Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.1% → 93.8%

Time: 2.0min

Alternatives: 4

Speedup: 2485.0×

• 33.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 25.1% 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: 93.8% accurate, 130.8× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \left(\frac{1}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}} \cdot \left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  -4.0
  (* (/ 1.0 (* (/ y-scale a) (/ x-scale b))) (* (/ b x-scale) (/ a y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * ((1.0 / ((y_45_scale / a) * (x_45_scale / b))) * ((b / x_45_scale) * (a / y_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 = (-4.0d0) * ((1.0d0 / ((y_45scale / a) * (x_45scale / b))) * ((b / x_45scale) * (a / y_45scale)))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * ((1.0 / ((y_45_scale / a) * (x_45_scale / b))) * ((b / x_45_scale) * (a / y_45_scale)));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return -4.0 * ((1.0 / ((y_45_scale / a) * (x_45_scale / b))) * ((b / x_45_scale) * (a / y_45_scale)))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(-4.0 * Float64(Float64(1.0 / Float64(Float64(y_45_scale / a) * Float64(x_45_scale / b))) * Float64(Float64(b / x_45_scale) * Float64(a / y_45_scale))))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = -4.0 * ((1.0 / ((y_45_scale / a) * (x_45_scale / b))) * ((b / x_45_scale) * (a / y_45_scale)));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(-4.0 * N[(N[(1.0 / N[(N[(y$45$scale / a), $MachinePrecision] * N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b / x$45$scale), $MachinePrecision] * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.0%

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

2. Taylor expanded in angle around 0 49.9%

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

   1. pow1 49.9%

      \[\leadsto -4 \cdot \frac{\color{blue}{{\left({a}^{2} \cdot {b}^{2}\right)}^{1}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]

   2. pow-prod-down 58.8%

      \[\leadsto -4 \cdot \frac{{\color{blue}{\left({\left(a \cdot b\right)}^{2}\right)}}^{1}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]

4. Applied egg-rr 58.8%

   \[\leadsto -4 \cdot \frac{\color{blue}{{\left({\left(a \cdot b\right)}^{2}\right)}^{1}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
5. Step-by-step derivation

   1. unpow1 58.8%

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

6. Simplified 58.8%

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

   1. unpow-prod-down 49.9%

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

   2. pow2 49.9%

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

   3. pow2 49.9%

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

   4. pow2 49.9%

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

   5. frac-times 48.8%

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

   6. unpow2 48.8%

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

   7. frac-times 60.2%

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

   8. add-sqr-sqrt 60.2%

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

   9. *-commutative 60.2%

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

   10. sqrt-prod 60.2%

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

   11. sqrt-prod 29.8%

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

   12. add-sqr-sqrt 43.1%

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

   13. times-frac 43.1%

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

   14. sqrt-prod 28.3%

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

   15. add-sqr-sqrt 45.0%

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

8. Applied egg-rr 95.8%

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

   1. *-commutative 95.8%

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

   2. clear-num 95.8%

      \[\leadsto -4 \cdot \left(\left(\color{blue}{\frac{1}{\frac{y-scale}{a}}} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)\right) \]

   3. clear-num 95.8%

      \[\leadsto -4 \cdot \left(\left(\frac{1}{\frac{y-scale}{a}} \cdot \color{blue}{\frac{1}{\frac{x-scale}{b}}}\right) \cdot \left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)\right) \]

   4. frac-times 96.0%

      \[\leadsto -4 \cdot \left(\color{blue}{\frac{1 \cdot 1}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}} \cdot \left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)\right) \]

   5. metadata-eval 96.0%

      \[\leadsto -4 \cdot \left(\frac{\color{blue}{1}}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}} \cdot \left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)\right) \]

10. Applied egg-rr 96.0%

    \[\leadsto -4 \cdot \left(\color{blue}{\frac{1}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}} \cdot \left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)\right) \]

11. Final simplification 96.0%

    \[\leadsto -4 \cdot \left(\frac{1}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}} \cdot \left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)\right) \]

Alternative 2: 78.6% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.0%

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

2. Taylor expanded in angle around 0 49.9%

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

   1. times-frac 48.8%

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

   2. unpow2 48.8%

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

   3. unpow2 48.8%

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

   4. unpow2 48.8%

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

   5. unpow2 48.8%

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

4. Simplified 48.8%

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

5. Taylor expanded in b around 0 48.8%

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

   1. unpow2 48.8%

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

   2. unpow2 48.8%

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

   3. times-frac 60.2%

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

7. Simplified 60.2%

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

8. Taylor expanded in a around 0 60.2%

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

   1. unpow2 60.2%

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

   2. unpow2 60.2%

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

   3. times-frac 75.1%

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

10. Simplified 75.1%

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

11. Final simplification 75.1%

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

Alternative 3: 94.1% accurate, 146.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b / x_45_scale) * (a / y_45_scale);
	return -4.0 * (t_0 * 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 / x_45scale) * (a / y_45scale)
    code = (-4.0d0) * (t_0 * 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 / x_45_scale) * (a / y_45_scale);
	return -4.0 * (t_0 * t_0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.0%

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

2. Taylor expanded in angle around 0 49.9%

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

   1. pow1 49.9%

      \[\leadsto -4 \cdot \frac{\color{blue}{{\left({a}^{2} \cdot {b}^{2}\right)}^{1}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]

   2. pow-prod-down 58.8%

      \[\leadsto -4 \cdot \frac{{\color{blue}{\left({\left(a \cdot b\right)}^{2}\right)}}^{1}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]

4. Applied egg-rr 58.8%

   \[\leadsto -4 \cdot \frac{\color{blue}{{\left({\left(a \cdot b\right)}^{2}\right)}^{1}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
5. Step-by-step derivation

   1. unpow1 58.8%

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

6. Simplified 58.8%

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

   1. unpow-prod-down 49.9%

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

   2. pow2 49.9%

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

   3. pow2 49.9%

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

   4. pow2 49.9%

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

   5. frac-times 48.8%

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

   6. unpow2 48.8%

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

   7. frac-times 60.2%

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

   8. add-sqr-sqrt 60.2%

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

   9. *-commutative 60.2%

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

   10. sqrt-prod 60.2%

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

   11. sqrt-prod 29.8%

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

   12. add-sqr-sqrt 43.1%

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

   13. times-frac 43.1%

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

   14. sqrt-prod 28.3%

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

   15. add-sqr-sqrt 45.0%

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

8. Applied egg-rr 95.8%

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

9. Final simplification 95.8%

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

Alternative 4: 35.8% accurate, 2485.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale) :precision binary64 0.0)

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

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
    code = 0.0d0
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 0.0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return 0.0
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 25.0%

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

   1. fma-neg 26.3%

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

3. Simplified 21.0%

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

4. Taylor expanded in b around 0 22.4%

   \[\leadsto \color{blue}{4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2} \cdot {x-scale}^{2}} + -4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
5. Step-by-step derivation

   1. *-commutative 22.4%

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

   2. *-commutative 22.4%

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

   3. *-commutative 22.4%

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

   4. distribute-lft-out 22.4%

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

6. Simplified 33.0%

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

7. Final simplification 33.0%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023178 
(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))))