Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.5% → 94.2%

Time: 1.9min

Alternatives: 3

Speedup: 146.2×

• 32.4% 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: 25.5% 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: 94.2% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

3. Simplified 17.2%

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

4. Taylor expanded in angle around 0 49.2%

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

   1. *-un-lft-identity 49.2%

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

   2. pow-prod-down 60.6%

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

   3. *-commutative 60.6%

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

   4. unpow2 60.6%

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

   5. times-frac 64.7%

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

   6. *-commutative 64.7%

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

   7. pow-prod-down 82.1%

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

   8. *-commutative 82.1%

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

6. Applied egg-rr 82.1%

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

   1. *-commutative 82.1%

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

8. Simplified 82.1%

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

   1. un-div-inv 82.2%

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

   2. associate-/r* 76.7%

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

   3. unpow2 76.7%

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

   4. pow-prod-down 60.6%

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

   5. associate-/l* 62.6%

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

   6. unpow2 62.6%

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

   7. associate-/l* 70.7%

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

   8. add-sqr-sqrt 34.4%

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

   9. associate-/r* 34.4%

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

10. Applied egg-rr 46.1%

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

11. Final simplification 46.1%

    \[\leadsto -4 \cdot \frac{\frac{a}{\frac{\frac{x-scale \cdot y-scale}{b}}{\sqrt{a}}}}{\frac{\frac{x-scale \cdot y-scale}{b}}{\sqrt{a}}} \]

Alternative 2: 86.8% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

3. Simplified 17.2%

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

4. Taylor expanded in angle around 0 49.2%

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

   1. *-un-lft-identity 49.2%

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

   2. pow-prod-down 60.6%

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

   3. *-commutative 60.6%

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

   4. unpow2 60.6%

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

   5. times-frac 64.7%

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

   6. *-commutative 64.7%

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

   7. pow-prod-down 82.1%

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

   8. *-commutative 82.1%

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

6. Applied egg-rr 82.1%

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

   1. *-commutative 82.1%

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

8. Simplified 82.1%

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

   1. un-div-inv 82.2%

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

   2. associate-/r* 76.7%

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

   3. unpow2 76.7%

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

   4. pow-prod-down 60.6%

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

   5. associate-/l* 62.6%

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

   6. unpow2 62.6%

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

   7. associate-/l* 70.7%

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

   8. add-sqr-sqrt 34.4%

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

   9. associate-/r* 34.4%

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

10. Applied egg-rr 46.1%

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

    1. associate-/l/ 44.7%

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

    2. add-sqr-sqrt 44.6%

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

    3. times-frac 46.0%

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

    4. associate-/l* 46.0%

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

    5. add-sqr-sqrt 46.1%

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

    6. associate-/l* 46.1%

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

    7. add-sqr-sqrt 94.1%

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

    8. associate-/l* 91.2%

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

    9. associate-/l* 88.0%

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

    10. associate-/r/ 86.9%

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

    11. *-commutative 86.9%

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

    12. associate-*l/ 85.7%

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

    13. times-frac 85.9%

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

    14. *-commutative 85.9%

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

    15. associate-*r/ 89.2%

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

12. Applied egg-rr 87.7%

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

    1. associate-*l* 87.4%

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

    2. *-commutative 87.4%

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

    3. associate-/l/ 86.7%

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

    4. associate-/r/ 87.7%

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

    5. associate-/l* 84.8%

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

    6. associate-/r* 84.9%

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

    7. associate-*r/ 87.8%

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

    8. associate-/l* 89.6%

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

    9. associate-/r/ 88.2%

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

14. Simplified 88.2%

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

15. Final simplification 88.2%

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

Alternative 3: 93.5% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

3. Simplified 17.2%

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

4. Taylor expanded in angle around 0 49.2%

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

   1. *-un-lft-identity 49.2%

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

   2. pow-prod-down 60.6%

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

   3. *-commutative 60.6%

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

   4. unpow2 60.6%

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

   5. times-frac 64.7%

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

   6. *-commutative 64.7%

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

   7. pow-prod-down 82.1%

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

   8. *-commutative 82.1%

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

6. Applied egg-rr 82.1%

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

   1. *-commutative 82.1%

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

8. Simplified 82.1%

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

   1. un-div-inv 82.2%

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

   2. associate-/r* 76.7%

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

   3. unpow2 76.7%

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

   4. pow-prod-down 60.6%

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

   5. associate-/l* 62.6%

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

   6. unpow2 62.6%

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

   7. associate-/l* 70.7%

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

   8. add-sqr-sqrt 34.4%

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

   9. associate-/r* 34.4%

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

10. Applied egg-rr 46.1%

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

    1. associate-/l/ 44.7%

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

    2. add-sqr-sqrt 44.6%

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

    3. times-frac 46.0%

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

    4. associate-/l* 46.0%

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

    5. add-sqr-sqrt 46.1%

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

    6. associate-/l* 44.7%

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

    7. associate-/l* 44.7%

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

    8. add-sqr-sqrt 91.2%

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

    9. associate-/l* 92.6%

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

    10. frac-times 76.7%

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

    11. swap-sqr 60.6%

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

    12. swap-sqr 49.2%

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

    13. frac-times 51.3%

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

    14. frac-times 63.0%

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

    15. frac-times 77.6%

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

    16. unswap-sqr 93.6%

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

12. Applied egg-rr 93.9%

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

13. Final simplification 93.9%

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

Reproduce

herbie shell --seed 2023326 
(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))))