b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 49.1%

Time: 1.2min

Alternatives: 4

Speedup: 2908.0×

• could not determine a ground truth

  1. a = -2.0177330564033466e-268
  2. b = -2.6299436537917647e+286
  3. angle = 1.31852817046116e+298
  4. x-scale = 2.0487220691625383e-170
  5. y-scale = -6.559936397014789e-67

Specification

Math

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

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)))

Initial Program: 0.1% accurate, 1.0× speedup

Math

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

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)))

Alternative 1: 49.1% accurate, 103.7× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 4.4 \cdot 10^{-250}:\\ \;\;\;\;0\\ \mathbf{elif}\;b\_m \leq 3.5 \cdot 10^{-18}:\\ \;\;\;\;y-scale\_m \cdot b\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot a\_m\right) \cdot x-scale\_m\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b_m 4.4e-250)
   0.0
   (if (<= b_m 3.5e-18) (* y-scale_m b_m) (* (* (* 0.25 a_m) x-scale_m) 4.0))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b_m <= 4.4e-250) {
		tmp = 0.0;
	} else if (b_m <= 3.5e-18) {
		tmp = y_45_scale_m * b_m;
	} else {
		tmp = ((0.25 * a_m) * x_45_scale_m) * 4.0;
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (b_m <= 4.4d-250) then
        tmp = 0.0d0
    else if (b_m <= 3.5d-18) then
        tmp = y_45scale_m * b_m
    else
        tmp = ((0.25d0 * a_m) * x_45scale_m) * 4.0d0
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b_m <= 4.4e-250) {
		tmp = 0.0;
	} else if (b_m <= 3.5e-18) {
		tmp = y_45_scale_m * b_m;
	} else {
		tmp = ((0.25 * a_m) * x_45_scale_m) * 4.0;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b_m <= 4.4e-250:
		tmp = 0.0
	elif b_m <= 3.5e-18:
		tmp = y_45_scale_m * b_m
	else:
		tmp = ((0.25 * a_m) * x_45_scale_m) * 4.0
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b_m <= 4.4e-250)
		tmp = 0.0;
	elseif (b_m <= 3.5e-18)
		tmp = Float64(y_45_scale_m * b_m);
	else
		tmp = Float64(Float64(Float64(0.25 * a_m) * x_45_scale_m) * 4.0);
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b_m <= 4.4e-250)
		tmp = 0.0;
	elseif (b_m <= 3.5e-18)
		tmp = y_45_scale_m * b_m;
	else
		tmp = ((0.25 * a_m) * x_45_scale_m) * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b$95$m, 4.4e-250], 0.0, If[LessEqual[b$95$m, 3.5e-18], N[(y$45$scale$95$m * b$95$m), $MachinePrecision], N[(N[(N[(0.25 * a$95$m), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision] * 4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.4e-250

   1. Initial program 0.2%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 18.4%

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

   9. Taylor expanded in b around 0

      \[\leadsto 0 \]
   10. Step-by-step derivation

   11. Applied rewrites 33.3%

       \[\leadsto 0 \]

   if 4.4e-250 < b < 3.4999999999999999e-18

   1. Initial program 0.1%

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

   3. Taylor expanded in x-scale around inf

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

   5. Applied rewrites 10.4%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 48.4%

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

   10. Applied rewrites 48.6%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot b\right) \cdot 0.25 \]

   11. Taylor expanded in b around 0

       \[\leadsto b \cdot y-scale \]
   12. Step-by-step derivation

   13. Applied rewrites 48.6%

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

   if 3.4999999999999999e-18 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 38.2

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

   5. Applied rewrites 38.2%

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

   7. Applied rewrites 38.4%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-250}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-18}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot a\right) \cdot x-scale\right) \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.1% accurate, 161.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 4.4 \cdot 10^{-250}:\\ \;\;\;\;0\\ \mathbf{elif}\;b\_m \leq 3.5 \cdot 10^{-18}:\\ \;\;\;\;y-scale\_m \cdot b\_m\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot a\_m\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b_m 4.4e-250)
   0.0
   (if (<= b_m 3.5e-18) (* y-scale_m b_m) (* x-scale_m a_m))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b_m <= 4.4e-250) {
		tmp = 0.0;
	} else if (b_m <= 3.5e-18) {
		tmp = y_45_scale_m * b_m;
	} else {
		tmp = x_45_scale_m * a_m;
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (b_m <= 4.4d-250) then
        tmp = 0.0d0
    else if (b_m <= 3.5d-18) then
        tmp = y_45scale_m * b_m
    else
        tmp = x_45scale_m * a_m
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b_m <= 4.4e-250) {
		tmp = 0.0;
	} else if (b_m <= 3.5e-18) {
		tmp = y_45_scale_m * b_m;
	} else {
		tmp = x_45_scale_m * a_m;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b_m <= 4.4e-250:
		tmp = 0.0
	elif b_m <= 3.5e-18:
		tmp = y_45_scale_m * b_m
	else:
		tmp = x_45_scale_m * a_m
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b_m <= 4.4e-250)
		tmp = 0.0;
	elseif (b_m <= 3.5e-18)
		tmp = Float64(y_45_scale_m * b_m);
	else
		tmp = Float64(x_45_scale_m * a_m);
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b_m <= 4.4e-250)
		tmp = 0.0;
	elseif (b_m <= 3.5e-18)
		tmp = y_45_scale_m * b_m;
	else
		tmp = x_45_scale_m * a_m;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b$95$m, 4.4e-250], 0.0, If[LessEqual[b$95$m, 3.5e-18], N[(y$45$scale$95$m * b$95$m), $MachinePrecision], N[(x$45$scale$95$m * a$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.4e-250

   1. Initial program 0.2%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 18.4%

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

   9. Taylor expanded in b around 0

      \[\leadsto 0 \]
   10. Step-by-step derivation

   11. Applied rewrites 33.3%

       \[\leadsto 0 \]

   if 4.4e-250 < b < 3.4999999999999999e-18

   1. Initial program 0.1%

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

   3. Taylor expanded in x-scale around inf

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

   5. Applied rewrites 10.4%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 48.4%

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

   10. Applied rewrites 48.6%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot b\right) \cdot 0.25 \]

   11. Taylor expanded in b around 0

       \[\leadsto b \cdot y-scale \]
   12. Step-by-step derivation

   13. Applied rewrites 48.6%

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

   if 3.4999999999999999e-18 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 38.2

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

   5. Applied rewrites 38.2%

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

   7. Applied rewrites 38.4%

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

   8. Taylor expanded in a around 0

      \[\leadsto a \cdot \color{blue}{x-scale} \]
   9. Step-by-step derivation

   10. Applied rewrites 38.4%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-250}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-18}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.9% accurate, 242.0× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 5.1 \cdot 10^{-70}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot b\_m\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 5.1e-70) 0.0 (* y-scale_m b_m)))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 5.1e-70) {
		tmp = 0.0;
	} else {
		tmp = y_45_scale_m * b_m;
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (x_45scale_m <= 5.1d-70) then
        tmp = 0.0d0
    else
        tmp = y_45scale_m * b_m
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 5.1e-70) {
		tmp = 0.0;
	} else {
		tmp = y_45_scale_m * b_m;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if x_45_scale_m <= 5.1e-70:
		tmp = 0.0
	else:
		tmp = y_45_scale_m * b_m
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 5.1e-70)
		tmp = 0.0;
	else
		tmp = Float64(y_45_scale_m * b_m);
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (x_45_scale_m <= 5.1e-70)
		tmp = 0.0;
	else
		tmp = y_45_scale_m * b_m;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 5.1e-70], 0.0, N[(y$45$scale$95$m * b$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 5.10000000000000025e-70

   1. Initial program 0.1%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in b around 0

      \[\leadsto 0 \]
   10. Step-by-step derivation

   11. Applied rewrites 36.3%

       \[\leadsto 0 \]

   if 5.10000000000000025e-70 < x-scale

   1. Initial program 0.2%

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

   3. Taylor expanded in x-scale around inf

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 36.7%

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

   10. Applied rewrites 37.0%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot b\right) \cdot 0.25 \]

   11. Taylor expanded in b around 0

       \[\leadsto b \cdot y-scale \]
   12. Step-by-step derivation

   13. Applied rewrites 37.0%

       \[\leadsto b \cdot y-scale \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 5.1 \cdot 10^{-70}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 30.3% accurate, 2908.0× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ 0 \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m) :precision binary64 0.0)

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.0;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = 0.0d0
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.0;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return 0.0

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return 0.0
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := 0.0

Derivation

1. Initial program 0.1%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 4.5%

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

6. Taylor expanded in x-scale around 0

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

8. Applied rewrites 16.8%

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

9. Taylor expanded in b around 0

   \[\leadsto 0 \]
10. Step-by-step derivation

11. Applied rewrites 31.7%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(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))))