Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.8% → 95.0%

Time: 1.3min

Alternatives: 4

Speedup: 2485.0×

• 34.2% 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: 24.8% 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: 95.0% accurate, 7.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \sqrt{a \cdot b}\\ \mathbf{if}\;x-scale \leq 4.6 \cdot 10^{-158}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{\frac{y-scale}{\frac{b}{x-scale}}} \cdot \frac{a}{y-scale \cdot \frac{x-scale}{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{t_0}{x-scale} \cdot \frac{t_0}{y-scale}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sqrt (* a b))))
   (if (<= x-scale 4.6e-158)
     (*
      -4.0
      (* (/ a (/ y-scale (/ b x-scale))) (/ a (* y-scale (/ x-scale b)))))
     (* -4.0 (pow (* (/ t_0 x-scale) (/ t_0 y-scale)) 2.0)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sqrt((a * b));
	double tmp;
	if (x_45_scale <= 4.6e-158) {
		tmp = -4.0 * ((a / (y_45_scale / (b / x_45_scale))) * (a / (y_45_scale * (x_45_scale / b))));
	} else {
		tmp = -4.0 * pow(((t_0 / x_45_scale) * (t_0 / y_45_scale)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
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
    real(8) :: tmp
    t_0 = sqrt((a * b))
    if (x_45scale <= 4.6d-158) then
        tmp = (-4.0d0) * ((a / (y_45scale / (b / x_45scale))) * (a / (y_45scale * (x_45scale / b))))
    else
        tmp = (-4.0d0) * (((t_0 / x_45scale) * (t_0 / y_45scale)) ** 2.0d0)
    end if
    code = tmp
end function

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.sqrt((a * b));
	double tmp;
	if (x_45_scale <= 4.6e-158) {
		tmp = -4.0 * ((a / (y_45_scale / (b / x_45_scale))) * (a / (y_45_scale * (x_45_scale / b))));
	} else {
		tmp = -4.0 * Math.pow(((t_0 / x_45_scale) * (t_0 / y_45_scale)), 2.0);
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.sqrt((a * b))
	tmp = 0
	if x_45_scale <= 4.6e-158:
		tmp = -4.0 * ((a / (y_45_scale / (b / x_45_scale))) * (a / (y_45_scale * (x_45_scale / b))))
	else:
		tmp = -4.0 * math.pow(((t_0 / x_45_scale) * (t_0 / y_45_scale)), 2.0)
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sqrt(Float64(a * b))
	tmp = 0.0
	if (x_45_scale <= 4.6e-158)
		tmp = Float64(-4.0 * Float64(Float64(a / Float64(y_45_scale / Float64(b / x_45_scale))) * Float64(a / Float64(y_45_scale * Float64(x_45_scale / b)))));
	else
		tmp = Float64(-4.0 * (Float64(Float64(t_0 / x_45_scale) * Float64(t_0 / y_45_scale)) ^ 2.0));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sqrt((a * b));
	tmp = 0.0;
	if (x_45_scale <= 4.6e-158)
		tmp = -4.0 * ((a / (y_45_scale / (b / x_45_scale))) * (a / (y_45_scale * (x_45_scale / b))));
	else
		tmp = -4.0 * (((t_0 / x_45_scale) * (t_0 / y_45_scale)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sqrt[N[(a * b), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$45$scale, 4.6e-158], N[(-4.0 * N[(N[(a / N[(y$45$scale / N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a / N[(y$45$scale * N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(N[(t$95$0 / x$45$scale), $MachinePrecision] * N[(t$95$0 / y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 4.5999999999999998e-158

   1. Initial program 21.1%

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

   2. Taylor expanded in angle around 0 40.3%

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

      1. associate-/l* 42.6%

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

      2. unpow2 42.6%

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

      3. unpow2 42.6%

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

      4. unpow2 42.6%

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

      5. unswap-sqr 58.2%

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

      6. unpow2 58.2%

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

   4. Simplified 58.2%

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

      1. times-frac 74.2%

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

   6. Applied egg-rr 74.2%

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

      1. times-frac 92.2%

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

      2. associate-/l* 90.5%

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

      3. associate-/l* 96.1%

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

   8. Applied egg-rr 96.1%

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

      1. div-inv 96.1%

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

      2. clear-num 96.2%

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

   10. Applied egg-rr 96.2%

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

   if 4.5999999999999998e-158 < x-scale

   1. Initial program 24.7%

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

   2. Taylor expanded in angle around 0 49.4%

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

      1. associate-/l* 51.4%

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

      2. unpow2 51.4%

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

      3. unpow2 51.4%

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

      4. unpow2 51.4%

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

      5. unswap-sqr 61.0%

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

      6. unpow2 61.0%

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

   4. Simplified 61.0%

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

      1. times-frac 74.3%

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

   6. Applied egg-rr 74.3%

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

      1. frac-times 61.0%

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

      2. unpow2 61.0%

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

      3. unpow-prod-down 51.4%

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

      4. pow2 51.4%

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

      5. pow2 51.4%

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

      6. frac-times 58.7%

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

      7. times-frac 66.2%

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

      8. associate-/l* 71.3%

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

      9. associate-/l* 73.5%

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

   8. Applied egg-rr 73.5%

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

      1. associate-/r/ 76.6%

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

      2. associate-/r/ 81.7%

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

   10. Simplified 81.7%

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

       1. add-sqr-sqrt 81.7%

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

       2. pow2 81.7%

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

   12. Applied egg-rr 54.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 4.6 \cdot 10^{-158}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{\frac{y-scale}{\frac{b}{x-scale}}} \cdot \frac{a}{y-scale \cdot \frac{x-scale}{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{\sqrt{a \cdot b}}{x-scale} \cdot \frac{\sqrt{a \cdot b}}{y-scale}\right)}^{2}\\ \end{array} \]

Alternative 2: 82.9% accurate, 146.2× speedup

Math

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

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* -4.0 (* (* (/ a y-scale) (/ b y-scale)) (* (/ b x-scale) (/ a x-scale)))))

C

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

Fortran

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
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) * (b / y_45scale)) * ((b / x_45scale) * (a / x_45scale)))
end function

Java

a = Math.abs(a);
b = Math.abs(b);
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) * (b / y_45_scale)) * ((b / x_45_scale) * (a / x_45_scale)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(-4.0 * N[(N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b / x$45$scale), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.4%

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

2. Taylor expanded in angle around 0 43.8%

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

   1. associate-/l* 45.9%

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

   2. unpow2 45.9%

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

   3. unpow2 45.9%

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

   4. unpow2 45.9%

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

   5. unswap-sqr 59.3%

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

   6. unpow2 59.3%

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

4. Simplified 59.3%

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

   1. times-frac 74.3%

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

6. Applied egg-rr 74.3%

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

   1. frac-times 59.3%

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

   2. unpow2 59.3%

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

   3. unpow-prod-down 45.9%

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

   4. pow2 45.9%

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

   5. pow2 45.9%

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

   6. frac-times 52.5%

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

   7. times-frac 61.8%

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

   8. associate-/l* 66.8%

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

   9. associate-/l* 73.3%

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

8. Applied egg-rr 73.3%

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

   1. associate-/r/ 75.9%

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

   2. associate-/r/ 83.4%

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

10. Simplified 83.4%

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

11. Final simplification 83.4%

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

Alternative 3: 94.3% accurate, 146.2× speedup

Math

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

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* -4.0 (* (/ a (/ y-scale (/ b x-scale))) (/ a (* y-scale (/ x-scale b))))))

C

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

Fortran

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
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 / (b / x_45scale))) * (a / (y_45scale * (x_45scale / b))))
end function

Java

a = Math.abs(a);
b = Math.abs(b);
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 / (b / x_45_scale))) * (a / (y_45_scale * (x_45_scale / b))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(-4.0 * N[(N[(a / N[(y$45$scale / N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a / N[(y$45$scale * N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.4%

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

2. Taylor expanded in angle around 0 43.8%

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

   1. associate-/l* 45.9%

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

   2. unpow2 45.9%

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

   3. unpow2 45.9%

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

   4. unpow2 45.9%

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

   5. unswap-sqr 59.3%

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

   6. unpow2 59.3%

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

4. Simplified 59.3%

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

   1. times-frac 74.3%

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

6. Applied egg-rr 74.3%

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

   1. times-frac 92.0%

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

   2. associate-/l* 89.9%

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

   3. associate-/l* 93.8%

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

8. Applied egg-rr 93.8%

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

   1. div-inv 93.8%

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

   2. clear-num 93.8%

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

10. Applied egg-rr 93.8%

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

11. Final simplification 93.8%

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

Alternative 4: 35.3% accurate, 2485.0× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ 0 \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale) :precision binary64 0.0)

C

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

Fortran

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 22.4%

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

   1. fma-neg 23.8%

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

3. Simplified 20.2%

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

4. Taylor expanded in b around 0 21.3%

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

   1. *-commutative 21.3%

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

   2. *-commutative 21.3%

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

   3. *-commutative 21.3%

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

   4. distribute-lft-out 21.3%

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

6. Simplified 31.9%

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

7. Final simplification 31.9%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023208 
(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))))