Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.6% → 94.0%

Time: 24.0s

Alternatives: 12

Speedup: 1905.0×

• 34.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

Initial Program: 24.6% accurate, 1.0× speedup

Math

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

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
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

Alternative 1: 94.0% accurate, 13.6× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 2.05 \cdot 10^{+223}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{x-scale\_m} \cdot \frac{b}{y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{a \cdot b}{y-scale \cdot x-scale\_m}\right)}^{2} \cdot -4\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (if (<= x-scale_m 2.05e+223)
   (* -4.0 (pow (* (/ a x-scale_m) (/ b y-scale)) 2.0))
   (* (pow (/ (* a b) (* y-scale x-scale_m)) 2.0) -4.0)))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (x_45_scale_m <= 2.05e+223) {
		tmp = -4.0 * pow(((a / x_45_scale_m) * (b / y_45_scale)), 2.0);
	} else {
		tmp = pow(((a * b) / (y_45_scale * x_45_scale_m)), 2.0) * -4.0;
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (x_45scale_m <= 2.05d+223) then
        tmp = (-4.0d0) * (((a / x_45scale_m) * (b / y_45scale)) ** 2.0d0)
    else
        tmp = (((a * b) / (y_45scale * x_45scale_m)) ** 2.0d0) * (-4.0d0)
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (x_45_scale_m <= 2.05e+223) {
		tmp = -4.0 * Math.pow(((a / x_45_scale_m) * (b / y_45_scale)), 2.0);
	} else {
		tmp = Math.pow(((a * b) / (y_45_scale * x_45_scale_m)), 2.0) * -4.0;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	tmp = 0
	if x_45_scale_m <= 2.05e+223:
		tmp = -4.0 * math.pow(((a / x_45_scale_m) * (b / y_45_scale)), 2.0)
	else:
		tmp = math.pow(((a * b) / (y_45_scale * x_45_scale_m)), 2.0) * -4.0
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (x_45_scale_m <= 2.05e+223)
		tmp = Float64(-4.0 * (Float64(Float64(a / x_45_scale_m) * Float64(b / y_45_scale)) ^ 2.0));
	else
		tmp = Float64((Float64(Float64(a * b) / Float64(y_45_scale * x_45_scale_m)) ^ 2.0) * -4.0);
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0;
	if (x_45_scale_m <= 2.05e+223)
		tmp = -4.0 * (((a / x_45_scale_m) * (b / y_45_scale)) ^ 2.0);
	else
		tmp = (((a * b) / (y_45_scale * x_45_scale_m)) ^ 2.0) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[x$45$scale$95$m, 2.05e+223], N[(-4.0 * N[Power[N[(N[(a / x$45$scale$95$m), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(a * b), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.05e223

   1. Initial program 27.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -4 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{b \cdot b}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

      9. times-frac N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{\frac{b}{{y-scale}^{2}}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{b}{{x-scale}^{2}}}\right) \]

      15. unpow2 N/A

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

      16. lower-*.f64 54.5

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 78.7%

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

   9. Applied rewrites 94.1%

      \[\leadsto \color{blue}{{\left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right)}^{2} \cdot -4} \]
   10. Step-by-step derivation

   11. Applied rewrites 93.7%

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

   if 2.05e223 < x-scale

   1. Initial program 49.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -4 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{b \cdot b}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

      9. times-frac N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{\frac{b}{{y-scale}^{2}}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{b}{{x-scale}^{2}}}\right) \]

      15. unpow2 N/A

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

      16. lower-*.f64 59.4

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

   5. Applied rewrites 59.4%

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

   7. Applied rewrites 88.4%

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

   9. Applied rewrites 95.8%

      \[\leadsto \color{blue}{{\left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right)}^{2} \cdot -4} \]
   10. Step-by-step derivation

   11. Applied rewrites 92.0%

       \[\leadsto {\left(\frac{b \cdot a}{x-scale \cdot y-scale}\right)}^{2} \cdot -4 \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 2.05 \cdot 10^{+223}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{a \cdot b}{y-scale \cdot x-scale}\right)}^{2} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.2% accurate, 14.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{\frac{b}{y-scale}}{x-scale\_m} \cdot a\\ \mathbf{if}\;x-scale\_m \leq 2.05 \cdot 10^{+223}:\\ \;\;\;\;\left(t\_0 \cdot -4\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{a \cdot b}{y-scale \cdot x-scale\_m}\right)}^{2} \cdot -4\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (* (/ (/ b y-scale) x-scale_m) a)))
   (if (<= x-scale_m 2.05e+223)
     (* (* t_0 -4.0) t_0)
     (* (pow (/ (* a b) (* y-scale x-scale_m)) 2.0) -4.0))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = ((b / y_45_scale) / x_45_scale_m) * a;
	double tmp;
	if (x_45_scale_m <= 2.05e+223) {
		tmp = (t_0 * -4.0) * t_0;
	} else {
		tmp = pow(((a * b) / (y_45_scale * x_45_scale_m)), 2.0) * -4.0;
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((b / y_45scale) / x_45scale_m) * a
    if (x_45scale_m <= 2.05d+223) then
        tmp = (t_0 * (-4.0d0)) * t_0
    else
        tmp = (((a * b) / (y_45scale * x_45scale_m)) ** 2.0d0) * (-4.0d0)
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = ((b / y_45_scale) / x_45_scale_m) * a;
	double tmp;
	if (x_45_scale_m <= 2.05e+223) {
		tmp = (t_0 * -4.0) * t_0;
	} else {
		tmp = Math.pow(((a * b) / (y_45_scale * x_45_scale_m)), 2.0) * -4.0;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = ((b / y_45_scale) / x_45_scale_m) * a
	tmp = 0
	if x_45_scale_m <= 2.05e+223:
		tmp = (t_0 * -4.0) * t_0
	else:
		tmp = math.pow(((a * b) / (y_45_scale * x_45_scale_m)), 2.0) * -4.0
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(Float64(b / y_45_scale) / x_45_scale_m) * a)
	tmp = 0.0
	if (x_45_scale_m <= 2.05e+223)
		tmp = Float64(Float64(t_0 * -4.0) * t_0);
	else
		tmp = Float64((Float64(Float64(a * b) / Float64(y_45_scale * x_45_scale_m)) ^ 2.0) * -4.0);
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = ((b / y_45_scale) / x_45_scale_m) * a;
	tmp = 0.0;
	if (x_45_scale_m <= 2.05e+223)
		tmp = (t_0 * -4.0) * t_0;
	else
		tmp = (((a * b) / (y_45_scale * x_45_scale_m)) ^ 2.0) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(b / y$45$scale), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 2.05e+223], N[(N[(t$95$0 * -4.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Power[N[(N[(a * b), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.05e223

   1. Initial program 27.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -4 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{b \cdot b}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

      9. times-frac N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{\frac{b}{{y-scale}^{2}}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{b}{{x-scale}^{2}}}\right) \]

      15. unpow2 N/A

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

      16. lower-*.f64 54.5

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 78.7%

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

   9. Applied rewrites 94.1%

      \[\leadsto \color{blue}{{\left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right)}^{2} \cdot -4} \]
   10. Step-by-step derivation

   11. Applied rewrites 94.1%

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

   if 2.05e223 < x-scale

   1. Initial program 49.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -4 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{b \cdot b}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

      9. times-frac N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{\frac{b}{{y-scale}^{2}}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{b}{{x-scale}^{2}}}\right) \]

      15. unpow2 N/A

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

      16. lower-*.f64 59.4

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

   5. Applied rewrites 59.4%

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

   7. Applied rewrites 88.4%

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

   9. Applied rewrites 95.8%

      \[\leadsto \color{blue}{{\left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right)}^{2} \cdot -4} \]
   10. Step-by-step derivation

   11. Applied rewrites 92.0%

       \[\leadsto {\left(\frac{b \cdot a}{x-scale \cdot y-scale}\right)}^{2} \cdot -4 \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 2.05 \cdot 10^{+223}:\\ \;\;\;\;\left(\left(\frac{\frac{b}{y-scale}}{x-scale} \cdot a\right) \cdot -4\right) \cdot \left(\frac{\frac{b}{y-scale}}{x-scale} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{a \cdot b}{y-scale \cdot x-scale}\right)}^{2} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 29.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{\left(\frac{\left(a \cdot b\right) \cdot b}{y-scale \cdot x-scale\_m} \cdot a\right) \cdot -4}{y-scale \cdot x-scale\_m}\\ t_1 := \frac{b}{y-scale \cdot x-scale\_m}\\ \mathbf{if}\;a \leq 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+153}:\\ \;\;\;\;\left(t\_1 \cdot t\_1\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0
         (/
          (* (* (/ (* (* a b) b) (* y-scale x-scale_m)) a) -4.0)
          (* y-scale x-scale_m)))
        (t_1 (/ b (* y-scale x-scale_m))))
   (if (<= a 1e-134)
     t_0
     (if (<= a 3.5e+153) (* (* t_1 t_1) (* (* a a) -4.0)) t_0))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (((((a * b) * b) / (y_45_scale * x_45_scale_m)) * a) * -4.0) / (y_45_scale * x_45_scale_m);
	double t_1 = b / (y_45_scale * x_45_scale_m);
	double tmp;
	if (a <= 1e-134) {
		tmp = t_0;
	} else if (a <= 3.5e+153) {
		tmp = (t_1 * t_1) * ((a * a) * -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((((a * b) * b) / (y_45scale * x_45scale_m)) * a) * (-4.0d0)) / (y_45scale * x_45scale_m)
    t_1 = b / (y_45scale * x_45scale_m)
    if (a <= 1d-134) then
        tmp = t_0
    else if (a <= 3.5d+153) then
        tmp = (t_1 * t_1) * ((a * a) * (-4.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (((((a * b) * b) / (y_45_scale * x_45_scale_m)) * a) * -4.0) / (y_45_scale * x_45_scale_m);
	double t_1 = b / (y_45_scale * x_45_scale_m);
	double tmp;
	if (a <= 1e-134) {
		tmp = t_0;
	} else if (a <= 3.5e+153) {
		tmp = (t_1 * t_1) * ((a * a) * -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = (((((a * b) * b) / (y_45_scale * x_45_scale_m)) * a) * -4.0) / (y_45_scale * x_45_scale_m)
	t_1 = b / (y_45_scale * x_45_scale_m)
	tmp = 0
	if a <= 1e-134:
		tmp = t_0
	elif a <= 3.5e+153:
		tmp = (t_1 * t_1) * ((a * a) * -4.0)
	else:
		tmp = t_0
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(Float64(Float64(Float64(Float64(a * b) * b) / Float64(y_45_scale * x_45_scale_m)) * a) * -4.0) / Float64(y_45_scale * x_45_scale_m))
	t_1 = Float64(b / Float64(y_45_scale * x_45_scale_m))
	tmp = 0.0
	if (a <= 1e-134)
		tmp = t_0;
	elseif (a <= 3.5e+153)
		tmp = Float64(Float64(t_1 * t_1) * Float64(Float64(a * a) * -4.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = (((((a * b) * b) / (y_45_scale * x_45_scale_m)) * a) * -4.0) / (y_45_scale * x_45_scale_m);
	t_1 = b / (y_45_scale * x_45_scale_m);
	tmp = 0.0;
	if (a <= 1e-134)
		tmp = t_0;
	elseif (a <= 3.5e+153)
		tmp = (t_1 * t_1) * ((a * a) * -4.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * -4.0), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1e-134], t$95$0, If[LessEqual[a, 3.5e+153], N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.00000000000000004e-134 or 3.4999999999999999e153 < a

   1. Initial program 26.6%

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \frac{\color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{x-scale \cdot y-scale}}}{y-scale \cdot x-scale} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{x-scale \cdot y-scale} \cdot -4}}{y-scale \cdot x-scale} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{x-scale \cdot y-scale} \cdot -4}}{y-scale \cdot x-scale} \]

      3. times-frac N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{{a}^{2}}{x-scale} \cdot \frac{{b}^{2}}{y-scale}\right)} \cdot -4}{y-scale \cdot x-scale} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{{a}^{2}}{x-scale} \cdot \frac{{b}^{2}}{y-scale}\right)} \cdot -4}{y-scale \cdot x-scale} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\frac{{a}^{2}}{x-scale}} \cdot \frac{{b}^{2}}{y-scale}\right) \cdot -4}{y-scale \cdot x-scale} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{a \cdot a}}{x-scale} \cdot \frac{{b}^{2}}{y-scale}\right) \cdot -4}{y-scale \cdot x-scale} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\color{blue}{a \cdot a}}{x-scale} \cdot \frac{{b}^{2}}{y-scale}\right) \cdot -4}{y-scale \cdot x-scale} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\left(\frac{a \cdot a}{x-scale} \cdot \color{blue}{\frac{{b}^{2}}{y-scale}}\right) \cdot -4}{y-scale \cdot x-scale} \]

      9. unpow2 N/A

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

      10. lower-*.f64 62.3

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

   7. Applied rewrites 62.3%

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

   9. Applied rewrites 83.5%

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

   if 1.00000000000000004e-134 < a < 3.4999999999999999e153

   1. Initial program 37.8%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -4 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{b \cdot b}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

      9. times-frac N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{\frac{b}{{y-scale}^{2}}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{b}{{x-scale}^{2}}}\right) \]

      15. unpow2 N/A

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

      16. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

   7. Applied rewrites 94.3%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 10^{-134}:\\ \;\;\;\;\frac{\left(\frac{\left(a \cdot b\right) \cdot b}{y-scale \cdot x-scale} \cdot a\right) \cdot -4}{y-scale \cdot x-scale}\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+153}:\\ \;\;\;\;\left(\frac{b}{y-scale \cdot x-scale} \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(a \cdot b\right) \cdot b}{y-scale \cdot x-scale} \cdot a\right) \cdot -4}{y-scale \cdot x-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 29.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{b}{y-scale \cdot x-scale\_m}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \mathbf{if}\;x-scale\_m \leq 1.3 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x-scale\_m \leq 2.05 \cdot 10^{+136}:\\ \;\;\;\;\frac{\left(\left(\frac{b}{x-scale\_m \cdot x-scale\_m} \cdot -4\right) \cdot a\right) \cdot \left(a \cdot b\right)}{y-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* y-scale x-scale_m)))
        (t_1 (* (* t_0 t_0) (* (* a a) -4.0))))
   (if (<= x-scale_m 1.3e-42)
     t_1
     (if (<= x-scale_m 2.05e+136)
       (/
        (* (* (* (/ b (* x-scale_m x-scale_m)) -4.0) a) (* a b))
        (* y-scale y-scale))
       t_1))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale_m);
	double t_1 = (t_0 * t_0) * ((a * a) * -4.0);
	double tmp;
	if (x_45_scale_m <= 1.3e-42) {
		tmp = t_1;
	} else if (x_45_scale_m <= 2.05e+136) {
		tmp = ((((b / (x_45_scale_m * x_45_scale_m)) * -4.0) * a) * (a * b)) / (y_45_scale * y_45_scale);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b / (y_45scale * x_45scale_m)
    t_1 = (t_0 * t_0) * ((a * a) * (-4.0d0))
    if (x_45scale_m <= 1.3d-42) then
        tmp = t_1
    else if (x_45scale_m <= 2.05d+136) then
        tmp = ((((b / (x_45scale_m * x_45scale_m)) * (-4.0d0)) * a) * (a * b)) / (y_45scale * y_45scale)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale_m);
	double t_1 = (t_0 * t_0) * ((a * a) * -4.0);
	double tmp;
	if (x_45_scale_m <= 1.3e-42) {
		tmp = t_1;
	} else if (x_45_scale_m <= 2.05e+136) {
		tmp = ((((b / (x_45_scale_m * x_45_scale_m)) * -4.0) * a) * (a * b)) / (y_45_scale * y_45_scale);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = b / (y_45_scale * x_45_scale_m)
	t_1 = (t_0 * t_0) * ((a * a) * -4.0)
	tmp = 0
	if x_45_scale_m <= 1.3e-42:
		tmp = t_1
	elif x_45_scale_m <= 2.05e+136:
		tmp = ((((b / (x_45_scale_m * x_45_scale_m)) * -4.0) * a) * (a * b)) / (y_45_scale * y_45_scale)
	else:
		tmp = t_1
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(b / Float64(y_45_scale * x_45_scale_m))
	t_1 = Float64(Float64(t_0 * t_0) * Float64(Float64(a * a) * -4.0))
	tmp = 0.0
	if (x_45_scale_m <= 1.3e-42)
		tmp = t_1;
	elseif (x_45_scale_m <= 2.05e+136)
		tmp = Float64(Float64(Float64(Float64(Float64(b / Float64(x_45_scale_m * x_45_scale_m)) * -4.0) * a) * Float64(a * b)) / Float64(y_45_scale * y_45_scale));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = b / (y_45_scale * x_45_scale_m);
	t_1 = (t_0 * t_0) * ((a * a) * -4.0);
	tmp = 0.0;
	if (x_45_scale_m <= 1.3e-42)
		tmp = t_1;
	elseif (x_45_scale_m <= 2.05e+136)
		tmp = ((((b / (x_45_scale_m * x_45_scale_m)) * -4.0) * a) * (a * b)) / (y_45_scale * y_45_scale);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 1.3e-42], t$95$1, If[LessEqual[x$45$scale$95$m, 2.05e+136], N[(N[(N[(N[(N[(b / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] * a), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.3e-42 or 2.0499999999999999e136 < x-scale

   1. Initial program 30.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -4 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{b \cdot b}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

      9. times-frac N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{\frac{b}{{y-scale}^{2}}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{b}{{x-scale}^{2}}}\right) \]

      15. unpow2 N/A

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

      16. lower-*.f64 54.5

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 81.0%

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

   if 1.3e-42 < x-scale < 2.0499999999999999e136

   1. Initial program 24.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -4 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{b \cdot b}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

      9. times-frac N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{\frac{b}{{y-scale}^{2}}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{b}{{x-scale}^{2}}}\right) \]

      15. unpow2 N/A

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

      16. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

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

   7. Applied rewrites 69.1%

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

   9. Applied rewrites 83.4%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.3 \cdot 10^{-42}:\\ \;\;\;\;\left(\frac{b}{y-scale \cdot x-scale} \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \mathbf{elif}\;x-scale \leq 2.05 \cdot 10^{+136}:\\ \;\;\;\;\frac{\left(\left(\frac{b}{x-scale \cdot x-scale} \cdot -4\right) \cdot a\right) \cdot \left(a \cdot b\right)}{y-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{y-scale \cdot x-scale} \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.1% accurate, 29.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \left(\frac{b}{\left(y-scale \cdot x-scale\_m\right) \cdot x-scale\_m} \cdot \frac{b}{y-scale}\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \mathbf{if}\;y-scale \leq 6.6 \cdot 10^{-159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y-scale \leq 1.4 \cdot 10^{+76}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\frac{-4 \cdot a}{\left(y-scale \cdot y-scale\right) \cdot x-scale\_m} \cdot \frac{a}{x-scale\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (* (/ b (* (* y-scale x-scale_m) x-scale_m)) (/ b y-scale))
          (* (* a a) -4.0))))
   (if (<= y-scale 6.6e-159)
     t_0
     (if (<= y-scale 1.4e+76)
       (*
        (* b b)
        (* (/ (* -4.0 a) (* (* y-scale y-scale) x-scale_m)) (/ a x-scale_m)))
       t_0))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = ((b / ((y_45_scale * x_45_scale_m) * x_45_scale_m)) * (b / y_45_scale)) * ((a * a) * -4.0);
	double tmp;
	if (y_45_scale <= 6.6e-159) {
		tmp = t_0;
	} else if (y_45_scale <= 1.4e+76) {
		tmp = (b * b) * (((-4.0 * a) / ((y_45_scale * y_45_scale) * x_45_scale_m)) * (a / x_45_scale_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((b / ((y_45scale * x_45scale_m) * x_45scale_m)) * (b / y_45scale)) * ((a * a) * (-4.0d0))
    if (y_45scale <= 6.6d-159) then
        tmp = t_0
    else if (y_45scale <= 1.4d+76) then
        tmp = (b * b) * ((((-4.0d0) * a) / ((y_45scale * y_45scale) * x_45scale_m)) * (a / x_45scale_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = ((b / ((y_45_scale * x_45_scale_m) * x_45_scale_m)) * (b / y_45_scale)) * ((a * a) * -4.0);
	double tmp;
	if (y_45_scale <= 6.6e-159) {
		tmp = t_0;
	} else if (y_45_scale <= 1.4e+76) {
		tmp = (b * b) * (((-4.0 * a) / ((y_45_scale * y_45_scale) * x_45_scale_m)) * (a / x_45_scale_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = ((b / ((y_45_scale * x_45_scale_m) * x_45_scale_m)) * (b / y_45_scale)) * ((a * a) * -4.0)
	tmp = 0
	if y_45_scale <= 6.6e-159:
		tmp = t_0
	elif y_45_scale <= 1.4e+76:
		tmp = (b * b) * (((-4.0 * a) / ((y_45_scale * y_45_scale) * x_45_scale_m)) * (a / x_45_scale_m))
	else:
		tmp = t_0
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(Float64(b / Float64(Float64(y_45_scale * x_45_scale_m) * x_45_scale_m)) * Float64(b / y_45_scale)) * Float64(Float64(a * a) * -4.0))
	tmp = 0.0
	if (y_45_scale <= 6.6e-159)
		tmp = t_0;
	elseif (y_45_scale <= 1.4e+76)
		tmp = Float64(Float64(b * b) * Float64(Float64(Float64(-4.0 * a) / Float64(Float64(y_45_scale * y_45_scale) * x_45_scale_m)) * Float64(a / x_45_scale_m)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = ((b / ((y_45_scale * x_45_scale_m) * x_45_scale_m)) * (b / y_45_scale)) * ((a * a) * -4.0);
	tmp = 0.0;
	if (y_45_scale <= 6.6e-159)
		tmp = t_0;
	elseif (y_45_scale <= 1.4e+76)
		tmp = (b * b) * (((-4.0 * a) / ((y_45_scale * y_45_scale) * x_45_scale_m)) * (a / x_45_scale_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(b / N[(N[(y$45$scale * x$45$scale$95$m), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, 6.6e-159], t$95$0, If[LessEqual[y$45$scale, 1.4e+76], N[(N[(b * b), $MachinePrecision] * N[(N[(N[(-4.0 * a), $MachinePrecision] / N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(a / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 6.6000000000000003e-159 or 1.3999999999999999e76 < y-scale

   1. Initial program 27.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -4 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{b \cdot b}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

      9. times-frac N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{\frac{b}{{y-scale}^{2}}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{b}{{x-scale}^{2}}}\right) \]

      15. unpow2 N/A

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

      16. lower-*.f64 53.9

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

   5. Applied rewrites 53.9%

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

   7. Applied rewrites 80.3%

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

   9. Applied rewrites 70.9%

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

   if 6.6000000000000003e-159 < y-scale < 1.3999999999999999e76

   1. Initial program 42.5%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 58.2%

      \[\leadsto \frac{-4 \cdot \left(a \cdot a\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(\color{blue}{b} \cdot b\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 83.0%

       \[\leadsto \left(\frac{-4 \cdot a}{\left(y-scale \cdot y-scale\right) \cdot x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(b \cdot b\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 6.6 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{b}{\left(y-scale \cdot x-scale\right) \cdot x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \mathbf{elif}\;y-scale \leq 1.4 \cdot 10^{+76}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\frac{-4 \cdot a}{\left(y-scale \cdot y-scale\right) \cdot x-scale} \cdot \frac{a}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{\left(y-scale \cdot x-scale\right) \cdot x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.6% accurate, 29.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{\frac{b}{y-scale}}{x-scale\_m} \cdot a\\ \left(t\_0 \cdot -4\right) \cdot t\_0 \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (* (/ (/ b y-scale) x-scale_m) a))) (* (* t_0 -4.0) t_0)))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = ((b / y_45_scale) / x_45_scale_m) * a;
	return (t_0 * -4.0) * t_0;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = ((b / y_45scale) / x_45scale_m) * a
    code = (t_0 * (-4.0d0)) * t_0
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = ((b / y_45_scale) / x_45_scale_m) * a;
	return (t_0 * -4.0) * t_0;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = ((b / y_45_scale) / x_45_scale_m) * a
	return (t_0 * -4.0) * t_0

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(Float64(b / y_45_scale) / x_45_scale_m) * a)
	return Float64(Float64(t_0 * -4.0) * t_0)
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = ((b / y_45_scale) / x_45_scale_m) * a;
	tmp = (t_0 * -4.0) * t_0;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(b / y$45$scale), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision] * a), $MachinePrecision]}, N[(N[(t$95$0 * -4.0), $MachinePrecision] * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 29.7%

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

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto -4 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   5. unpow2 N/A

      \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   8. *-commutative N/A

      \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{b \cdot b}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

   9. times-frac N/A

      \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

   11. lower-/.f64 N/A

       \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{\frac{b}{{y-scale}^{2}}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

   12. unpow2 N/A

       \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

   14. lower-/.f64 N/A

       \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{b}{{x-scale}^{2}}}\right) \]

   15. unpow2 N/A

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

   16. lower-*.f64 55.0

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

5. Applied rewrites 55.0%

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

7. Applied rewrites 79.6%

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

9. Applied rewrites 94.3%

   \[\leadsto \color{blue}{{\left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right)}^{2} \cdot -4} \]
10. Step-by-step derivation

11. Applied rewrites 94.3%

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

12. Final simplification 94.3%

    \[\leadsto \left(\left(\frac{\frac{b}{y-scale}}{x-scale} \cdot a\right) \cdot -4\right) \cdot \left(\frac{\frac{b}{y-scale}}{x-scale} \cdot a\right) \]
13. Add Preprocessing

Alternative 7: 55.0% accurate, 32.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{b}{y-scale \cdot x-scale\_m}\\ \mathbf{if}\;b \leq 3.2 \cdot 10^{-264}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* y-scale x-scale_m))))
   (if (<= b 3.2e-264) 0.0 (* (* t_0 t_0) (* (* a a) -4.0)))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale_m);
	double tmp;
	if (b <= 3.2e-264) {
		tmp = 0.0;
	} else {
		tmp = (t_0 * t_0) * ((a * a) * -4.0);
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b / (y_45scale * x_45scale_m)
    if (b <= 3.2d-264) then
        tmp = 0.0d0
    else
        tmp = (t_0 * t_0) * ((a * a) * (-4.0d0))
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale_m);
	double tmp;
	if (b <= 3.2e-264) {
		tmp = 0.0;
	} else {
		tmp = (t_0 * t_0) * ((a * a) * -4.0);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = b / (y_45_scale * x_45_scale_m)
	tmp = 0
	if b <= 3.2e-264:
		tmp = 0.0
	else:
		tmp = (t_0 * t_0) * ((a * a) * -4.0)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(b / Float64(y_45_scale * x_45_scale_m))
	tmp = 0.0
	if (b <= 3.2e-264)
		tmp = 0.0;
	else
		tmp = Float64(Float64(t_0 * t_0) * Float64(Float64(a * a) * -4.0));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = b / (y_45_scale * x_45_scale_m);
	tmp = 0.0;
	if (b <= 3.2e-264)
		tmp = 0.0;
	else
		tmp = (t_0 * t_0) * ((a * a) * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3.2e-264], 0.0, N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.19999999999999995e-264

   1. Initial program 31.5%

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

   4. Applied rewrites 28.4%

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

   5. Taylor expanded in b around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{0} \]

      3. mul0-rgt 46.8

         \[\leadsto \color{blue}{0} \]

   7. Applied rewrites 46.8%

      \[\leadsto \color{blue}{0} \]

   if 3.19999999999999995e-264 < b

   1. Initial program 27.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -4 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{b \cdot b}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

      9. times-frac N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{\frac{b}{{y-scale}^{2}}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{b}{{x-scale}^{2}}}\right) \]

      15. unpow2 N/A

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

      16. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   7. Applied rewrites 83.8%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-264}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{y-scale \cdot x-scale} \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.9% accurate, 32.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 1.05 \cdot 10^{+150}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\frac{-4 \cdot a}{\left(y-scale \cdot y-scale\right) \cdot x-scale\_m} \cdot \frac{a}{x-scale\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot b}{\left(y-scale \cdot x-scale\_m\right) \cdot \left(y-scale \cdot x-scale\_m\right)} \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (if (<= y-scale 1.05e+150)
   (*
    (* b b)
    (* (/ (* -4.0 a) (* (* y-scale y-scale) x-scale_m)) (/ a x-scale_m)))
   (*
    (/ (* b b) (* (* y-scale x-scale_m) (* y-scale x-scale_m)))
    (* (* a a) -4.0))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 1.05e+150) {
		tmp = (b * b) * (((-4.0 * a) / ((y_45_scale * y_45_scale) * x_45_scale_m)) * (a / x_45_scale_m));
	} else {
		tmp = ((b * b) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * ((a * a) * -4.0);
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (y_45scale <= 1.05d+150) then
        tmp = (b * b) * ((((-4.0d0) * a) / ((y_45scale * y_45scale) * x_45scale_m)) * (a / x_45scale_m))
    else
        tmp = ((b * b) / ((y_45scale * x_45scale_m) * (y_45scale * x_45scale_m))) * ((a * a) * (-4.0d0))
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 1.05e+150) {
		tmp = (b * b) * (((-4.0 * a) / ((y_45_scale * y_45_scale) * x_45_scale_m)) * (a / x_45_scale_m));
	} else {
		tmp = ((b * b) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * ((a * a) * -4.0);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	tmp = 0
	if y_45_scale <= 1.05e+150:
		tmp = (b * b) * (((-4.0 * a) / ((y_45_scale * y_45_scale) * x_45_scale_m)) * (a / x_45_scale_m))
	else:
		tmp = ((b * b) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * ((a * a) * -4.0)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 1.05e+150)
		tmp = Float64(Float64(b * b) * Float64(Float64(Float64(-4.0 * a) / Float64(Float64(y_45_scale * y_45_scale) * x_45_scale_m)) * Float64(a / x_45_scale_m)));
	else
		tmp = Float64(Float64(Float64(b * b) / Float64(Float64(y_45_scale * x_45_scale_m) * Float64(y_45_scale * x_45_scale_m))) * Float64(Float64(a * a) * -4.0));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0;
	if (y_45_scale <= 1.05e+150)
		tmp = (b * b) * (((-4.0 * a) / ((y_45_scale * y_45_scale) * x_45_scale_m)) * (a / x_45_scale_m));
	else
		tmp = ((b * b) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * ((a * a) * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[y$45$scale, 1.05e+150], N[(N[(b * b), $MachinePrecision] * N[(N[(N[(-4.0 * a), $MachinePrecision] / N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(a / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.04999999999999999e150

   1. Initial program 26.8%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 63.1%

      \[\leadsto \frac{-4 \cdot \left(a \cdot a\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(\color{blue}{b} \cdot b\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 66.9%

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

   if 1.04999999999999999e150 < y-scale

   1. Initial program 44.3%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -4 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{b \cdot b}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

      9. times-frac N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{\frac{b}{{y-scale}^{2}}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{b}{{x-scale}^{2}}}\right) \]

      15. unpow2 N/A

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

      16. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 65.5%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.05 \cdot 10^{+150}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\frac{-4 \cdot a}{\left(y-scale \cdot y-scale\right) \cdot x-scale} \cdot \frac{a}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.8% accurate, 35.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot b}{\left(y-scale \cdot x-scale\_m\right) \cdot \left(y-scale \cdot x-scale\_m\right)} \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (if (<= b 1.7e-123)
   0.0
   (*
    (/ (* b b) (* (* y-scale x-scale_m) (* y-scale x-scale_m)))
    (* (* a a) -4.0))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b <= 1.7e-123) {
		tmp = 0.0;
	} else {
		tmp = ((b * b) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * ((a * a) * -4.0);
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b <= 1.7d-123) then
        tmp = 0.0d0
    else
        tmp = ((b * b) / ((y_45scale * x_45scale_m) * (y_45scale * x_45scale_m))) * ((a * a) * (-4.0d0))
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b <= 1.7e-123) {
		tmp = 0.0;
	} else {
		tmp = ((b * b) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * ((a * a) * -4.0);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	tmp = 0
	if b <= 1.7e-123:
		tmp = 0.0
	else:
		tmp = ((b * b) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * ((a * a) * -4.0)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (b <= 1.7e-123)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(b * b) / Float64(Float64(y_45_scale * x_45_scale_m) * Float64(y_45_scale * x_45_scale_m))) * Float64(Float64(a * a) * -4.0));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0;
	if (b <= 1.7e-123)
		tmp = 0.0;
	else
		tmp = ((b * b) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * ((a * a) * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[b, 1.7e-123], 0.0, N[(N[(N[(b * b), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.7e-123

   1. Initial program 36.5%

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in b around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{0} \]

      3. mul0-rgt 50.3

         \[\leadsto \color{blue}{0} \]

   7. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0} \]

   if 1.7e-123 < b

   1. Initial program 17.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -4 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot {a}^{2}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{b \cdot b}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

      9. times-frac N/A

         \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\frac{b}{{y-scale}^{2}} \cdot \frac{b}{{x-scale}^{2}}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{\frac{b}{{y-scale}^{2}}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b}{{x-scale}^{2}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{b}{{x-scale}^{2}}}\right) \]

      15. unpow2 N/A

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

      16. lower-*.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 71.0%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.9% accurate, 35.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-152}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot -4}{\left(y-scale \cdot x-scale\_m\right) \cdot \left(y-scale \cdot x-scale\_m\right)} \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (if (<= b 8.8e-152)
   0.0
   (*
    (/ (* (* a a) -4.0) (* (* y-scale x-scale_m) (* y-scale x-scale_m)))
    (* b b))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b <= 8.8e-152) {
		tmp = 0.0;
	} else {
		tmp = (((a * a) * -4.0) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * (b * b);
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b <= 8.8d-152) then
        tmp = 0.0d0
    else
        tmp = (((a * a) * (-4.0d0)) / ((y_45scale * x_45scale_m) * (y_45scale * x_45scale_m))) * (b * b)
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b <= 8.8e-152) {
		tmp = 0.0;
	} else {
		tmp = (((a * a) * -4.0) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * (b * b);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	tmp = 0
	if b <= 8.8e-152:
		tmp = 0.0
	else:
		tmp = (((a * a) * -4.0) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * (b * b)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (b <= 8.8e-152)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(Float64(a * a) * -4.0) / Float64(Float64(y_45_scale * x_45_scale_m) * Float64(y_45_scale * x_45_scale_m))) * Float64(b * b));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0;
	if (b <= 8.8e-152)
		tmp = 0.0;
	else
		tmp = (((a * a) * -4.0) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[b, 8.8e-152], 0.0, N[(N[(N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.79999999999999939e-152

   1. Initial program 35.3%

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

   4. Applied rewrites 31.3%

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

   5. Taylor expanded in b around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{0} \]

      3. mul0-rgt 49.6

         \[\leadsto \color{blue}{0} \]

   7. Applied rewrites 49.6%

      \[\leadsto \color{blue}{0} \]

   if 8.79999999999999939e-152 < b

   1. Initial program 20.7%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 71.5%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-152}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot -4}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.3% accurate, 35.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \frac{\left(\frac{a \cdot b}{y-scale \cdot x-scale\_m} \cdot \left(a \cdot b\right)\right) \cdot -4}{y-scale \cdot x-scale\_m} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (/
  (* (* (/ (* a b) (* y-scale x-scale_m)) (* a b)) -4.0)
  (* y-scale x-scale_m)))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return ((((a * b) / (y_45_scale * x_45_scale_m)) * (a * b)) * -4.0) / (y_45_scale * x_45_scale_m);
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = ((((a * b) / (y_45scale * x_45scale_m)) * (a * b)) * (-4.0d0)) / (y_45scale * x_45scale_m)
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return ((((a * b) / (y_45_scale * x_45_scale_m)) * (a * b)) * -4.0) / (y_45_scale * x_45_scale_m);
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	return ((((a * b) / (y_45_scale * x_45_scale_m)) * (a * b)) * -4.0) / (y_45_scale * x_45_scale_m)

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	return Float64(Float64(Float64(Float64(Float64(a * b) / Float64(y_45_scale * x_45_scale_m)) * Float64(a * b)) * -4.0) / Float64(y_45_scale * x_45_scale_m))
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = ((((a * b) / (y_45_scale * x_45_scale_m)) * (a * b)) * -4.0) / (y_45_scale * x_45_scale_m);
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := N[(N[(N[(N[(N[(a * b), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.7%

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

4. Applied rewrites 33.2%

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

5. Taylor expanded in angle around 0

   \[\leadsto \frac{\color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{x-scale \cdot y-scale}}}{y-scale \cdot x-scale} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{x-scale \cdot y-scale} \cdot -4}}{y-scale \cdot x-scale} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{x-scale \cdot y-scale} \cdot -4}}{y-scale \cdot x-scale} \]

   3. times-frac N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{{a}^{2}}{x-scale} \cdot \frac{{b}^{2}}{y-scale}\right)} \cdot -4}{y-scale \cdot x-scale} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{{a}^{2}}{x-scale} \cdot \frac{{b}^{2}}{y-scale}\right)} \cdot -4}{y-scale \cdot x-scale} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\left(\color{blue}{\frac{{a}^{2}}{x-scale}} \cdot \frac{{b}^{2}}{y-scale}\right) \cdot -4}{y-scale \cdot x-scale} \]

   6. unpow2 N/A

      \[\leadsto \frac{\left(\frac{\color{blue}{a \cdot a}}{x-scale} \cdot \frac{{b}^{2}}{y-scale}\right) \cdot -4}{y-scale \cdot x-scale} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\left(\frac{\color{blue}{a \cdot a}}{x-scale} \cdot \frac{{b}^{2}}{y-scale}\right) \cdot -4}{y-scale \cdot x-scale} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\left(\frac{a \cdot a}{x-scale} \cdot \color{blue}{\frac{{b}^{2}}{y-scale}}\right) \cdot -4}{y-scale \cdot x-scale} \]

   9. unpow2 N/A

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

   10. lower-*.f64 66.1

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

7. Applied rewrites 66.1%

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

9. Applied rewrites 91.4%

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

10. Final simplification 91.4%

    \[\leadsto \frac{\left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot \left(a \cdot b\right)\right) \cdot -4}{y-scale \cdot x-scale} \]
11. Add Preprocessing

Alternative 12: 35.2% accurate, 1905.0× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ 0 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := 0.0

Derivation

1. Initial program 29.7%

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

4. Applied rewrites 26.8%

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

5. Taylor expanded in b around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{0} \]

   3. mul0-rgt 40.7

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 40.7%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024273 
(FPCore (a b angle x-scale y-scale)
  :name "Simplification of discriminant from scale-rotated-ellipse"
  :precision binary64
  (- (* (/ (/ (* (* (* 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 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) (PI)))) (cos (* (/ angle 180.0) (PI)))) x-scale) y-scale)) (* (* 4.0 (/ (/ (+ (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))))