* a_m (* (* (* y-scale_m angle) (sqrt 8.0)) (sqrt 0.0))))
   (* 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 <= 1.9e-51) {
		tmp = -0.25 * (a_m * (((y_45_scale_m * angle) * sqrt(8.0)) * sqrt(0.0)));
	} 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 (y_45scale_m <= 1.9d-51) then
        tmp = (-0.25d0) * (a_m * (((y_45scale_m * angle) * sqrt(8.0d0)) * sqrt(0.0d0)))
    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 (y_45_scale_m <= 1.9e-51) {
		tmp = -0.25 * (a_m * (((y_45_scale_m * angle) * Math.sqrt(8.0)) * Math.sqrt(0.0)));
	} 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 y_45_scale_m <= 1.9e-51:
		tmp = -0.25 * (a_m * (((y_45_scale_m * angle) * math.sqrt(8.0)) * math.sqrt(0.0)))
	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 <= 1.9e-51)
		tmp = Float64(-0.25 * Float64(a_m * Float64(Float64(Float64(y_45_scale_m * angle) * sqrt(8.0)) * sqrt(0.0))));
	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 (y_45_scale_m <= 1.9e-51)
		tmp = -0.25 * (a_m * (((y_45_scale_m * angle) * sqrt(8.0)) * sqrt(0.0)));
	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[y$45$scale$95$m, 1.9e-51], N[(-0.25 * N[(a$95$m * N[(N[(N[(y$45$scale$95$m * angle), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.0], $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 < 1.90000000000000001e-51

   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 a around inf 2.1%

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

   5. Simplified 2.1%

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

   6. Taylor expanded in angle around 0 2.2%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(a \cdot \left(angle \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right)\right) \cdot \sqrt{\left(-3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{y-scale}^{2}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}\right) - 0.5 \cdot \left({y-scale}^{2} \cdot \left(-2 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}} - -3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}} + 0.0001234567901234568 \cdot \frac{{\pi}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)}\right)} \]

   7. Taylor expanded in x-scale around -inf 6.1%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(a \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - 0.5 \cdot \left({y-scale}^{2} \cdot \left(-6.17283950617284 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{y-scale}^{2}} + 0.0001234567901234568 \cdot \frac{{\pi}^{2}}{{y-scale}^{2}}\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 6.1%

         \[\leadsto \color{blue}{\left(-0.25 \cdot \left(a \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right)\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - 0.5 \cdot \left({y-scale}^{2} \cdot \left(-6.17283950617284 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{y-scale}^{2}} + 0.0001234567901234568 \cdot \frac{{\pi}^{2}}{{y-scale}^{2}}\right)\right)}} \]

      2. *-commutative 6.1%

         \[\leadsto \left(-0.25 \cdot \left(a \cdot \left(angle \cdot \color{blue}{\left(\sqrt{8} \cdot y-scale\right)}\right)\right)\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - 0.5 \cdot \left({y-scale}^{2} \cdot \left(-6.17283950617284 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{y-scale}^{2}} + 0.0001234567901234568 \cdot \frac{{\pi}^{2}}{{y-scale}^{2}}\right)\right)} \]

      3. fmm-def 3.3%

         \[\leadsto \left(-0.25 \cdot \left(a \cdot \left(angle \cdot \left(\sqrt{8} \cdot y-scale\right)\right)\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\pi}^{2}, -0.5 \cdot \left({y-scale}^{2} \cdot \left(-6.17283950617284 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{y-scale}^{2}} + 0.0001234567901234568 \cdot \frac{{\pi}^{2}}{{y-scale}^{2}}\right)\right)\right)}} \]

      4. *-commutative 3.3%

         \[\leadsto \left(-0.25 \cdot \left(a \cdot \left(angle \cdot \left(\sqrt{8} \cdot y-scale\right)\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\pi}^{2}, -\color{blue}{\left({y-scale}^{2} \cdot \left(-6.17283950617284 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{y-scale}^{2}} + 0.0001234567901234568 \cdot \frac{{\pi}^{2}}{{y-scale}^{2}}\right)\right) \cdot 0.5}\right)} \]

   9. Simplified 3.3%

      \[\leadsto \color{blue}{\left(-0.25 \cdot \left(a \cdot \left(angle \cdot \left(\sqrt{8} \cdot y-scale\right)\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\pi}^{2}, \left({y-scale}^{2} \cdot \left(\frac{{\pi}^{2}}{{y-scale}^{2}} \cdot 6.17283950617284 \cdot 10^{-5}\right)\right) \cdot -0.5\right)}} \]

   10. Taylor expanded in a around 0 30.2%

       \[\leadsto \color{blue}{-0.25 \cdot \left(\left(a \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{-3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2}}\right)} \]
   11. Step-by-step derivation

       1. associate-*l* 32.1%

          \[\leadsto -0.25 \cdot \color{blue}{\left(a \cdot \left(\left(angle \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{-3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2}}\right)\right)} \]

       2. associate-*r* 32.6%

          \[\leadsto -0.25 \cdot \left(a \cdot \left(\color{blue}{\left(\left(angle \cdot y-scale\right) \cdot \sqrt{8}\right)} \cdot \sqrt{-3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2}}\right)\right) \]

       3. distribute-rgt-out 32.6%

          \[\leadsto -0.25 \cdot \left(a \cdot \left(\left(\left(angle \cdot y-scale\right) \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{{\pi}^{2} \cdot \left(-3.08641975308642 \cdot 10^{-5} + 3.08641975308642 \cdot 10^{-5}\right)}}\right)\right) \]

       4. metadata-eval 32.6%

          \[\leadsto -0.25 \cdot \left(a \cdot \left(\left(\left(angle \cdot y-scale\right) \cdot \sqrt{8}\right) \cdot \sqrt{{\pi}^{2} \cdot \color{blue}{0}}\right)\right) \]

       5. mul0-rgt 32.6%

          \[\leadsto -0.25 \cdot \left(a \cdot \left(\left(\left(angle \cdot y-scale\right) \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{0}}\right)\right) \]

   12. Simplified 32.6%

       \[\leadsto \color{blue}{-0.25 \cdot \left(a \cdot \left(\left(\left(angle \cdot y-scale\right) \cdot \sqrt{8}\right) \cdot \sqrt{0}\right)\right)} \]

   if 1.90000000000000001e-51 < y-scale

   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 35.7%

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

      1. associate-*r* 35.7%

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

      2. *-commutative 35.7%

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

   5. Simplified 35.7%

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

      1. sqrt-unprod 35.9%

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

      2. metadata-eval 35.9%

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

      3. metadata-eval 35.9%

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

   7. Applied egg-rr 35.9%

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

   8. Taylor expanded in a around 0 35.9%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.9 \cdot 10^{-51}:\\ \;\;\;\;-0.25 \cdot \left(a \cdot \left(\left(\left(y-scale \cdot angle\right) \cdot \sqrt{8}\right) \cdot \sqrt{0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 37.6% accurate, 12.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 2.2 \cdot 10^{-127}:\\ \;\;\;\;\left(e^{\mathsf{log1p}\left(a\_m \cdot 0.25\right)} + -1\right) \cdot \left(x-scale\_m \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

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 2.2e-127)
   (* (+ (exp (log1p (* a_m 0.25))) -1.0) (* 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 (y_45_scale_m <= 2.2e-127) {
		tmp = (exp(log1p((a_m * 0.25))) + -1.0) * (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 (y_45_scale_m <= 2.2e-127) {
		tmp = (Math.exp(Math.log1p((a_m * 0.25))) + -1.0) * (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 y_45_scale_m <= 2.2e-127:
		tmp = (math.exp(math.log1p((a_m * 0.25))) + -1.0) * (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 (y_45_scale_m <= 2.2e-127)
		tmp = Float64(Float64(exp(log1p(Float64(a_m * 0.25))) + -1.0) * 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[y$45$scale$95$m, 2.2e-127], N[(N[(N[Exp[N[Log[1 + N[(a$95$m * 0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.2000000000000001e-127

   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 18.0%

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

      1. associate-*r* 18.0%

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

      2. *-commutative 18.0%

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

   5. Simplified 18.0%

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

      1. sqrt-unprod 18.2%

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

      2. metadata-eval 18.2%

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

      3. metadata-eval 18.2%

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

   7. Applied egg-rr 18.2%

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

      1. expm1-log1p-u 15.3%

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

      2. expm1-undefine 24.4%

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

   9. Applied egg-rr 24.4%

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

   if 2.2000000000000001e-127 < y-scale

   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 32.3%

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

      1. associate-*r* 32.3%

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

      2. *-commutative 32.3%

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

   5. Simplified 32.3%

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

      1. sqrt-unprod 32.5%

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

      2. metadata-eval 32.5%

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

      3. metadata-eval 32.5%

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

   7. Applied egg-rr 32.5%

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

   8. Taylor expanded in a around 0 32.5%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.2 \cdot 10^{-127}:\\ \;\;\;\;\left(e^{\mathsf{log1p}\left(a \cdot 0.25\right)} + -1\right) \cdot \left(x-scale \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 34.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\left(a\_m \cdot 0.25\right) \cdot \left(e^{\mathsf{log1p}\left(x-scale\_m \cdot 4\right)} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

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 5e-78)
   (* (* a_m 0.25) (+ (exp (log1p (* x-scale_m 4.0))) -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 <= 5e-78) {
		tmp = (a_m * 0.25) * (exp(log1p((x_45_scale_m * 4.0))) + -1.0);
	} 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 <= 5e-78) {
		tmp = (a_m * 0.25) * (Math.exp(Math.log1p((x_45_scale_m * 4.0))) + -1.0);
	} 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 <= 5e-78:
		tmp = (a_m * 0.25) * (math.exp(math.log1p((x_45_scale_m * 4.0))) + -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 <= 5e-78)
		tmp = Float64(Float64(a_m * 0.25) * Float64(exp(log1p(Float64(x_45_scale_m * 4.0))) + -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, 5e-78], N[(N[(a$95$m * 0.25), $MachinePrecision] * N[(N[Exp[N[Log[1 + N[(x$45$scale$95$m * 4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.9999999999999996e-78

   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 23.3%

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

      1. associate-*r* 23.3%

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

      2. *-commutative 23.3%

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

   5. Simplified 23.3%

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

      1. expm1-log1p-u 21.3%

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

      2. expm1-undefine 30.0%

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

      3. sqrt-unprod 30.0%

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

      4. metadata-eval 30.0%

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

      5. metadata-eval 30.0%

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

   7. Applied egg-rr 30.0%

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

   if 4.9999999999999996e-78 < b

   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 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. sqrt-unprod 23.3%

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

      2. metadata-eval 23.3%

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

      3. metadata-eval 23.3%

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

   7. Applied egg-rr 23.3%

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

   8. 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 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\left(a \cdot 0.25\right) \cdot \left(e^{\mathsf{log1p}\left(x-scale \cdot 4\right)} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.5% accurate, 13.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 9.2 \cdot 10^{-113}:\\ \;\;\;\;e^{\mathsf{log1p}\left(a\_m \cdot x-scale\_m\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

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 9.2e-113)
   (+ (exp (log1p (* 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 (y_45_scale_m <= 9.2e-113) {
		tmp = exp(log1p((a_m * x_45_scale_m))) + -1.0;
	} 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 <= 9.2e-113) {
		tmp = Math.exp(Math.log1p((a_m * x_45_scale_m))) + -1.0;
	} 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 y_45_scale_m <= 9.2e-113:
		tmp = math.exp(math.log1p((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 (y_45_scale_m <= 9.2e-113)
		tmp = Float64(exp(log1p(Float64(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[y$45$scale$95$m, 9.2e-113], N[(N[Exp[N[Log[1 + N[(a$95$m * x$45$scale$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 9.20000000000000032e-113

   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 18.2%

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

      1. associate-*r* 18.2%

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

      2. *-commutative 18.2%

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

   5. Simplified 18.2%

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

      1. associate-*l* 18.2%

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

      2. associate-*l* 18.2%

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

      3. add-exp-log 16.6%

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

      4. associate-*l* 16.6%

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

      5. sqrt-unprod 16.6%

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

      6. metadata-eval 16.6%

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

      7. metadata-eval 16.6%

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

   7. Applied egg-rr 16.6%

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

   8. Taylor expanded in a around 0 6.0%

      \[\leadsto e^{\color{blue}{\log a + \log x-scale}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 6.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\log a + \log x-scale}\right)\right)} \]

      2. expm1-undefine 5.7%

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

      3. sum-log 19.5%

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

      4. add-exp-log 26.5%

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

   10. Applied egg-rr 26.5%

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

   if 9.20000000000000032e-113 < y-scale

   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 32.9%

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

      1. associate-*r* 32.9%

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

      2. *-commutative 32.9%

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

   5. Simplified 32.9%

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

      1. sqrt-unprod 33.1%

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

      2. metadata-eval 33.1%

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

      3. metadata-eval 33.1%

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

   7. Applied egg-rr 33.1%

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

   8. Taylor expanded in a around 0 33.1%

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

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 9.2 \cdot 10^{-113}:\\ \;\;\;\;e^{\mathsf{log1p}\left(a \cdot x-scale\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 36.2% accurate, 13.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 6.5 \cdot 10^{-130}:\\ \;\;\;\;\log \left(1 + \mathsf{expm1}\left(a\_m \cdot x-scale\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

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 6.5e-130)
   (log (+ 1.0 (expm1 (* a_m x-scale_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 <= 6.5e-130) {
		tmp = log((1.0 + expm1((a_m * x_45_scale_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 <= 6.5e-130) {
		tmp = Math.log((1.0 + Math.expm1((a_m * x_45_scale_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 <= 6.5e-130:
		tmp = math.log((1.0 + math.expm1((a_m * x_45_scale_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 <= 6.5e-130)
		tmp = log(Float64(1.0 + expm1(Float64(a_m * x_45_scale_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, 6.5e-130], N[Log[N[(1.0 + N[(Exp[N[(a$95$m * x$45$scale$95$m), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 6.5000000000000002e-130

   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 18.0%

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

      1. associate-*r* 18.1%

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

      2. *-commutative 18.1%

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

   5. Simplified 18.1%

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

      1. associate-*l* 18.0%

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

      2. associate-*l* 18.0%

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

      3. add-exp-log 16.4%

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

      4. associate-*l* 16.4%

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

      5. sqrt-unprod 16.4%

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

      6. metadata-eval 16.4%

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

      7. metadata-eval 16.4%

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

   7. Applied egg-rr 16.4%

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

   8. Taylor expanded in a around 0 6.2%

      \[\leadsto e^{\color{blue}{\log a + \log x-scale}} \]
   9. Step-by-step derivation

      1. log1p-expm1-u 5.6%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{\log a + \log x-scale}\right)\right)} \]

      2. log1p-undefine 5.2%

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

      3. sum-log 17.0%

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

      4. add-exp-log 24.6%

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

   10. Applied egg-rr 24.6%

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

   if 6.5000000000000002e-130 < y-scale

   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 32.3%

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

      1. associate-*r* 32.3%

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

      2. *-commutative 32.3%

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

   5. Simplified 32.3%

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

      1. sqrt-unprod 32.5%

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

      2. metadata-eval 32.5%

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

      3. metadata-eval 32.5%

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

   7. Applied egg-rr 32.5%

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

   8. Taylor expanded in a around 0 32.5%

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

Alternative 6: 34.4% accurate, 13.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 2.8 \cdot 10^{-161}:\\ \;\;\;\;\sqrt[3]{{\left(a\_m \cdot x-scale\_m\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

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 2.8e-161)
   (cbrt (pow (* a_m x-scale_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 <= 2.8e-161) {
		tmp = cbrt(pow((a_m * x_45_scale_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 <= 2.8e-161) {
		tmp = Math.cbrt(Math.pow((a_m * x_45_scale_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 <= 2.8e-161)
		tmp = cbrt((Float64(a_m * x_45_scale_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, 2.8e-161], N[Power[N[Power[N[(a$95$m * x$45$scale$95$m), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.79999999999999992e-161

   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 18.7%

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

      1. associate-*r* 18.7%

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

      2. *-commutative 18.7%

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

   5. Simplified 18.7%

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

      1. associate-*l* 18.7%

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

      2. associate-*l* 18.7%

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

      3. add-exp-log 17.0%

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

      4. associate-*l* 17.0%

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

      5. sqrt-unprod 17.0%

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

      6. metadata-eval 17.0%

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

      7. metadata-eval 17.0%

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

   7. Applied egg-rr 17.0%

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

   8. Taylor expanded in a around 0 6.5%

      \[\leadsto e^{\color{blue}{\log a + \log x-scale}} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 5.4%

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

      2. pow3 5.4%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{\log a + \log x-scale}\right)}^{3}}} \]

      3. sum-log 16.5%

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

      4. add-exp-log 21.6%

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

   10. Applied egg-rr 21.6%

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

   if 2.79999999999999992e-161 < y-scale

   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 30.3%

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

      1. associate-*r* 30.3%

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

      2. *-commutative 30.3%

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

   5. Simplified 30.3%

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

      1. sqrt-unprod 30.6%

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

      2. metadata-eval 30.6%

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

      3. metadata-eval 30.6%

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

   7. Applied egg-rr 30.6%

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

   8. Taylor expanded in a around 0 30.6%

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

Alternative 7: 32.0% 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 0.1%

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

3. Taylor expanded in angle around 0 23.3%

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

   1. associate-*r* 23.3%

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

   2. *-commutative 23.3%

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

5. Simplified 23.3%

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

   1. sqrt-unprod 23.4%

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

   2. metadata-eval 23.4%

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

   3. metadata-eval 23.4%

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

7. Applied egg-rr 23.4%

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

8. Taylor expanded in a around 0 23.4%

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

Reproduce

herbie shell --seed 2024170 
(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))))