Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.9% → 94.2%

Time: 20.3s

Alternatives: 10

Speedup: 40.5×

• 33.1% 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: 24.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) (PI)))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

Alternative 1: 94.2% accurate, 26.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{\frac{b}{y-scale}}{x-scale} \cdot a\_m\\ t_1 := \frac{b \cdot a\_m}{y-scale \cdot x-scale}\\ \mathbf{if}\;a\_m \leq 8 \cdot 10^{-147}:\\ \;\;\;\;-4 \cdot \left(t\_1 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ (/ b y-scale) x-scale) a_m))
        (t_1 (/ (* b a_m) (* y-scale x-scale))))
   (if (<= a_m 8e-147) (* -4.0 (* t_1 t_1)) (* (* t_0 t_0) -4.0))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((b / y_45_scale) / x_45_scale) * a_m;
	double t_1 = (b * a_m) / (y_45_scale * x_45_scale);
	double tmp;
	if (a_m <= 8e-147) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = (t_0 * t_0) * -4.0;
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((b / y_45scale) / x_45scale) * a_m
    t_1 = (b * a_m) / (y_45scale * x_45scale)
    if (a_m <= 8d-147) then
        tmp = (-4.0d0) * (t_1 * t_1)
    else
        tmp = (t_0 * t_0) * (-4.0d0)
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((b / y_45_scale) / x_45_scale) * a_m;
	double t_1 = (b * a_m) / (y_45_scale * x_45_scale);
	double tmp;
	if (a_m <= 8e-147) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = (t_0 * t_0) * -4.0;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = ((b / y_45_scale) / x_45_scale) * a_m
	t_1 = (b * a_m) / (y_45_scale * x_45_scale)
	tmp = 0
	if a_m <= 8e-147:
		tmp = -4.0 * (t_1 * t_1)
	else:
		tmp = (t_0 * t_0) * -4.0
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(b / y_45_scale) / x_45_scale) * a_m)
	t_1 = Float64(Float64(b * a_m) / Float64(y_45_scale * x_45_scale))
	tmp = 0.0
	if (a_m <= 8e-147)
		tmp = Float64(-4.0 * Float64(t_1 * t_1));
	else
		tmp = Float64(Float64(t_0 * t_0) * -4.0);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = ((b / y_45_scale) / x_45_scale) * a_m;
	t_1 = (b * a_m) / (y_45_scale * x_45_scale);
	tmp = 0.0;
	if (a_m <= 8e-147)
		tmp = -4.0 * (t_1 * t_1);
	else
		tmp = (t_0 * t_0) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(b / y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] * a$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * a$95$m), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 8e-147], N[(-4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 7.9999999999999998e-147

   1. Initial program 25.3%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 42.3

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

   5. Applied rewrites 42.3%

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

   7. Applied rewrites 74.1%

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

   9. Applied rewrites 95.0%

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

   if 7.9999999999999998e-147 < a

   1. Initial program 17.1%

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

   3. Taylor expanded in 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 61.7

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

   5. Applied rewrites 61.7%

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

   7. Applied rewrites 72.8%

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

   9. Applied rewrites 96.1%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{-147}:\\ \;\;\;\;-4 \cdot \left(\frac{b \cdot a}{y-scale \cdot x-scale} \cdot \frac{b \cdot a}{y-scale \cdot x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{b}{y-scale}}{x-scale} \cdot a\right) \cdot \left(\frac{\frac{b}{y-scale}}{x-scale} \cdot a\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 29.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \left(\frac{b \cdot a\_m}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot a\_m\right)\right) \cdot -4\\ \mathbf{if}\;x-scale \leq 3.55 \cdot 10^{-147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x-scale \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\left(\frac{b \cdot a\_m}{\left(x-scale \cdot x-scale\right) \cdot y-scale} \cdot \frac{b \cdot a\_m}{y-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (*
           (/ (* b a_m) (* (* y-scale x-scale) (* y-scale x-scale)))
           (* b a_m))
          -4.0)))
   (if (<= x-scale 3.55e-147)
     t_0
     (if (<= x-scale 2e+129)
       (*
        (* (/ (* b a_m) (* (* x-scale x-scale) y-scale)) (/ (* b a_m) y-scale))
        -4.0)
       t_0))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((b * a_m) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * a_m)) * -4.0;
	double tmp;
	if (x_45_scale <= 3.55e-147) {
		tmp = t_0;
	} else if (x_45_scale <= 2e+129) {
		tmp = (((b * a_m) / ((x_45_scale * x_45_scale) * y_45_scale)) * ((b * a_m) / y_45_scale)) * -4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((b * a_m) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * (b * a_m)) * (-4.0d0)
    if (x_45scale <= 3.55d-147) then
        tmp = t_0
    else if (x_45scale <= 2d+129) then
        tmp = (((b * a_m) / ((x_45scale * x_45scale) * y_45scale)) * ((b * a_m) / y_45scale)) * (-4.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((b * a_m) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * a_m)) * -4.0;
	double tmp;
	if (x_45_scale <= 3.55e-147) {
		tmp = t_0;
	} else if (x_45_scale <= 2e+129) {
		tmp = (((b * a_m) / ((x_45_scale * x_45_scale) * y_45_scale)) * ((b * a_m) / y_45_scale)) * -4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = (((b * a_m) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * a_m)) * -4.0
	tmp = 0
	if x_45_scale <= 3.55e-147:
		tmp = t_0
	elif x_45_scale <= 2e+129:
		tmp = (((b * a_m) / ((x_45_scale * x_45_scale) * y_45_scale)) * ((b * a_m) / y_45_scale)) * -4.0
	else:
		tmp = t_0
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(Float64(b * a_m) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * Float64(b * a_m)) * -4.0)
	tmp = 0.0
	if (x_45_scale <= 3.55e-147)
		tmp = t_0;
	elseif (x_45_scale <= 2e+129)
		tmp = Float64(Float64(Float64(Float64(b * a_m) / Float64(Float64(x_45_scale * x_45_scale) * y_45_scale)) * Float64(Float64(b * a_m) / y_45_scale)) * -4.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = (((b * a_m) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * a_m)) * -4.0;
	tmp = 0.0;
	if (x_45_scale <= 3.55e-147)
		tmp = t_0;
	elseif (x_45_scale <= 2e+129)
		tmp = (((b * a_m) / ((x_45_scale * x_45_scale) * y_45_scale)) * ((b * a_m) / y_45_scale)) * -4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(N[(b * a$95$m), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * a$95$m), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[x$45$scale, 3.55e-147], t$95$0, If[LessEqual[x$45$scale, 2e+129], N[(N[(N[(N[(b * a$95$m), $MachinePrecision] / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b * a$95$m), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 3.55000000000000008e-147 or 2e129 < x-scale

   1. Initial program 22.1%

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

   3. Taylor expanded in 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 46.6

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

   5. Applied rewrites 46.6%

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

   7. Applied rewrites 71.0%

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

   9. Applied rewrites 78.3%

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

   11. Applied rewrites 78.3%

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

   if 3.55000000000000008e-147 < x-scale < 2e129

   1. Initial program 23.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 58.4

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

   5. Applied rewrites 58.4%

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

   7. Applied rewrites 82.2%

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

   9. Applied rewrites 95.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 3.55 \cdot 10^{-147}:\\ \;\;\;\;\left(\frac{b \cdot a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot a\right)\right) \cdot -4\\ \mathbf{elif}\;x-scale \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\left(\frac{b \cdot a}{\left(x-scale \cdot x-scale\right) \cdot y-scale} \cdot \frac{b \cdot a}{y-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b \cdot a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot a\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 29.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \left(\frac{b \cdot a\_m}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot a\_m\right)\right) \cdot -4\\ \mathbf{if}\;y-scale \leq 4.2 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y-scale \leq 3.1 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{b \cdot a\_m}{x-scale} \cdot \left(\frac{a\_m}{\left(y-scale \cdot y-scale\right) \cdot x-scale} \cdot b\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (*
           (/ (* b a_m) (* (* y-scale x-scale) (* y-scale x-scale)))
           (* b a_m))
          -4.0)))
   (if (<= y-scale 4.2e-132)
     t_0
     (if (<= y-scale 3.1e+154)
       (*
        (* (/ (* b a_m) x-scale) (* (/ a_m (* (* y-scale y-scale) x-scale)) b))
        -4.0)
       t_0))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((b * a_m) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * a_m)) * -4.0;
	double tmp;
	if (y_45_scale <= 4.2e-132) {
		tmp = t_0;
	} else if (y_45_scale <= 3.1e+154) {
		tmp = (((b * a_m) / x_45_scale) * ((a_m / ((y_45_scale * y_45_scale) * x_45_scale)) * b)) * -4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((b * a_m) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * (b * a_m)) * (-4.0d0)
    if (y_45scale <= 4.2d-132) then
        tmp = t_0
    else if (y_45scale <= 3.1d+154) then
        tmp = (((b * a_m) / x_45scale) * ((a_m / ((y_45scale * y_45scale) * x_45scale)) * b)) * (-4.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((b * a_m) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * a_m)) * -4.0;
	double tmp;
	if (y_45_scale <= 4.2e-132) {
		tmp = t_0;
	} else if (y_45_scale <= 3.1e+154) {
		tmp = (((b * a_m) / x_45_scale) * ((a_m / ((y_45_scale * y_45_scale) * x_45_scale)) * b)) * -4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = (((b * a_m) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * a_m)) * -4.0
	tmp = 0
	if y_45_scale <= 4.2e-132:
		tmp = t_0
	elif y_45_scale <= 3.1e+154:
		tmp = (((b * a_m) / x_45_scale) * ((a_m / ((y_45_scale * y_45_scale) * x_45_scale)) * b)) * -4.0
	else:
		tmp = t_0
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(Float64(b * a_m) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * Float64(b * a_m)) * -4.0)
	tmp = 0.0
	if (y_45_scale <= 4.2e-132)
		tmp = t_0;
	elseif (y_45_scale <= 3.1e+154)
		tmp = Float64(Float64(Float64(Float64(b * a_m) / x_45_scale) * Float64(Float64(a_m / Float64(Float64(y_45_scale * y_45_scale) * x_45_scale)) * b)) * -4.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = (((b * a_m) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * a_m)) * -4.0;
	tmp = 0.0;
	if (y_45_scale <= 4.2e-132)
		tmp = t_0;
	elseif (y_45_scale <= 3.1e+154)
		tmp = (((b * a_m) / x_45_scale) * ((a_m / ((y_45_scale * y_45_scale) * x_45_scale)) * b)) * -4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(N[(b * a$95$m), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * a$95$m), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[y$45$scale, 4.2e-132], t$95$0, If[LessEqual[y$45$scale, 3.1e+154], N[(N[(N[(N[(b * a$95$m), $MachinePrecision] / x$45$scale), $MachinePrecision] * N[(N[(a$95$m / N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.2000000000000002e-132 or 3.1000000000000001e154 < y-scale

   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 49.6

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

   5. Applied rewrites 49.6%

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

   7. Applied rewrites 73.4%

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

   9. Applied rewrites 80.2%

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

   11. Applied rewrites 80.2%

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

   if 4.2000000000000002e-132 < y-scale < 3.1000000000000001e154

   1. Initial program 19.0%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 48.5

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

   5. Applied rewrites 48.5%

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

   7. Applied rewrites 74.3%

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

   9. Applied rewrites 89.9%

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

   11. Applied rewrites 86.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 4.2 \cdot 10^{-132}:\\ \;\;\;\;\left(\frac{b \cdot a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot a\right)\right) \cdot -4\\ \mathbf{elif}\;y-scale \leq 3.1 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{b \cdot a}{x-scale} \cdot \left(\frac{a}{\left(y-scale \cdot y-scale\right) \cdot x-scale} \cdot b\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b \cdot a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot a\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 29.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{b}{y-scale \cdot x-scale}\\ t_1 := \left(\frac{b \cdot a\_m}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot a\_m\right)\right) \cdot -4\\ \mathbf{if}\;a\_m \leq 2.9 \cdot 10^{-154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a\_m \leq 1.6 \cdot 10^{+83}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot \left(\left(a\_m \cdot a\_m\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* y-scale x-scale)))
        (t_1
         (*
          (*
           (/ (* b a_m) (* (* y-scale x-scale) (* y-scale x-scale)))
           (* b a_m))
          -4.0)))
   (if (<= a_m 2.9e-154)
     t_1
     (if (<= a_m 1.6e+83) (* (* t_0 t_0) (* (* a_m a_m) -4.0)) t_1))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	double t_1 = (((b * a_m) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * a_m)) * -4.0;
	double tmp;
	if (a_m <= 2.9e-154) {
		tmp = t_1;
	} else if (a_m <= 1.6e+83) {
		tmp = (t_0 * t_0) * ((a_m * a_m) * -4.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b / (y_45scale * x_45scale)
    t_1 = (((b * a_m) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * (b * a_m)) * (-4.0d0)
    if (a_m <= 2.9d-154) then
        tmp = t_1
    else if (a_m <= 1.6d+83) then
        tmp = (t_0 * t_0) * ((a_m * a_m) * (-4.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	double t_1 = (((b * a_m) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * a_m)) * -4.0;
	double tmp;
	if (a_m <= 2.9e-154) {
		tmp = t_1;
	} else if (a_m <= 1.6e+83) {
		tmp = (t_0 * t_0) * ((a_m * a_m) * -4.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = b / (y_45_scale * x_45_scale)
	t_1 = (((b * a_m) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * a_m)) * -4.0
	tmp = 0
	if a_m <= 2.9e-154:
		tmp = t_1
	elif a_m <= 1.6e+83:
		tmp = (t_0 * t_0) * ((a_m * a_m) * -4.0)
	else:
		tmp = t_1
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(y_45_scale * x_45_scale))
	t_1 = Float64(Float64(Float64(Float64(b * a_m) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * Float64(b * a_m)) * -4.0)
	tmp = 0.0
	if (a_m <= 2.9e-154)
		tmp = t_1;
	elseif (a_m <= 1.6e+83)
		tmp = Float64(Float64(t_0 * t_0) * Float64(Float64(a_m * a_m) * -4.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = b / (y_45_scale * x_45_scale);
	t_1 = (((b * a_m) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * a_m)) * -4.0;
	tmp = 0.0;
	if (a_m <= 2.9e-154)
		tmp = t_1;
	elseif (a_m <= 1.6e+83)
		tmp = (t_0 * t_0) * ((a_m * a_m) * -4.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(b * a$95$m), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * a$95$m), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[a$95$m, 2.9e-154], t$95$1, If[LessEqual[a$95$m, 1.6e+83], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(N[(a$95$m * a$95$m), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.9e-154 or 1.5999999999999999e83 < a

   1. Initial program 21.9%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 45.2

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

   5. Applied rewrites 45.2%

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

   7. Applied rewrites 74.2%

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

   9. Applied rewrites 79.9%

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

   11. Applied rewrites 79.9%

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

   if 2.9e-154 < a < 1.5999999999999999e83

   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 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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 90.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.9 \cdot 10^{-154}:\\ \;\;\;\;\left(\frac{b \cdot a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot a\right)\right) \cdot -4\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+83}:\\ \;\;\;\;\left(\frac{b}{y-scale \cdot x-scale} \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b \cdot a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot a\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 35.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 3.5 \cdot 10^{-226}:\\ \;\;\;\;\left(\frac{b \cdot a\_m}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot \left(b \cdot a\_m\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b \cdot a\_m}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot \left(b \cdot a\_m\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 3.5e-226)
   (*
    (* (/ (* b a_m) (* (* (* y-scale x-scale) x-scale) y-scale)) (* b a_m))
    -4.0)
   (*
    (* (/ (* b a_m) (* (* (* y-scale x-scale) y-scale) x-scale)) (* b a_m))
    -4.0)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 3.5e-226) {
		tmp = (((b * a_m) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)) * (b * a_m)) * -4.0;
	} else {
		tmp = (((b * a_m) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (b * a_m)) * -4.0;
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (y_45scale <= 3.5d-226) then
        tmp = (((b * a_m) / (((y_45scale * x_45scale) * x_45scale) * y_45scale)) * (b * a_m)) * (-4.0d0)
    else
        tmp = (((b * a_m) / (((y_45scale * x_45scale) * y_45scale) * x_45scale)) * (b * a_m)) * (-4.0d0)
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 3.5e-226) {
		tmp = (((b * a_m) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)) * (b * a_m)) * -4.0;
	} else {
		tmp = (((b * a_m) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (b * a_m)) * -4.0;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if y_45_scale <= 3.5e-226:
		tmp = (((b * a_m) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)) * (b * a_m)) * -4.0
	else:
		tmp = (((b * a_m) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (b * a_m)) * -4.0
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 3.5e-226)
		tmp = Float64(Float64(Float64(Float64(b * a_m) / Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)) * Float64(b * a_m)) * -4.0);
	else
		tmp = Float64(Float64(Float64(Float64(b * a_m) / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * Float64(b * a_m)) * -4.0);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (y_45_scale <= 3.5e-226)
		tmp = (((b * a_m) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)) * (b * a_m)) * -4.0;
	else
		tmp = (((b * a_m) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (b * a_m)) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 3.5e-226], N[(N[(N[(N[(b * a$95$m), $MachinePrecision] / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(b * a$95$m), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(N[(b * a$95$m), $MachinePrecision] / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(b * a$95$m), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 3.5e-226

   1. Initial program 20.3%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 55.3

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

   5. Applied rewrites 55.3%

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

   7. Applied rewrites 75.4%

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

   9. Applied rewrites 82.4%

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

   11. Applied rewrites 79.8%

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

   if 3.5e-226 < y-scale

   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 41.7

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

   5. Applied rewrites 41.7%

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

   7. Applied rewrites 71.3%

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

   9. Applied rewrites 75.3%

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

   10. Taylor expanded in x-scale around 0

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

   12. Applied rewrites 73.7%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.5 \cdot 10^{-226}:\\ \;\;\;\;\left(\frac{b \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot \left(b \cdot a\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot \left(b \cdot a\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.7% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.3%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 49.4

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

5. Applied rewrites 49.4%

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

7. Applied rewrites 73.6%

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

9. Applied rewrites 93.8%

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

10. Final simplification 93.8%

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

Alternative 7: 92.1% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.3%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 49.4

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

5. Applied rewrites 49.4%

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

7. Applied rewrites 73.6%

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

9. Applied rewrites 93.8%

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

11. Applied rewrites 91.5%

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

12. Final simplification 91.5%

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

Alternative 8: 83.2% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.3%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 49.4

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

5. Applied rewrites 49.4%

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

7. Applied rewrites 73.6%

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

9. Applied rewrites 79.3%

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

11. Applied rewrites 79.3%

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

12. Final simplification 79.3%

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

Alternative 9: 79.8% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.3%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 49.4

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

5. Applied rewrites 49.4%

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

7. Applied rewrites 73.6%

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

9. Applied rewrites 79.3%

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

10. Taylor expanded in x-scale around 0

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

12. Applied rewrites 76.4%

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

13. Final simplification 76.4%

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

Alternative 10: 61.0% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.3%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 49.4

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

5. Applied rewrites 49.4%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 53.8%

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

9. Final simplification 53.8%

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

Reproduce

herbie shell --seed 2024288 
(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))))