Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.9% → 94.2%

Time: 2.5min

Alternatives: 3

Speedup: 2485.0×

• 34.4% 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: 24.9% 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, 130.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{a}{\frac{x-scale}{b}}}{y-scale}\\ t_1 := a \cdot \frac{b}{y-scale \cdot x-scale}\\ \mathbf{if}\;y-scale \leq 2.9 \cdot 10^{-259}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (/ a (/ x-scale b)) y-scale))
        (t_1 (* a (/ b (* y-scale x-scale)))))
   (if (<= y-scale 2.9e-259) (* -4.0 (* t_1 t_1)) (* -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 / b)) / y_45_scale;
	double t_1 = a * (b / (y_45_scale * x_45_scale));
	double tmp;
	if (y_45_scale <= 2.9e-259) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (a / (x_45scale / b)) / y_45scale
    t_1 = a * (b / (y_45scale * x_45scale))
    if (y_45scale <= 2.9d-259) then
        tmp = (-4.0d0) * (t_1 * t_1)
    else
        tmp = (-4.0d0) * (t_0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a / (x_45_scale / b)) / y_45_scale;
	double t_1 = a * (b / (y_45_scale * x_45_scale));
	double tmp;
	if (y_45_scale <= 2.9e-259) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (a / (x_45_scale / b)) / y_45_scale
	t_1 = a * (b / (y_45_scale * x_45_scale))
	tmp = 0
	if y_45_scale <= 2.9e-259:
		tmp = -4.0 * (t_1 * t_1)
	else:
		tmp = -4.0 * (t_0 * t_0)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a / Float64(x_45_scale / b)) / y_45_scale)
	t_1 = Float64(a * Float64(b / Float64(y_45_scale * x_45_scale)))
	tmp = 0.0
	if (y_45_scale <= 2.9e-259)
		tmp = Float64(-4.0 * Float64(t_1 * t_1));
	else
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (a / (x_45_scale / b)) / y_45_scale;
	t_1 = a * (b / (y_45_scale * x_45_scale));
	tmp = 0.0;
	if (y_45_scale <= 2.9e-259)
		tmp = -4.0 * (t_1 * t_1);
	else
		tmp = -4.0 * (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a / N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, 2.9e-259], N[(-4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.90000000000000009e-259

   1. Initial program 27.6%

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

   3. Simplified 25.3%

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

   4. Taylor expanded in angle around 0 51.8%

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

      1. times-frac 49.6%

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

      2. unpow2 49.6%

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

      3. unpow2 49.6%

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

      4. unpow2 49.6%

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

      5. unpow2 49.6%

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

   6. Simplified 49.6%

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

      1. expm1-log1p-u 49.5%

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

      2. expm1-udef 46.5%

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

      3. *-commutative 46.5%

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

      4. times-frac 58.3%

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

      5. pow2 58.3%

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

      6. times-frac 68.2%

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

      7. pow2 68.2%

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

   8. Applied egg-rr 68.2%

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

      1. expm1-def 79.1%

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

      2. expm1-log1p 79.5%

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

   10. Simplified 79.5%

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

   11. Taylor expanded in b around 0 51.8%

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

   13. Simplified 96.3%

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

       1. unpow2 96.3%

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

   15. Applied egg-rr 96.3%

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

   if 2.90000000000000009e-259 < y-scale

   1. Initial program 22.6%

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

   3. Simplified 20.0%

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

   4. Taylor expanded in angle around 0 49.0%

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

      1. *-commutative 49.0%

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

      2. times-frac 50.6%

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

      3. unpow2 50.6%

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

      4. unpow2 50.6%

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

      5. times-frac 58.7%

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

      6. unpow2 58.7%

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

      7. unpow2 58.7%

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

   6. Simplified 58.7%

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

      1. associate-*r/ 60.7%

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

      2. pow2 60.7%

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

      3. pow2 60.7%

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

      4. pow-prod-down 74.5%

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

   8. Applied egg-rr 74.5%

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

      1. associate-*l/ 72.4%

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

      2. *-commutative 72.4%

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

   10. Simplified 72.4%

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

       1. add-sqr-sqrt 72.3%

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

       2. sqrt-div 72.4%

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

       3. unpow2 72.4%

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

       4. sqrt-prod 42.2%

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

       5. add-sqr-sqrt 51.9%

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

       6. associate-/l* 50.3%

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

       7. sqrt-prod 50.2%

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

       8. add-sqr-sqrt 50.3%

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

       9. sqrt-div 50.3%

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

       10. unpow2 50.3%

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

       11. sqrt-prod 41.6%

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

       12. add-sqr-sqrt 70.9%

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

       13. associate-/l* 77.5%

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

       14. sqrt-prod 93.2%

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

       15. add-sqr-sqrt 93.3%

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

   12. Applied egg-rr 93.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.9 \cdot 10^{-259}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{a}{\frac{x-scale}{b}}}{y-scale} \cdot \frac{\frac{a}{\frac{x-scale}{b}}}{y-scale}\right)\\ \end{array} \]

Alternative 2: 93.9% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.3%

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

3. Simplified 22.9%

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

4. Taylor expanded in angle around 0 50.5%

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

   1. times-frac 50.8%

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

   2. unpow2 50.8%

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

   3. unpow2 50.8%

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

   4. unpow2 50.8%

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

   5. unpow2 50.8%

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

6. Simplified 50.8%

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

   1. expm1-log1p-u 50.8%

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

   2. expm1-udef 49.1%

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

   3. *-commutative 49.1%

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

   4. times-frac 60.6%

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

   5. pow2 60.6%

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

   6. times-frac 69.8%

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

   7. pow2 69.8%

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

8. Applied egg-rr 69.8%

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

   1. expm1-def 78.0%

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

   2. expm1-log1p 78.7%

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

10. Simplified 78.7%

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

11. Taylor expanded in b around 0 50.5%

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

13. Simplified 92.6%

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

    1. unpow2 92.6%

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

15. Applied egg-rr 92.6%

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

16. Final simplification 92.6%

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

Alternative 3: 35.3% accurate, 2485.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.3%

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

3. Simplified 24.3%

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

4. Taylor expanded in b around 0 23.5%

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

   1. distribute-rgt-out 23.5%

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

   2. metadata-eval 23.5%

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

   3. mul0-rgt 34.6%

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

6. Simplified 34.6%

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

7. Final simplification 34.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023279 
(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))))