Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.4% → 90.7%

Time: 52.2s

Alternatives: 5

Speedup: 40.5×

• 33.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = -1.3292481275496746e+308
  2. b = -1.9955853071597118e+302
  3. angle = -5.1118442769173696e+302
  4. x-scale = -2.262465316064051e-46
  5. y-scale = 8.284789788602635e-194

Specification

Math

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

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)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Initial Program: 24.4% accurate, 1.0× speedup

Math

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

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)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Alternative 1: 90.7% accurate, 35.9× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* y-scale x-scale)))) (* (* (* t_0 a) (* -4.0 a)) t_0)))

C

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

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) :: t_0
    t_0 = b / (y_45scale * x_45scale)
    code = ((t_0 * a) * ((-4.0d0) * a)) * t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.8%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
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. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 55.0

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

5. Applied rewrites 55.0%

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

7. Applied rewrites 79.2%

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

9. Applied rewrites 92.6%

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

10. Final simplification 92.6%

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

Alternative 2: 79.1% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-152}:\\ \;\;\;\;\frac{\left(\left(a \cdot a\right) \cdot -4\right) \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \cdot b\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{+155}:\\ \;\;\;\;\frac{b \cdot b}{y-scale \cdot x-scale} \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \left(-4 \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-4 \cdot a\right) \cdot b\right) \cdot a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= b 1.65e-152)
   (* (/ (* (* (* a a) -4.0) (/ b (* y-scale x-scale))) (* y-scale x-scale)) b)
   (if (<= b 1.75e+155)
     (*
      (/ (* b b) (* y-scale x-scale))
      (* (/ a (* y-scale x-scale)) (* -4.0 a)))
     (*
      (/ (* (* (* -4.0 a) b) a) (* (* y-scale x-scale) (* y-scale x-scale)))
      b))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 1.65e-152) {
		tmp = ((((a * a) * -4.0) * (b / (y_45_scale * x_45_scale))) / (y_45_scale * x_45_scale)) * b;
	} else if (b <= 1.75e+155) {
		tmp = ((b * b) / (y_45_scale * x_45_scale)) * ((a / (y_45_scale * x_45_scale)) * (-4.0 * a));
	} else {
		tmp = ((((-4.0 * a) * b) * a) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * 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 <= 1.65d-152) then
        tmp = ((((a * a) * (-4.0d0)) * (b / (y_45scale * x_45scale))) / (y_45scale * x_45scale)) * b
    else if (b <= 1.75d+155) then
        tmp = ((b * b) / (y_45scale * x_45scale)) * ((a / (y_45scale * x_45scale)) * ((-4.0d0) * a))
    else
        tmp = (((((-4.0d0) * a) * b) * a) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * 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 <= 1.65e-152) {
		tmp = ((((a * a) * -4.0) * (b / (y_45_scale * x_45_scale))) / (y_45_scale * x_45_scale)) * b;
	} else if (b <= 1.75e+155) {
		tmp = ((b * b) / (y_45_scale * x_45_scale)) * ((a / (y_45_scale * x_45_scale)) * (-4.0 * a));
	} else {
		tmp = ((((-4.0 * a) * b) * a) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b <= 1.65e-152:
		tmp = ((((a * a) * -4.0) * (b / (y_45_scale * x_45_scale))) / (y_45_scale * x_45_scale)) * b
	elif b <= 1.75e+155:
		tmp = ((b * b) / (y_45_scale * x_45_scale)) * ((a / (y_45_scale * x_45_scale)) * (-4.0 * a))
	else:
		tmp = ((((-4.0 * a) * b) * a) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b <= 1.65e-152)
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * -4.0) * Float64(b / Float64(y_45_scale * x_45_scale))) / Float64(y_45_scale * x_45_scale)) * b);
	elseif (b <= 1.75e+155)
		tmp = Float64(Float64(Float64(b * b) / Float64(y_45_scale * x_45_scale)) * Float64(Float64(a / Float64(y_45_scale * x_45_scale)) * Float64(-4.0 * a)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-4.0 * a) * b) * a) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b <= 1.65e-152)
		tmp = ((((a * a) * -4.0) * (b / (y_45_scale * x_45_scale))) / (y_45_scale * x_45_scale)) * b;
	elseif (b <= 1.75e+155)
		tmp = ((b * b) / (y_45_scale * x_45_scale)) * ((a / (y_45_scale * x_45_scale)) * (-4.0 * a));
	else
		tmp = ((((-4.0 * a) * b) * a) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b, 1.65e-152], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision] * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[b, 1.75e+155], N[(N[(N[(b * b), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-4.0 * a), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.64999999999999999e-152

   1. Initial program 30.3%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 61.1%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 68.8%

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

   12. Applied rewrites 80.8%

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

   if 1.64999999999999999e-152 < b < 1.74999999999999992e155

   1. Initial program 20.2%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

   7. Applied rewrites 82.5%

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

   9. Applied rewrites 90.5%

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

   if 1.74999999999999992e155 < b

   1. Initial program 0.0%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 50.3

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

   5. Applied rewrites 50.3%

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

   7. Applied rewrites 62.6%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 71.3%

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

   12. Applied rewrites 84.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-152}:\\ \;\;\;\;\frac{\left(\left(a \cdot a\right) \cdot -4\right) \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \cdot b\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{+155}:\\ \;\;\;\;\frac{b \cdot b}{y-scale \cdot x-scale} \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \left(-4 \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-4 \cdot a\right) \cdot b\right) \cdot a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.55 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left(\left(-4 \cdot a\right) \cdot b\right) \cdot a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot b\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{\left(\left(a \cdot a\right) \cdot -4\right) \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot b\right) \cdot \left(-4 \cdot a\right)\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a 3.55e-161)
   (* (/ (* (* (* -4.0 a) b) a) (* (* y-scale x-scale) (* y-scale x-scale))) b)
   (if (<= a 1.3e+156)
     (*
      (/ (* (* (* a a) -4.0) (/ b (* y-scale x-scale))) (* y-scale x-scale))
      b)
     (*
      (* (* (/ b (* (* (* y-scale x-scale) y-scale) x-scale)) b) (* -4.0 a))
      a))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 3.55e-161) {
		tmp = ((((-4.0 * a) * b) * a) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b;
	} else if (a <= 1.3e+156) {
		tmp = ((((a * a) * -4.0) * (b / (y_45_scale * x_45_scale))) / (y_45_scale * x_45_scale)) * b;
	} else {
		tmp = (((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * b) * (-4.0 * a)) * a;
	}
	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 (a <= 3.55d-161) then
        tmp = (((((-4.0d0) * a) * b) * a) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * b
    else if (a <= 1.3d+156) then
        tmp = ((((a * a) * (-4.0d0)) * (b / (y_45scale * x_45scale))) / (y_45scale * x_45scale)) * b
    else
        tmp = (((b / (((y_45scale * x_45scale) * y_45scale) * x_45scale)) * b) * ((-4.0d0) * a)) * a
    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 (a <= 3.55e-161) {
		tmp = ((((-4.0 * a) * b) * a) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b;
	} else if (a <= 1.3e+156) {
		tmp = ((((a * a) * -4.0) * (b / (y_45_scale * x_45_scale))) / (y_45_scale * x_45_scale)) * b;
	} else {
		tmp = (((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * b) * (-4.0 * a)) * a;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= 3.55e-161:
		tmp = ((((-4.0 * a) * b) * a) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b
	elif a <= 1.3e+156:
		tmp = ((((a * a) * -4.0) * (b / (y_45_scale * x_45_scale))) / (y_45_scale * x_45_scale)) * b
	else:
		tmp = (((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * b) * (-4.0 * a)) * a
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= 3.55e-161)
		tmp = Float64(Float64(Float64(Float64(Float64(-4.0 * a) * b) * a) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * b);
	elseif (a <= 1.3e+156)
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * -4.0) * Float64(b / Float64(y_45_scale * x_45_scale))) / Float64(y_45_scale * x_45_scale)) * b);
	else
		tmp = Float64(Float64(Float64(Float64(b / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * b) * Float64(-4.0 * a)) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a <= 3.55e-161)
		tmp = ((((-4.0 * a) * b) * a) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b;
	elseif (a <= 1.3e+156)
		tmp = ((((a * a) * -4.0) * (b / (y_45_scale * x_45_scale))) / (y_45_scale * x_45_scale)) * b;
	else
		tmp = (((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * b) * (-4.0 * a)) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, 3.55e-161], N[(N[(N[(N[(N[(-4.0 * a), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[a, 1.3e+156], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision] * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[(N[(N[(b / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 3.55e-161

   1. Initial program 30.4%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 52.7

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

   5. Applied rewrites 52.7%

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

   7. Applied rewrites 61.9%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 71.8%

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

   12. Applied rewrites 80.1%

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

   if 3.55e-161 < a < 1.30000000000000009e156

   1. Initial program 19.2%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

   7. Applied rewrites 68.0%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 77.6%

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

   12. Applied rewrites 93.5%

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

   if 1.30000000000000009e156 < a

   1. Initial program 0.0%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 42.7

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

   5. Applied rewrites 42.7%

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

   7. Applied rewrites 66.3%

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

   9. Applied rewrites 74.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.55 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left(\left(-4 \cdot a\right) \cdot b\right) \cdot a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot b\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{\left(\left(a \cdot a\right) \cdot -4\right) \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot b\right) \cdot \left(-4 \cdot a\right)\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 40.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((-4.0 * a) * b) * a) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * 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) * b) * a) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * 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) * b) * a) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.8%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
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. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 55.0

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

5. Applied rewrites 55.0%

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

7. Applied rewrites 62.2%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 71.4%

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

12. Applied rewrites 77.1%

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

13. Final simplification 77.1%

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

Alternative 5: 69.4% accurate, 40.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((a * a) * -4.0) * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * 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 = ((((a * a) * (-4.0d0)) * b) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.8%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
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. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 55.0

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

5. Applied rewrites 55.0%

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

7. Applied rewrites 62.2%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 71.4%

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

11. Final simplification 71.4%

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

Reproduce

herbie shell --seed 2024235 
(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))))