Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.6% → 90.0%

Time: 20.5s

Alternatives: 7

Speedup: 40.5×

• 33.6% 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: 25.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 90.0% accurate, 32.3× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* a b) (/ (* (/ -4.0 y-scale) (/ (* a b) x-scale)) (* y-scale x-scale))))

C

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

Fortran

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

Java

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

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (a * b) * (((-4.0 / y_45_scale) * ((a * b) / x_45_scale)) / (y_45_scale * x_45_scale))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.0%

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

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 54.6

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

5. Applied rewrites 54.6%

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

7. Applied rewrites 81.7%

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

9. Applied rewrites 86.4%

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

11. Applied rewrites 90.9%

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

Alternative 2: 83.0% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-169} \lor \neg \left(b \leq 9 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{\left(\left(b \cdot a\right) \cdot \frac{b \cdot a}{x-scale}\right) \cdot -4}{\left(y-scale \cdot x-scale\right) \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (or (<= b 1.08e-169) (not (<= b 9e+80)))
   (/ (* (* (* b a) (/ (* b a) x-scale)) -4.0) (* (* y-scale x-scale) y-scale))
   (*
    (* (/ (* -4.0 a) (* y-scale x-scale)) (/ a (* y-scale x-scale)))
    (* b b))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if ((b <= 1.08e-169) || !(b <= 9e+80)) {
		tmp = (((b * a) * ((b * a) / x_45_scale)) * -4.0) / ((y_45_scale * x_45_scale) * y_45_scale);
	} else {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if ((b <= 1.08d-169) .or. (.not. (b <= 9d+80))) then
        tmp = (((b * a) * ((b * a) / x_45scale)) * (-4.0d0)) / ((y_45scale * x_45scale) * y_45scale)
    else
        tmp = ((((-4.0d0) * a) / (y_45scale * x_45scale)) * (a / (y_45scale * x_45scale))) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if ((b <= 1.08e-169) || !(b <= 9e+80)) {
		tmp = (((b * a) * ((b * a) / x_45_scale)) * -4.0) / ((y_45_scale * x_45_scale) * y_45_scale);
	} else {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if (b <= 1.08e-169) or not (b <= 9e+80):
		tmp = (((b * a) * ((b * a) / x_45_scale)) * -4.0) / ((y_45_scale * x_45_scale) * y_45_scale)
	else:
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if ((b <= 1.08e-169) || !(b <= 9e+80))
		tmp = Float64(Float64(Float64(Float64(b * a) * Float64(Float64(b * a) / x_45_scale)) * -4.0) / Float64(Float64(y_45_scale * x_45_scale) * y_45_scale));
	else
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) / Float64(y_45_scale * x_45_scale)) * Float64(a / Float64(y_45_scale * x_45_scale))) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if ((b <= 1.08e-169) || ~((b <= 9e+80)))
		tmp = (((b * a) * ((b * a) / x_45_scale)) * -4.0) / ((y_45_scale * x_45_scale) * y_45_scale);
	else
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[Or[LessEqual[b, 1.08e-169], N[Not[LessEqual[b, 9e+80]], $MachinePrecision]], N[(N[(N[(N[(b * a), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0799999999999999e-169 or 9.00000000000000013e80 < b

   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-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 48.8

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

   5. Applied rewrites 48.8%

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

   7. Applied rewrites 73.5%

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

   9. Applied rewrites 79.3%

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

   if 1.0799999999999999e-169 < b < 9.00000000000000013e80

   1. Initial program 31.4%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 79.4%

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

   10. Applied rewrites 95.3%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-169} \lor \neg \left(b \leq 9 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{\left(\left(b \cdot a\right) \cdot \frac{b \cdot a}{x-scale}\right) \cdot -4}{\left(y-scale \cdot x-scale\right) \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-169}:\\ \;\;\;\;\frac{\left(\left(b \cdot a\right) \cdot \frac{b \cdot a}{x-scale}\right) \cdot -4}{\left(y-scale \cdot x-scale\right) \cdot y-scale}\\ \mathbf{elif}\;b \leq 2.85 \cdot 10^{+110}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a \cdot b\right) \cdot \left(\left(a \cdot b\right) \cdot -4\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= b 1.08e-169)
   (/ (* (* (* b a) (/ (* b a) x-scale)) -4.0) (* (* y-scale x-scale) y-scale))
   (if (<= b 2.85e+110)
     (*
      (* (/ (* -4.0 a) (* y-scale x-scale)) (/ a (* y-scale x-scale)))
      (* b b))
     (/
      (/ (* (* a b) (* (* a b) -4.0)) (* y-scale x-scale))
      (* y-scale x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 1.08e-169) {
		tmp = (((b * a) * ((b * a) / x_45_scale)) * -4.0) / ((y_45_scale * x_45_scale) * y_45_scale);
	} else if (b <= 2.85e+110) {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	} else {
		tmp = (((a * b) * ((a * b) * -4.0)) / (y_45_scale * x_45_scale)) / (y_45_scale * x_45_scale);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b <= 1.08d-169) then
        tmp = (((b * a) * ((b * a) / x_45scale)) * (-4.0d0)) / ((y_45scale * x_45scale) * y_45scale)
    else if (b <= 2.85d+110) then
        tmp = ((((-4.0d0) * a) / (y_45scale * x_45scale)) * (a / (y_45scale * x_45scale))) * (b * b)
    else
        tmp = (((a * b) * ((a * b) * (-4.0d0))) / (y_45scale * x_45scale)) / (y_45scale * x_45scale)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 1.08e-169) {
		tmp = (((b * a) * ((b * a) / x_45_scale)) * -4.0) / ((y_45_scale * x_45_scale) * y_45_scale);
	} else if (b <= 2.85e+110) {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	} else {
		tmp = (((a * b) * ((a * b) * -4.0)) / (y_45_scale * x_45_scale)) / (y_45_scale * x_45_scale);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b <= 1.08e-169:
		tmp = (((b * a) * ((b * a) / x_45_scale)) * -4.0) / ((y_45_scale * x_45_scale) * y_45_scale)
	elif b <= 2.85e+110:
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b)
	else:
		tmp = (((a * b) * ((a * b) * -4.0)) / (y_45_scale * x_45_scale)) / (y_45_scale * x_45_scale)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b <= 1.08e-169)
		tmp = Float64(Float64(Float64(Float64(b * a) * Float64(Float64(b * a) / x_45_scale)) * -4.0) / Float64(Float64(y_45_scale * x_45_scale) * y_45_scale));
	elseif (b <= 2.85e+110)
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) / Float64(y_45_scale * x_45_scale)) * Float64(a / Float64(y_45_scale * x_45_scale))) * Float64(b * b));
	else
		tmp = Float64(Float64(Float64(Float64(a * b) * Float64(Float64(a * b) * -4.0)) / Float64(y_45_scale * x_45_scale)) / Float64(y_45_scale * x_45_scale));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b <= 1.08e-169)
		tmp = (((b * a) * ((b * a) / x_45_scale)) * -4.0) / ((y_45_scale * x_45_scale) * y_45_scale);
	elseif (b <= 2.85e+110)
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	else
		tmp = (((a * b) * ((a * b) * -4.0)) / (y_45_scale * x_45_scale)) / (y_45_scale * x_45_scale);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b, 1.08e-169], N[(N[(N[(N[(b * a), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.85e+110], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * b), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.0799999999999999e-169

   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-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 50.6

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

   5. Applied rewrites 50.6%

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

   7. Applied rewrites 73.7%

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

   9. Applied rewrites 78.7%

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

   if 1.0799999999999999e-169 < b < 2.8500000000000001e110

   1. Initial program 28.7%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 76.9%

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

   10. Applied rewrites 94.3%

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

   if 2.8500000000000001e110 < b

   1. Initial program 0.2%

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

   3. Taylor expanded in 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-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 39.4

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

   5. Applied rewrites 39.4%

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

   7. Applied rewrites 80.7%

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

   9. Applied rewrites 83.1%

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

   11. Applied rewrites 83.1%

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

Alternative 4: 75.3% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{-172}:\\ \;\;\;\;\frac{\left(b \cdot \left(b \cdot \left(\frac{a}{x-scale} \cdot a\right)\right)\right) \cdot -4}{\left(y-scale \cdot x-scale\right) \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= b 3.9e-172)
   (/ (* (* b (* b (* (/ a x-scale) a))) -4.0) (* (* y-scale x-scale) y-scale))
   (*
    (* (/ (* -4.0 a) (* y-scale x-scale)) (/ a (* y-scale x-scale)))
    (* b b))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 3.9e-172) {
		tmp = ((b * (b * ((a / x_45_scale) * a))) * -4.0) / ((y_45_scale * x_45_scale) * y_45_scale);
	} else {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b <= 3.9d-172) then
        tmp = ((b * (b * ((a / x_45scale) * a))) * (-4.0d0)) / ((y_45scale * x_45scale) * y_45scale)
    else
        tmp = ((((-4.0d0) * a) / (y_45scale * x_45scale)) * (a / (y_45scale * x_45scale))) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 3.9e-172) {
		tmp = ((b * (b * ((a / x_45_scale) * a))) * -4.0) / ((y_45_scale * x_45_scale) * y_45_scale);
	} else {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b <= 3.9e-172:
		tmp = ((b * (b * ((a / x_45_scale) * a))) * -4.0) / ((y_45_scale * x_45_scale) * y_45_scale)
	else:
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b <= 3.9e-172)
		tmp = Float64(Float64(Float64(b * Float64(b * Float64(Float64(a / x_45_scale) * a))) * -4.0) / Float64(Float64(y_45_scale * x_45_scale) * y_45_scale));
	else
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) / Float64(y_45_scale * x_45_scale)) * Float64(a / Float64(y_45_scale * x_45_scale))) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b <= 3.9e-172)
		tmp = ((b * (b * ((a / x_45_scale) * a))) * -4.0) / ((y_45_scale * x_45_scale) * y_45_scale);
	else
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b, 3.9e-172], N[(N[(N[(b * N[(b * N[(N[(a / x$45$scale), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.89999999999999973e-172

   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-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 50.6

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

   5. Applied rewrites 50.6%

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

   7. Applied rewrites 73.7%

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

   9. Applied rewrites 70.8%

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

   if 3.89999999999999973e-172 < b

   1. Initial program 18.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 64.7%

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

   10. Applied rewrites 79.2%

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

Alternative 5: 63.6% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 5.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot a\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale} \cdot \frac{a}{\left(x-scale \cdot x-scale\right) \cdot y-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale 5.2e-186)
   (* (/ (* -4.0 (* a a)) (* (* y-scale x-scale) (* y-scale x-scale))) (* b b))
   (*
    (* (/ (* -4.0 a) y-scale) (/ a (* (* x-scale x-scale) y-scale)))
    (* b b))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 5.2e-186) {
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * b);
	} else {
		tmp = (((-4.0 * a) / y_45_scale) * (a / ((x_45_scale * x_45_scale) * y_45_scale))) * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (x_45scale <= 5.2d-186) then
        tmp = (((-4.0d0) * (a * a)) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * (b * b)
    else
        tmp = ((((-4.0d0) * a) / y_45scale) * (a / ((x_45scale * x_45scale) * y_45scale))) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 5.2e-186) {
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * b);
	} else {
		tmp = (((-4.0 * a) / y_45_scale) * (a / ((x_45_scale * x_45_scale) * y_45_scale))) * (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if x_45_scale <= 5.2e-186:
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * b)
	else:
		tmp = (((-4.0 * a) / y_45_scale) * (a / ((x_45_scale * x_45_scale) * y_45_scale))) * (b * b)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (x_45_scale <= 5.2e-186)
		tmp = Float64(Float64(Float64(-4.0 * Float64(a * a)) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * Float64(b * b));
	else
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) / y_45_scale) * Float64(a / Float64(Float64(x_45_scale * x_45_scale) * y_45_scale))) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (x_45_scale <= 5.2e-186)
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * b);
	else
		tmp = (((-4.0 * a) / y_45_scale) * (a / ((x_45_scale * x_45_scale) * y_45_scale))) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[x$45$scale, 5.2e-186], N[(N[(N[(-4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(a / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 5.19999999999999986e-186

   1. Initial program 20.2%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 55.8%

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

   if 5.19999999999999986e-186 < x-scale

   1. Initial program 24.5%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 64.8%

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

   10. Applied rewrites 72.7%

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

Alternative 6: 75.7% accurate, 35.9× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* (/ (* -4.0 a) (* y-scale x-scale)) (/ a (* y-scale x-scale))) (* b b)))

C

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

Fortran

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

Java

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

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.0%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 47.6%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 59.4%

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

10. Applied rewrites 73.9%

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

Alternative 7: 61.9% accurate, 40.5× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (/ (* -4.0 (* a a)) (* (* y-scale x-scale) (* y-scale x-scale))) (* b b)))

C

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

Fortran

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

Java

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

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * b)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.0%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 47.6%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 59.4%

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

Reproduce

herbie shell --seed 2024340 
(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))))