Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.2% → 93.6%

Time: 24.5s

Alternatives: 10

Speedup: 1905.0×

• Could not converge on a ground truth

  1. a = 1.9303152666505922e+114
  2. b = 4.9899608537417775e+116
  3. angle = 3.7256074371027597e+33
  4. x-scale = 3.1911227450646933e+230
  5. y-scale = 7.895988465556659e-249

• 34.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 24.2% 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: 93.6% accurate, 13.0× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := {\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale\_m}\right)}^{2} \cdot -4\\ \mathbf{if}\;y-scale\_m \leq 3.35 \cdot 10^{-161}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y-scale\_m \leq 3.7 \cdot 10^{+213}:\\ \;\;\;\;{\left(\frac{a \cdot b}{y-scale\_m \cdot x-scale}\right)}^{2} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0 (* (pow (* (/ b x-scale) (/ a y-scale_m)) 2.0) -4.0)))
   (if (<= y-scale_m 3.35e-161)
     t_0
     (if (<= y-scale_m 3.7e+213)
       (* (pow (/ (* a b) (* y-scale_m x-scale)) 2.0) -4.0)
       t_0))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = pow(((b / x_45_scale) * (a / y_45_scale_m)), 2.0) * -4.0;
	double tmp;
	if (y_45_scale_m <= 3.35e-161) {
		tmp = t_0;
	} else if (y_45_scale_m <= 3.7e+213) {
		tmp = pow(((a * b) / (y_45_scale_m * x_45_scale)), 2.0) * -4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((b / x_45scale) * (a / y_45scale_m)) ** 2.0d0) * (-4.0d0)
    if (y_45scale_m <= 3.35d-161) then
        tmp = t_0
    else if (y_45scale_m <= 3.7d+213) then
        tmp = (((a * b) / (y_45scale_m * x_45scale)) ** 2.0d0) * (-4.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = Math.pow(((b / x_45_scale) * (a / y_45_scale_m)), 2.0) * -4.0;
	double tmp;
	if (y_45_scale_m <= 3.35e-161) {
		tmp = t_0;
	} else if (y_45_scale_m <= 3.7e+213) {
		tmp = Math.pow(((a * b) / (y_45_scale_m * x_45_scale)), 2.0) * -4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	t_0 = math.pow(((b / x_45_scale) * (a / y_45_scale_m)), 2.0) * -4.0
	tmp = 0
	if y_45_scale_m <= 3.35e-161:
		tmp = t_0
	elif y_45_scale_m <= 3.7e+213:
		tmp = math.pow(((a * b) / (y_45_scale_m * x_45_scale)), 2.0) * -4.0
	else:
		tmp = t_0
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64((Float64(Float64(b / x_45_scale) * Float64(a / y_45_scale_m)) ^ 2.0) * -4.0)
	tmp = 0.0
	if (y_45_scale_m <= 3.35e-161)
		tmp = t_0;
	elseif (y_45_scale_m <= 3.7e+213)
		tmp = Float64((Float64(Float64(a * b) / Float64(y_45_scale_m * x_45_scale)) ^ 2.0) * -4.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = (((b / x_45_scale) * (a / y_45_scale_m)) ^ 2.0) * -4.0;
	tmp = 0.0;
	if (y_45_scale_m <= 3.35e-161)
		tmp = t_0;
	elseif (y_45_scale_m <= 3.7e+213)
		tmp = (((a * b) / (y_45_scale_m * x_45_scale)) ^ 2.0) * -4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[Power[N[(N[(b / x$45$scale), $MachinePrecision] * N[(a / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 3.35e-161], t$95$0, If[LessEqual[y$45$scale$95$m, 3.7e+213], N[(N[Power[N[(N[(a * b), $MachinePrecision] / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 3.35e-161 or 3.69999999999999993e213 < y-scale

   1. Initial program 24.8%

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

   7. Applied rewrites 77.5%

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

   9. Applied rewrites 93.1%

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

   11. Applied rewrites 93.1%

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

   if 3.35e-161 < y-scale < 3.69999999999999993e213

   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 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 64.0

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

   5. Applied rewrites 64.0%

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

   7. Applied rewrites 75.0%

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

   9. Applied rewrites 94.4%

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

   11. Applied rewrites 95.7%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.35 \cdot 10^{-161}:\\ \;\;\;\;{\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)}^{2} \cdot -4\\ \mathbf{elif}\;y-scale \leq 3.7 \cdot 10^{+213}:\\ \;\;\;\;{\left(\frac{a \cdot b}{y-scale \cdot x-scale}\right)}^{2} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)}^{2} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{t\_2 \cdot \left(t\_1 \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)\right)}{x-scale}}{y-scale\_m}\\ \mathbf{if}\;t\_3 \cdot t\_3 - \frac{\frac{{\left(t\_1 \cdot b\right)}^{2} + {\left(t\_2 \cdot a\right)}^{2}}{y-scale\_m}}{y-scale\_m} \cdot \left(\frac{\frac{{\left(t\_2 \cdot b\right)}^{2} + {\left(t\_1 \cdot a\right)}^{2}}{x-scale}}{x-scale} \cdot 4\right) \leq 0:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot -4}{\left(y-scale\_m \cdot x-scale\right) \cdot \left(y-scale\_m \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{b}{x-scale \cdot x-scale} \cdot b\right) \cdot a\right) \cdot \frac{-4}{y-scale\_m \cdot y-scale\_m}\right) \cdot a\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0 (* (PI) (/ angle 180.0)))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* t_2 (* t_1 (* (- (pow b 2.0) (pow a 2.0)) 2.0))) x-scale)
          y-scale_m)))
   (if (<=
        (-
         (* t_3 t_3)
         (*
          (/
           (/ (+ (pow (* t_1 b) 2.0) (pow (* t_2 a) 2.0)) y-scale_m)
           y-scale_m)
          (*
           (/ (/ (+ (pow (* t_2 b) 2.0) (pow (* t_1 a) 2.0)) x-scale) x-scale)
           4.0)))
        0.0)
     (*
      (/ (* (* a a) -4.0) (* (* y-scale_m x-scale) (* y-scale_m x-scale)))
      (* b b))
     (*
      (*
       (* (* (/ b (* x-scale x-scale)) b) a)
       (/ -4.0 (* y-scale_m y-scale_m)))
      a))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 #s(literal 4 binary64) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale))) < 0.0

   1. Initial program 77.1%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 76.8%

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

   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 88.3%

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

   if 0.0 < (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 #s(literal 4 binary64) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)))

   1. Initial program 0.7%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 49.3

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

   5. Applied rewrites 49.3%

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

   7. Applied rewrites 69.0%

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

   9. Applied rewrites 80.8%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 61.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)\right)}{x-scale}}{y-scale} - \frac{\frac{{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot b\right)}^{2} + {\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot a\right)}^{2}}{y-scale}}{y-scale} \cdot \left(\frac{\frac{{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot b\right)}^{2} + {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot a\right)}^{2}}{x-scale}}{x-scale} \cdot 4\right) \leq 0:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot -4}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{b}{x-scale \cdot x-scale} \cdot b\right) \cdot a\right) \cdot \frac{-4}{y-scale \cdot y-scale}\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 14.2× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ -4 \cdot {\left(\frac{\frac{b}{y-scale\_m} \cdot a}{x-scale}\right)}^{2} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (* -4.0 (pow (/ (* (/ b y-scale_m) a) x-scale) 2.0)))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return -4.0 * pow((((b / y_45_scale_m) * a) / x_45_scale), 2.0);
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    code = (-4.0d0) * ((((b / y_45scale_m) * a) / x_45scale) ** 2.0d0)
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return -4.0 * Math.pow((((b / y_45_scale_m) * a) / x_45_scale), 2.0);
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return -4.0 * math.pow((((b / y_45_scale_m) * a) / x_45_scale), 2.0)

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(-4.0 * (Float64(Float64(Float64(b / y_45_scale_m) * a) / x_45_scale) ^ 2.0))
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = -4.0 * ((((b / y_45_scale_m) * a) / x_45_scale) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 23.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 56.5

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

5. Applied rewrites 56.5%

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

7. Applied rewrites 76.8%

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

9. Applied rewrites 93.5%

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

11. Applied rewrites 96.3%

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

12. Final simplification 96.3%

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

Alternative 4: 65.9% accurate, 27.2× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 6.8 \cdot 10^{-114}:\\ \;\;\;\;\left(\left(\left(\frac{b}{x-scale \cdot x-scale} \cdot b\right) \cdot a\right) \cdot \frac{-4}{y-scale\_m \cdot y-scale\_m}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\frac{a}{y-scale\_m \cdot x-scale} \cdot \frac{-4 \cdot a}{y-scale\_m \cdot x-scale}\right)\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (if (<= (/ angle 180.0) 6.8e-114)
   (*
    (* (* (* (/ b (* x-scale x-scale)) b) a) (/ -4.0 (* y-scale_m y-scale_m)))
    a)
   (*
    (* b b)
    (* (/ a (* y-scale_m x-scale)) (/ (* -4.0 a) (* y-scale_m x-scale))))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if ((angle / 180.0) <= 6.8e-114) {
		tmp = ((((b / (x_45_scale * x_45_scale)) * b) * a) * (-4.0 / (y_45_scale_m * y_45_scale_m))) * a;
	} else {
		tmp = (b * b) * ((a / (y_45_scale_m * x_45_scale)) * ((-4.0 * a) / (y_45_scale_m * x_45_scale)));
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if ((angle / 180.0d0) <= 6.8d-114) then
        tmp = ((((b / (x_45scale * x_45scale)) * b) * a) * ((-4.0d0) / (y_45scale_m * y_45scale_m))) * a
    else
        tmp = (b * b) * ((a / (y_45scale_m * x_45scale)) * (((-4.0d0) * a) / (y_45scale_m * x_45scale)))
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if ((angle / 180.0) <= 6.8e-114) {
		tmp = ((((b / (x_45_scale * x_45_scale)) * b) * a) * (-4.0 / (y_45_scale_m * y_45_scale_m))) * a;
	} else {
		tmp = (b * b) * ((a / (y_45_scale_m * x_45_scale)) * ((-4.0 * a) / (y_45_scale_m * x_45_scale)));
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	tmp = 0
	if (angle / 180.0) <= 6.8e-114:
		tmp = ((((b / (x_45_scale * x_45_scale)) * b) * a) * (-4.0 / (y_45_scale_m * y_45_scale_m))) * a
	else:
		tmp = (b * b) * ((a / (y_45_scale_m * x_45_scale)) * ((-4.0 * a) / (y_45_scale_m * x_45_scale)))
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 6.8e-114)
		tmp = Float64(Float64(Float64(Float64(Float64(b / Float64(x_45_scale * x_45_scale)) * b) * a) * Float64(-4.0 / Float64(y_45_scale_m * y_45_scale_m))) * a);
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(a / Float64(y_45_scale_m * x_45_scale)) * Float64(Float64(-4.0 * a) / Float64(y_45_scale_m * x_45_scale))));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0;
	if ((angle / 180.0) <= 6.8e-114)
		tmp = ((((b / (x_45_scale * x_45_scale)) * b) * a) * (-4.0 / (y_45_scale_m * y_45_scale_m))) * a;
	else
		tmp = (b * b) * ((a / (y_45_scale_m * x_45_scale)) * ((-4.0 * a) / (y_45_scale_m * x_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 6.8e-114], N[(N[(N[(N[(N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision] * N[(-4.0 / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(N[(a / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * a), $MachinePrecision] / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 6.79999999999999962e-114

   1. Initial program 24.9%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 56.4

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

   5. Applied rewrites 56.4%

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

   7. Applied rewrites 76.6%

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

   9. Applied rewrites 86.7%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 67.6%

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

   if 6.79999999999999962e-114 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 21.1%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 52.0%

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

   7. Applied rewrites 52.0%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 61.0%

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

   12. Applied rewrites 77.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 6.8 \cdot 10^{-114}:\\ \;\;\;\;\left(\left(\left(\frac{b}{x-scale \cdot x-scale} \cdot b\right) \cdot a\right) \cdot \frac{-4}{y-scale \cdot y-scale}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \frac{-4 \cdot a}{y-scale \cdot x-scale}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 29.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot \left(\frac{a}{y-scale\_m \cdot x-scale} \cdot \frac{-4 \cdot a}{y-scale\_m \cdot x-scale}\right)\\ t_1 := \frac{b}{y-scale\_m \cdot x-scale}\\ \mathbf{if}\;a \leq 1.55 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+160}:\\ \;\;\;\;\left(t\_1 \cdot t\_1\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0
         (*
          (* b b)
          (*
           (/ a (* y-scale_m x-scale))
           (/ (* -4.0 a) (* y-scale_m x-scale)))))
        (t_1 (/ b (* y-scale_m x-scale))))
   (if (<= a 1.55e-156)
     t_0
     (if (<= a 9.5e+160) (* (* t_1 t_1) (* (* a a) -4.0)) t_0))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (b * b) * ((a / (y_45_scale_m * x_45_scale)) * ((-4.0 * a) / (y_45_scale_m * x_45_scale)));
	double t_1 = b / (y_45_scale_m * x_45_scale);
	double tmp;
	if (a <= 1.55e-156) {
		tmp = t_0;
	} else if (a <= 9.5e+160) {
		tmp = (t_1 * t_1) * ((a * a) * -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) * ((a / (y_45scale_m * x_45scale)) * (((-4.0d0) * a) / (y_45scale_m * x_45scale)))
    t_1 = b / (y_45scale_m * x_45scale)
    if (a <= 1.55d-156) then
        tmp = t_0
    else if (a <= 9.5d+160) then
        tmp = (t_1 * t_1) * ((a * a) * (-4.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (b * b) * ((a / (y_45_scale_m * x_45_scale)) * ((-4.0 * a) / (y_45_scale_m * x_45_scale)));
	double t_1 = b / (y_45_scale_m * x_45_scale);
	double tmp;
	if (a <= 1.55e-156) {
		tmp = t_0;
	} else if (a <= 9.5e+160) {
		tmp = (t_1 * t_1) * ((a * a) * -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	t_0 = (b * b) * ((a / (y_45_scale_m * x_45_scale)) * ((-4.0 * a) / (y_45_scale_m * x_45_scale)))
	t_1 = b / (y_45_scale_m * x_45_scale)
	tmp = 0
	if a <= 1.55e-156:
		tmp = t_0
	elif a <= 9.5e+160:
		tmp = (t_1 * t_1) * ((a * a) * -4.0)
	else:
		tmp = t_0
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64(Float64(b * b) * Float64(Float64(a / Float64(y_45_scale_m * x_45_scale)) * Float64(Float64(-4.0 * a) / Float64(y_45_scale_m * x_45_scale))))
	t_1 = Float64(b / Float64(y_45_scale_m * x_45_scale))
	tmp = 0.0
	if (a <= 1.55e-156)
		tmp = t_0;
	elseif (a <= 9.5e+160)
		tmp = Float64(Float64(t_1 * t_1) * Float64(Float64(a * a) * -4.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = (b * b) * ((a / (y_45_scale_m * x_45_scale)) * ((-4.0 * a) / (y_45_scale_m * x_45_scale)));
	t_1 = b / (y_45_scale_m * x_45_scale);
	tmp = 0.0;
	if (a <= 1.55e-156)
		tmp = t_0;
	elseif (a <= 9.5e+160)
		tmp = (t_1 * t_1) * ((a * a) * -4.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * N[(N[(a / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * a), $MachinePrecision] / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.55e-156], t$95$0, If[LessEqual[a, 9.5e+160], 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 < 1.5499999999999999e-156 or 9.5000000000000006e160 < a

   1. Initial program 23.8%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 42.2%

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

   7. Applied rewrites 43.3%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 55.7%

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

   12. Applied rewrites 75.7%

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

   if 1.5499999999999999e-156 < a < 9.5000000000000006e160

   1. Initial program 23.3%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 95.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.55 \cdot 10^{-156}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \frac{-4 \cdot a}{y-scale \cdot x-scale}\right)\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+160}:\\ \;\;\;\;\left(\frac{b}{y-scale \cdot x-scale} \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \frac{-4 \cdot a}{y-scale \cdot x-scale}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 29.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot \left(\frac{a}{y-scale\_m \cdot x-scale} \cdot \frac{-4 \cdot a}{y-scale\_m \cdot x-scale}\right)\\ \mathbf{if}\;a \leq 1.55 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+160}:\\ \;\;\;\;\left(\frac{\frac{b}{y-scale\_m \cdot x-scale}}{y-scale\_m \cdot x-scale} \cdot b\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0
         (*
          (* b b)
          (*
           (/ a (* y-scale_m x-scale))
           (/ (* -4.0 a) (* y-scale_m x-scale))))))
   (if (<= a 1.55e-156)
     t_0
     (if (<= a 9.5e+160)
       (*
        (* (/ (/ b (* y-scale_m x-scale)) (* y-scale_m x-scale)) b)
        (* (* a a) -4.0))
       t_0))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (b * b) * ((a / (y_45_scale_m * x_45_scale)) * ((-4.0 * a) / (y_45_scale_m * x_45_scale)));
	double tmp;
	if (a <= 1.55e-156) {
		tmp = t_0;
	} else if (a <= 9.5e+160) {
		tmp = (((b / (y_45_scale_m * x_45_scale)) / (y_45_scale_m * x_45_scale)) * b) * ((a * a) * -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) * ((a / (y_45scale_m * x_45scale)) * (((-4.0d0) * a) / (y_45scale_m * x_45scale)))
    if (a <= 1.55d-156) then
        tmp = t_0
    else if (a <= 9.5d+160) then
        tmp = (((b / (y_45scale_m * x_45scale)) / (y_45scale_m * x_45scale)) * b) * ((a * a) * (-4.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (b * b) * ((a / (y_45_scale_m * x_45_scale)) * ((-4.0 * a) / (y_45_scale_m * x_45_scale)));
	double tmp;
	if (a <= 1.55e-156) {
		tmp = t_0;
	} else if (a <= 9.5e+160) {
		tmp = (((b / (y_45_scale_m * x_45_scale)) / (y_45_scale_m * x_45_scale)) * b) * ((a * a) * -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	t_0 = (b * b) * ((a / (y_45_scale_m * x_45_scale)) * ((-4.0 * a) / (y_45_scale_m * x_45_scale)))
	tmp = 0
	if a <= 1.55e-156:
		tmp = t_0
	elif a <= 9.5e+160:
		tmp = (((b / (y_45_scale_m * x_45_scale)) / (y_45_scale_m * x_45_scale)) * b) * ((a * a) * -4.0)
	else:
		tmp = t_0
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64(Float64(b * b) * Float64(Float64(a / Float64(y_45_scale_m * x_45_scale)) * Float64(Float64(-4.0 * a) / Float64(y_45_scale_m * x_45_scale))))
	tmp = 0.0
	if (a <= 1.55e-156)
		tmp = t_0;
	elseif (a <= 9.5e+160)
		tmp = Float64(Float64(Float64(Float64(b / Float64(y_45_scale_m * x_45_scale)) / Float64(y_45_scale_m * x_45_scale)) * b) * Float64(Float64(a * a) * -4.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = (b * b) * ((a / (y_45_scale_m * x_45_scale)) * ((-4.0 * a) / (y_45_scale_m * x_45_scale)));
	tmp = 0.0;
	if (a <= 1.55e-156)
		tmp = t_0;
	elseif (a <= 9.5e+160)
		tmp = (((b / (y_45_scale_m * x_45_scale)) / (y_45_scale_m * x_45_scale)) * b) * ((a * a) * -4.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * N[(N[(a / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * a), $MachinePrecision] / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.55e-156], t$95$0, If[LessEqual[a, 9.5e+160], N[(N[(N[(N[(b / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.5499999999999999e-156 or 9.5000000000000006e160 < a

   1. Initial program 23.8%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 42.2%

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

   7. Applied rewrites 43.3%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 55.7%

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

   12. Applied rewrites 75.7%

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

   if 1.5499999999999999e-156 < a < 9.5000000000000006e160

   1. Initial program 23.3%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 95.6%

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

   9. Applied rewrites 93.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.55 \cdot 10^{-156}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \frac{-4 \cdot a}{y-scale \cdot x-scale}\right)\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+160}:\\ \;\;\;\;\left(\frac{\frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \cdot b\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \frac{-4 \cdot a}{y-scale \cdot x-scale}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.8% accurate, 32.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-158}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-36}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot -4}{\left(y-scale\_m \cdot x-scale\right) \cdot \left(y-scale\_m \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b \cdot b}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\_m\right) \cdot y-scale\_m} \cdot \left(-4 \cdot a\right)\right) \cdot a\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (if (<= b 6.5e-158)
   0.0
   (if (<= b 3e-36)
     (*
      (/ (* (* a a) -4.0) (* (* y-scale_m x-scale) (* y-scale_m x-scale)))
      (* b b))
     (*
      (*
       (/ (* b b) (* (* (* x-scale x-scale) y-scale_m) y-scale_m))
       (* -4.0 a))
      a))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (b <= 6.5e-158) {
		tmp = 0.0;
	} else if (b <= 3e-36) {
		tmp = (((a * a) * -4.0) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	} else {
		tmp = (((b * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * (-4.0 * a)) * a;
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (b <= 6.5d-158) then
        tmp = 0.0d0
    else if (b <= 3d-36) then
        tmp = (((a * a) * (-4.0d0)) / ((y_45scale_m * x_45scale) * (y_45scale_m * x_45scale))) * (b * b)
    else
        tmp = (((b * b) / (((x_45scale * x_45scale) * y_45scale_m) * y_45scale_m)) * ((-4.0d0) * a)) * a
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (b <= 6.5e-158) {
		tmp = 0.0;
	} else if (b <= 3e-36) {
		tmp = (((a * a) * -4.0) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	} else {
		tmp = (((b * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * (-4.0 * a)) * a;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	tmp = 0
	if b <= 6.5e-158:
		tmp = 0.0
	elif b <= 3e-36:
		tmp = (((a * a) * -4.0) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b)
	else:
		tmp = (((b * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * (-4.0 * a)) * a
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0
	if (b <= 6.5e-158)
		tmp = 0.0;
	elseif (b <= 3e-36)
		tmp = Float64(Float64(Float64(Float64(a * a) * -4.0) / Float64(Float64(y_45_scale_m * x_45_scale) * Float64(y_45_scale_m * x_45_scale))) * Float64(b * b));
	else
		tmp = Float64(Float64(Float64(Float64(b * b) / Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * Float64(-4.0 * a)) * a);
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0;
	if (b <= 6.5e-158)
		tmp = 0.0;
	elseif (b <= 3e-36)
		tmp = (((a * a) * -4.0) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	else
		tmp = (((b * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * (-4.0 * a)) * a;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := If[LessEqual[b, 6.5e-158], 0.0, If[LessEqual[b, 3e-36], N[(N[(N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(y$45$scale$95$m * x$45$scale), $MachinePrecision] * N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] / N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale$95$m), $MachinePrecision] * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 6.49999999999999971e-158

   1. Initial program 29.2%

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

   4. Applied rewrites 23.2%

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

   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%

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

   if 6.49999999999999971e-158 < b < 3.0000000000000002e-36

   1. Initial program 19.2%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 48.3%

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

   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 82.9%

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

   if 3.0000000000000002e-36 < b

   1. Initial program 12.7%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 56.1

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

   5. Applied rewrites 56.1%

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

   7. Applied rewrites 73.8%

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

   9. Applied rewrites 85.3%

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

   11. Applied rewrites 61.8%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-158}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-36}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot -4}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b \cdot b}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale} \cdot \left(-4 \cdot a\right)\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.7% accurate, 35.9× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-158}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot -4}{\left(y-scale\_m \cdot x-scale\right) \cdot \left(y-scale\_m \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (if (<= b 6.5e-158)
   0.0
   (*
    (/ (* (* a a) -4.0) (* (* y-scale_m x-scale) (* y-scale_m x-scale)))
    (* b b))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (b <= 6.5e-158) {
		tmp = 0.0;
	} else {
		tmp = (((a * a) * -4.0) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (b <= 6.5d-158) then
        tmp = 0.0d0
    else
        tmp = (((a * a) * (-4.0d0)) / ((y_45scale_m * x_45scale) * (y_45scale_m * x_45scale))) * (b * b)
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (b <= 6.5e-158) {
		tmp = 0.0;
	} else {
		tmp = (((a * a) * -4.0) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	tmp = 0
	if b <= 6.5e-158:
		tmp = 0.0
	else:
		tmp = (((a * a) * -4.0) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b)
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0
	if (b <= 6.5e-158)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(Float64(a * a) * -4.0) / Float64(Float64(y_45_scale_m * x_45_scale) * Float64(y_45_scale_m * x_45_scale))) * Float64(b * b));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0;
	if (b <= 6.5e-158)
		tmp = 0.0;
	else
		tmp = (((a * a) * -4.0) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := If[LessEqual[b, 6.5e-158], 0.0, N[(N[(N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(y$45$scale$95$m * x$45$scale), $MachinePrecision] * N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.49999999999999971e-158

   1. Initial program 29.2%

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

   4. Applied rewrites 23.2%

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

   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%

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

   if 6.49999999999999971e-158 < b

   1. Initial program 14.3%

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

   3. Taylor expanded in 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 44.7%

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

   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 61.2%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-158}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot -4}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.8% accurate, 35.9× speedup

Math

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

FPCore

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

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (b / (y_45_scale_m * x_45_scale)) * a;
	return (t_0 * -4.0) * t_0;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    t_0 = (b / (y_45scale_m * x_45scale)) * a
    code = (t_0 * (-4.0d0)) * t_0
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (b / (y_45_scale_m * x_45_scale)) * a;
	return (t_0 * -4.0) * t_0;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	t_0 = (b / (y_45_scale_m * x_45_scale)) * a
	return (t_0 * -4.0) * t_0

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64(Float64(b / Float64(y_45_scale_m * x_45_scale)) * a)
	return Float64(Float64(t_0 * -4.0) * t_0)
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = (b / (y_45_scale_m * x_45_scale)) * a;
	tmp = (t_0 * -4.0) * t_0;
end

Wolfram

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

Derivation

1. Initial program 23.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 56.5

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

5. Applied rewrites 56.5%

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

7. Applied rewrites 76.8%

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

9. Applied rewrites 93.5%

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

11. Applied rewrites 93.5%

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

12. Final simplification 93.5%

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

Alternative 10: 34.7% accurate, 1905.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.6%

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

4. Applied rewrites 20.9%

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

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 38.0

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

7. Applied rewrites 38.0%

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

Reproduce

herbie shell --seed 2024276 
(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))))