Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.9% → 94.1%

Time: 2.1min

Alternatives: 7

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: 94.1% accurate, 107.8× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := \frac{a}{y-scale} \cdot \frac{b}{x-scale}\\ t_1 := \frac{1}{\frac{x-scale \cdot y-scale}{a \cdot b}}\\ \mathbf{if}\;b \leq 1.6 \cdot 10^{-173}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ a y-scale) (/ b x-scale)))
        (t_1 (/ 1.0 (/ (* x-scale y-scale) (* a b)))))
   (if (<= b 1.6e-173) (* -4.0 (* t_1 t_1)) (* -4.0 (* t_0 t_0)))))

C

b = abs(b);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a / y_45_scale) * (b / x_45_scale);
	double t_1 = 1.0 / ((x_45_scale * y_45_scale) / (a * b));
	double tmp;
	if (b <= 1.6e-173) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Fortran

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) :: t_1
    real(8) :: tmp
    t_0 = (a / y_45scale) * (b / x_45scale)
    t_1 = 1.0d0 / ((x_45scale * y_45scale) / (a * b))
    if (b <= 1.6d-173) then
        tmp = (-4.0d0) * (t_1 * t_1)
    else
        tmp = (-4.0d0) * (t_0 * t_0)
    end if
    code = tmp
end function

Java

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 = (a / y_45_scale) * (b / x_45_scale);
	double t_1 = 1.0 / ((x_45_scale * y_45_scale) / (a * b));
	double tmp;
	if (b <= 1.6e-173) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (a / y_45_scale) * (b / x_45_scale)
	t_1 = 1.0 / ((x_45_scale * y_45_scale) / (a * b))
	tmp = 0
	if b <= 1.6e-173:
		tmp = -4.0 * (t_1 * t_1)
	else:
		tmp = -4.0 * (t_0 * t_0)
	return tmp

Julia

b = abs(b)
function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale))
	t_1 = Float64(1.0 / Float64(Float64(x_45_scale * y_45_scale) / Float64(a * b)))
	tmp = 0.0
	if (b <= 1.6e-173)
		tmp = Float64(-4.0 * Float64(t_1 * t_1));
	else
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (a / y_45_scale) * (b / x_45_scale);
	t_1 = 1.0 / ((x_45_scale * y_45_scale) / (a * b));
	tmp = 0.0;
	if (b <= 1.6e-173)
		tmp = -4.0 * (t_1 * t_1);
	else
		tmp = -4.0 * (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.6e-173], 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 b < 1.6e-173

   1. Initial program 29.3%

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

   3. Simplified 27.2%

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

   4. Taylor expanded in angle around 0 51.4%

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

      1. times-frac 49.3%

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

      2. unpow2 49.3%

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

      3. unpow2 49.3%

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

      4. unpow2 49.3%

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

      5. unpow2 49.3%

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

   6. Simplified 49.3%

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

      1. pow2 49.3%

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

      2. pow2 49.3%

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

      3. frac-times 51.4%

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

      4. pow2 51.4%

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

      5. pow2 51.4%

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

      6. clear-num 51.4%

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

      7. pow-prod-down 59.7%

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

      8. pow-prod-down 81.0%

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

   8. Applied egg-rr 81.0%

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

      1. add-sqr-sqrt 81.0%

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

      2. sqrt-div 81.1%

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

      3. metadata-eval 81.1%

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

      4. sqrt-div 81.1%

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

      5. unpow2 81.1%

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

      6. sqrt-prod 42.3%

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

      7. add-sqr-sqrt 59.2%

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

      8. sqrt-pow1 59.7%

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

      9. metadata-eval 59.7%

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

      10. pow1 59.7%

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

      11. sqrt-div 59.7%

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

      12. metadata-eval 59.7%

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

      13. sqrt-div 59.7%

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

      14. unpow2 59.7%

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

      15. sqrt-prod 30.2%

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

      16. add-sqr-sqrt 58.2%

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

      17. sqrt-pow1 92.1%

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

      18. metadata-eval 92.1%

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

      19. pow1 92.1%

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

   10. Applied egg-rr 92.1%

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

   if 1.6e-173 < b

   1. Initial program 18.4%

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

   3. Simplified 15.3%

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

   4. Taylor expanded in angle around 0 48.9%

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

      1. pow2 48.9%

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

      2. pow2 48.9%

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

      3. frac-times 50.8%

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

      4. pow2 50.8%

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

      5. times-frac 65.7%

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

      6. pow2 65.7%

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

      7. times-frac 77.4%

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

   6. Applied egg-rr 77.4%

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

      1. unswap-sqr 95.3%

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

   8. Simplified 95.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-173}:\\ \;\;\;\;-4 \cdot \left(\frac{1}{\frac{x-scale \cdot y-scale}{a \cdot b}} \cdot \frac{1}{\frac{x-scale \cdot y-scale}{a \cdot b}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\\ \end{array} \]

Alternative 2: 93.5% accurate, 21.8× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ -4 \cdot \frac{1}{\frac{1}{{\left(\frac{\frac{x-scale \cdot y-scale}{a}}{b}\right)}^{-2}}} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* -4.0 (/ 1.0 (/ 1.0 (pow (/ (/ (* x-scale y-scale) a) b) -2.0)))))

C

b = abs(b);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * (1.0 / (1.0 / pow((((x_45_scale * y_45_scale) / a) / b), -2.0)));
}

Fortran

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) * (1.0d0 / (1.0d0 / ((((x_45scale * y_45scale) / a) / b) ** (-2.0d0))))
end function

Java

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 * (1.0 / (1.0 / Math.pow((((x_45_scale * y_45_scale) / a) / b), -2.0)));
}

Python

b = abs(b)
def code(a, b, angle, x_45_scale, y_45_scale):
	return -4.0 * (1.0 / (1.0 / math.pow((((x_45_scale * y_45_scale) / a) / b), -2.0)))

Julia

b = abs(b)
function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(-4.0 * Float64(1.0 / Float64(1.0 / (Float64(Float64(Float64(x_45_scale * y_45_scale) / a) / b) ^ -2.0))))
end

MATLAB

b = abs(b)
function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = -4.0 * (1.0 / (1.0 / ((((x_45_scale * y_45_scale) / a) / b) ^ -2.0)));
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(-4.0 * N[(1.0 / N[(1.0 / N[Power[N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / a), $MachinePrecision] / b), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $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%

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

   1. times-frac 49.9%

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

   2. unpow2 49.9%

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

   3. unpow2 49.9%

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

   4. unpow2 49.9%

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

   5. unpow2 49.9%

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

6. Simplified 49.9%

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

   1. pow2 49.9%

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

   2. pow2 49.9%

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

   3. frac-times 50.5%

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

   4. pow2 50.5%

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

   5. pow2 50.5%

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

   6. clear-num 50.5%

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

   7. pow-prod-down 61.0%

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

   8. pow-prod-down 78.3%

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

8. Applied egg-rr 78.3%

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

   1. clear-num 78.3%

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

   2. inv-pow 78.3%

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

   3. unpow2 78.3%

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

   4. unpow2 78.3%

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

   5. frac-times 91.0%

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

   6. frac-times 83.0%

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

   7. frac-times 91.5%

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

   8. pow2 91.5%

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

10. Applied egg-rr 91.5%

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

12. Simplified 91.7%

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

    1. *-commutative 91.7%

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

    2. times-frac 91.0%

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

    3. associate-/r* 93.5%

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

    4. *-commutative 93.5%

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

14. Applied egg-rr 93.5%

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

15. Final simplification 93.5%

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

Alternative 3: 94.0% accurate, 118.0× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := \frac{a}{y-scale} \cdot \frac{b}{x-scale}\\ t_1 := \frac{x-scale \cdot y-scale}{a \cdot b}\\ \mathbf{if}\;b \leq 10^{-172}:\\ \;\;\;\;-4 \cdot \frac{1}{t_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ a y-scale) (/ b x-scale)))
        (t_1 (/ (* x-scale y-scale) (* a b))))
   (if (<= b 1e-172) (* -4.0 (/ 1.0 (* t_1 t_1))) (* -4.0 (* t_0 t_0)))))

C

b = abs(b);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a / y_45_scale) * (b / x_45_scale);
	double t_1 = (x_45_scale * y_45_scale) / (a * b);
	double tmp;
	if (b <= 1e-172) {
		tmp = -4.0 * (1.0 / (t_1 * t_1));
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Fortran

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) :: t_1
    real(8) :: tmp
    t_0 = (a / y_45scale) * (b / x_45scale)
    t_1 = (x_45scale * y_45scale) / (a * b)
    if (b <= 1d-172) then
        tmp = (-4.0d0) * (1.0d0 / (t_1 * t_1))
    else
        tmp = (-4.0d0) * (t_0 * t_0)
    end if
    code = tmp
end function

Java

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 = (a / y_45_scale) * (b / x_45_scale);
	double t_1 = (x_45_scale * y_45_scale) / (a * b);
	double tmp;
	if (b <= 1e-172) {
		tmp = -4.0 * (1.0 / (t_1 * t_1));
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (a / y_45_scale) * (b / x_45_scale)
	t_1 = (x_45_scale * y_45_scale) / (a * b)
	tmp = 0
	if b <= 1e-172:
		tmp = -4.0 * (1.0 / (t_1 * t_1))
	else:
		tmp = -4.0 * (t_0 * t_0)
	return tmp

Julia

b = abs(b)
function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale))
	t_1 = Float64(Float64(x_45_scale * y_45_scale) / Float64(a * b))
	tmp = 0.0
	if (b <= 1e-172)
		tmp = Float64(-4.0 * Float64(1.0 / Float64(t_1 * t_1)));
	else
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (a / y_45_scale) * (b / x_45_scale);
	t_1 = (x_45_scale * y_45_scale) / (a * b);
	tmp = 0.0;
	if (b <= 1e-172)
		tmp = -4.0 * (1.0 / (t_1 * t_1));
	else
		tmp = -4.0 * (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1e-172], N[(-4.0 * N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1e-172

   1. Initial program 29.3%

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

   3. Simplified 27.2%

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

   4. Taylor expanded in angle around 0 51.4%

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

      1. times-frac 49.3%

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

      2. unpow2 49.3%

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

      3. unpow2 49.3%

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

      4. unpow2 49.3%

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

      5. unpow2 49.3%

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

   6. Simplified 49.3%

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

      1. pow2 49.3%

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

      2. pow2 49.3%

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

      3. frac-times 51.4%

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

      4. pow2 51.4%

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

      5. pow2 51.4%

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

      6. clear-num 51.4%

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

      7. pow-prod-down 59.7%

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

      8. pow-prod-down 81.0%

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

   8. Applied egg-rr 81.0%

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

      1. add-sqr-sqrt 80.9%

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

      2. sqrt-div 81.0%

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

      3. unpow2 81.0%

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

      4. sqrt-prod 42.3%

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

      5. add-sqr-sqrt 59.2%

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

      6. sqrt-pow1 59.7%

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

      7. metadata-eval 59.7%

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

      8. pow1 59.7%

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

      9. sqrt-div 59.7%

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

      10. unpow2 59.7%

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

      11. sqrt-prod 30.2%

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

      12. add-sqr-sqrt 58.2%

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

      13. sqrt-pow1 92.0%

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

      14. metadata-eval 92.0%

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

      15. pow1 92.0%

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

   10. Applied egg-rr 92.0%

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

   if 1e-172 < b

   1. Initial program 18.4%

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

   3. Simplified 15.3%

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

   4. Taylor expanded in angle around 0 48.9%

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

      1. pow2 48.9%

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

      2. pow2 48.9%

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

      3. frac-times 50.8%

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

      4. pow2 50.8%

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

      5. times-frac 65.7%

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

      6. pow2 65.7%

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

      7. times-frac 77.4%

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

   6. Applied egg-rr 77.4%

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

      1. unswap-sqr 95.3%

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

   8. Simplified 95.3%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-172}:\\ \;\;\;\;-4 \cdot \frac{1}{\frac{x-scale \cdot y-scale}{a \cdot b} \cdot \frac{x-scale \cdot y-scale}{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\\ \end{array} \]

Alternative 4: 93.8% accurate, 130.8× speedup

Math

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

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ x-scale b) (/ y-scale a)))) (* -4.0 (/ 1.0 (* t_0 t_0)))))

C

b = abs(b);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (x_45_scale / b) * (y_45_scale / a);
	return -4.0 * (1.0 / (t_0 * t_0));
}

Fortran

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
    t_0 = (x_45scale / b) * (y_45scale / a)
    code = (-4.0d0) * (1.0d0 / (t_0 * t_0))
end function

Java

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 = (x_45_scale / b) * (y_45_scale / a);
	return -4.0 * (1.0 / (t_0 * t_0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(x$45$scale / b), $MachinePrecision] * N[(y$45$scale / a), $MachinePrecision]), $MachinePrecision]}, N[(-4.0 * N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $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%

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

   1. times-frac 49.9%

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

   2. unpow2 49.9%

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

   3. unpow2 49.9%

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

   4. unpow2 49.9%

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

   5. unpow2 49.9%

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

6. Simplified 49.9%

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

   1. pow2 49.9%

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

   2. pow2 49.9%

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

   3. frac-times 50.5%

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

   4. pow2 50.5%

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

   5. pow2 50.5%

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

   6. clear-num 50.5%

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

   7. pow-prod-down 61.0%

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

   8. pow-prod-down 78.3%

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

8. Applied egg-rr 78.3%

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

   1. clear-num 78.3%

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

   2. inv-pow 78.3%

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

   3. unpow2 78.3%

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

   4. unpow2 78.3%

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

   5. frac-times 91.0%

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

   6. frac-times 83.0%

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

   7. frac-times 91.5%

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

   8. pow2 91.5%

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

10. Applied egg-rr 91.5%

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

12. Simplified 91.7%

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

    1. pow-flip 91.6%

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

    2. *-commutative 91.6%

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

    3. times-frac 91.0%

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

    4. metadata-eval 91.0%

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

    5. pow2 91.0%

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

    6. times-frac 83.2%

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

    7. *-commutative 83.2%

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

    8. times-frac 91.6%

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

    9. *-commutative 91.6%

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

14. Applied egg-rr 91.6%

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

15. Final simplification 91.6%

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

Alternative 5: 91.2% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Fortran

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) * ((b / x_45scale) * ((a / y_45scale) * ((a / y_45scale) * (b / x_45scale))))
end function

Java

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 * ((b / x_45_scale) * ((a / y_45_scale) * ((a / y_45_scale) * (b / x_45_scale))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. times-frac 49.9%

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

   2. unpow2 49.9%

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

   3. unpow2 49.9%

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

   4. unpow2 49.9%

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

   5. unpow2 49.9%

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

6. Simplified 49.9%

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

   1. pow2 49.9%

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

   2. associate-*l/ 50.8%

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

   3. pow2 50.8%

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

   4. times-frac 60.9%

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

8. Applied egg-rr 60.9%

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

   1. unswap-sqr 74.1%

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

   2. frac-times 91.2%

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

   3. associate-*l/ 89.8%

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

   4. associate-*l/ 91.5%

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

   5. associate-*r* 89.2%

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

10. Applied egg-rr 89.2%

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

11. Final simplification 89.2%

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

Alternative 6: 93.9% accurate, 146.2× speedup

Math

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

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ a y-scale) (/ b x-scale)))) (* -4.0 (* t_0 t_0))))

C

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

Fortran

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
    t_0 = (a / y_45scale) * (b / x_45scale)
    code = (-4.0d0) * (t_0 * t_0)
end function

Java

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 = (a / y_45_scale) * (b / x_45_scale);
	return -4.0 * (t_0 * t_0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $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%

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

   1. pow2 50.5%

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

   2. pow2 50.5%

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

   3. frac-times 49.9%

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

   4. pow2 49.9%

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

   5. times-frac 59.3%

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

   6. pow2 59.3%

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

   7. times-frac 74.8%

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

6. Applied egg-rr 74.8%

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

   1. unswap-sqr 91.5%

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

8. Simplified 91.5%

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

9. Final simplification 91.5%

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

Alternative 7: 35.3% accurate, 2485.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Julia

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

MATLAB

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

Wolfram

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 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. *-commutative 23.5%

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

   3. metadata-eval 23.5%

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

   4. 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))))