Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.0% → 93.5%

Time: 1.6min

Alternatives: 2

Speedup: 2485.0×

• 34.7% 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: 24.0% 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: 93.5% accurate, 118.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{\left(x-scale \cdot y-scale\right) \cdot \frac{1}{b}}\\ -4 \cdot \left(t_0 \cdot t_0\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* (* x-scale y-scale) (/ 1.0 b))))) (* -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 = a / ((x_45_scale * y_45_scale) * (1.0 / b));
	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 = a / ((x_45scale * y_45scale) * (1.0d0 / b))
    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 = a / ((x_45_scale * y_45_scale) * (1.0 / b));
	return -4.0 * (t_0 * t_0);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a / ((x_45_scale * y_45_scale) * (1.0 / b))
	return -4.0 * (t_0 * t_0)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(Float64(x_45_scale * y_45_scale) * Float64(1.0 / b)))
	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 = a / ((x_45_scale * y_45_scale) * (1.0 / b));
	tmp = -4.0 * (t_0 * t_0);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 27.6%

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

3. Simplified 23.2%

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

4. Taylor expanded in angle around 0 49.1%

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

   1. associate-/l* 49.3%

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

6. Simplified 49.3%

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

   1. expm1-log1p-u 49.3%

      \[\leadsto -4 \cdot \frac{{a}^{2}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x-scale}^{2} \cdot {y-scale}^{2}}{{b}^{2}}\right)\right)}} \]

   2. expm1-udef 48.7%

      \[\leadsto -4 \cdot \frac{{a}^{2}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{x-scale}^{2} \cdot {y-scale}^{2}}{{b}^{2}}\right)} - 1}} \]

   3. div-inv 48.7%

      \[\leadsto -4 \cdot \frac{{a}^{2}}{e^{\mathsf{log1p}\left(\color{blue}{\left({x-scale}^{2} \cdot {y-scale}^{2}\right) \cdot \frac{1}{{b}^{2}}}\right)} - 1} \]

   4. pow-prod-down 59.2%

      \[\leadsto -4 \cdot \frac{{a}^{2}}{e^{\mathsf{log1p}\left(\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \frac{1}{{b}^{2}}\right)} - 1} \]

   5. pow-flip 59.2%

      \[\leadsto -4 \cdot \frac{{a}^{2}}{e^{\mathsf{log1p}\left({\left(x-scale \cdot y-scale\right)}^{2} \cdot \color{blue}{{b}^{\left(-2\right)}}\right)} - 1} \]

   6. metadata-eval 59.2%

      \[\leadsto -4 \cdot \frac{{a}^{2}}{e^{\mathsf{log1p}\left({\left(x-scale \cdot y-scale\right)}^{2} \cdot {b}^{\color{blue}{-2}}\right)} - 1} \]

8. Applied egg-rr 59.2%

   \[\leadsto -4 \cdot \frac{{a}^{2}}{\color{blue}{e^{\mathsf{log1p}\left({\left(x-scale \cdot y-scale\right)}^{2} \cdot {b}^{-2}\right)} - 1}} \]
9. Step-by-step derivation

   1. expm1-def 60.6%

      \[\leadsto -4 \cdot \frac{{a}^{2}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(x-scale \cdot y-scale\right)}^{2} \cdot {b}^{-2}\right)\right)}} \]

   2. expm1-log1p 60.8%

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

10. Simplified 60.8%

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

    1. add-sqr-sqrt 60.7%

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

    2. sqrt-div 60.7%

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

    3. unpow2 60.7%

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

    4. sqrt-prod 30.0%

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

    5. add-sqr-sqrt 46.9%

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

    6. sqrt-prod 46.9%

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

    7. unpow2 46.9%

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

    8. sqrt-prod 27.1%

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

    9. add-sqr-sqrt 46.7%

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

    10. sqrt-pow1 46.6%

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

    11. metadata-eval 46.6%

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

    12. unpow-1 46.6%

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

    13. sqrt-div 46.6%

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

    14. unpow2 46.6%

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

    15. sqrt-prod 23.7%

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

    16. add-sqr-sqrt 50.9%

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

    17. sqrt-prod 51.8%

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

12. Applied egg-rr 94.7%

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

13. Final simplification 94.7%

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

Alternative 2: 34.0% 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 27.6%

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

3. Simplified 23.3%

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

4. Taylor expanded in b around 0 29.1%

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

   1. distribute-rgt-out 29.1%

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

   2. metadata-eval 29.1%

      \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{0} \]

   3. mul0-rgt 40.4%

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

6. Simplified 40.4%

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

7. Final simplification 40.4%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023333 
(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))))