Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.4% → 93.8%

Time: 1.3min

Alternatives: 4

Speedup: 2485.0×

• 34.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t_0\\ t_2 := \cos t_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t_1\right) \cdot t_2}{x-scale}}{y-scale}\\ t_3 \cdot t_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t_1\right)}^{2} + {\left(b \cdot t_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t_2\right)}^{2} + {\left(b \cdot t_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Initial Program: 25.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t_0\\ t_2 := \cos t_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t_1\right) \cdot t_2}{x-scale}}{y-scale}\\ t_3 \cdot t_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t_1\right)}^{2} + {\left(b \cdot t_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t_2\right)}^{2} + {\left(b \cdot t_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Alternative 1: 93.8% accurate, 22.2× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := \frac{a}{y-scale} \cdot \frac{b}{x-scale}\\ \mathbf{if}\;b \leq 1.76 \cdot 10^{+34}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot a}{y-scale \cdot x-scale}\right)}^{2}\\ \end{array} \end{array} \]

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))))
   (if (<= b 1.76e+34)
     (* -4.0 (* t_0 t_0))
     (* -4.0 (pow (/ (* b a) (* y-scale x-scale)) 2.0)))))

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 tmp;
	if (b <= 1.76e+34) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * pow(((b * a) / (y_45_scale * x_45_scale)), 2.0);
	}
	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) :: tmp
    t_0 = (a / y_45scale) * (b / x_45scale)
    if (b <= 1.76d+34) then
        tmp = (-4.0d0) * (t_0 * t_0)
    else
        tmp = (-4.0d0) * (((b * a) / (y_45scale * x_45scale)) ** 2.0d0)
    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 tmp;
	if (b <= 1.76e+34) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * Math.pow(((b * a) / (y_45_scale * x_45_scale)), 2.0);
	}
	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)
	tmp = 0
	if b <= 1.76e+34:
		tmp = -4.0 * (t_0 * t_0)
	else:
		tmp = -4.0 * math.pow(((b * a) / (y_45_scale * x_45_scale)), 2.0)
	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))
	tmp = 0.0
	if (b <= 1.76e+34)
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	else
		tmp = Float64(-4.0 * (Float64(Float64(b * a) / Float64(y_45_scale * x_45_scale)) ^ 2.0));
	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);
	tmp = 0.0;
	if (b <= 1.76e+34)
		tmp = -4.0 * (t_0 * t_0);
	else
		tmp = -4.0 * (((b * a) / (y_45_scale * x_45_scale)) ^ 2.0);
	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]}, If[LessEqual[b, 1.76e+34], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(N[(b * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.75999999999999995e34

   1. Initial program 30.8%

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

   2. Taylor expanded in angle around 0 46.6%

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

      1. times-frac 46.6%

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

      2. unpow2 46.6%

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

      3. unpow2 46.6%

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

      4. unpow2 46.6%

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

      5. unpow2 46.6%

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

   4. Simplified 46.6%

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

      1. associate-*r/ 46.2%

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

      2. times-frac 55.3%

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

      3. pow2 55.3%

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

   6. Applied egg-rr 55.3%

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

      1. add-sqr-sqrt 55.3%

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

      2. pow2 55.3%

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

      3. associate-/l* 54.4%

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

      4. frac-times 54.4%

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

      5. un-div-inv 54.2%

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

      6. sqrt-prod 54.2%

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

      7. sqrt-prod 31.7%

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

      8. add-sqr-sqrt 44.4%

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

      9. frac-times 44.4%

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

      10. clear-num 44.4%

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

      11. times-frac 44.4%

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

      12. sqrt-prod 29.2%

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

      13. add-sqr-sqrt 46.2%

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

   8. Applied egg-rr 92.4%

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

   if 1.75999999999999995e34 < b

   1. Initial program 5.8%

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

   2. Taylor expanded in angle around 0 48.0%

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

      1. div-inv 48.0%

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

      2. pow-prod-down 63.8%

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

      3. pow-prod-down 79.3%

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

      4. pow-flip 79.3%

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

      5. metadata-eval 79.3%

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

   4. Applied egg-rr 79.3%

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

   5. Taylor expanded in a around 0 48.0%

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

      1. unpow2 48.0%

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

      2. unpow2 48.0%

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

      3. unpow2 48.0%

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

      4. unpow2 48.0%

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

      5. times-frac 49.5%

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

      6. associate-/r* 51.6%

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

      7. times-frac 61.0%

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

      8. associate-*r/ 69.2%

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

      9. associate-*l/ 69.3%

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

      10. swap-sqr 86.9%

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

      11. unpow2 86.9%

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

      12. associate-*l/ 90.6%

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

      13. associate-*r/ 96.0%

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

      14. associate-/r* 94.2%

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

      15. *-rgt-identity 94.2%

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

      16. *-commutative 94.2%

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

      17. associate-*r/ 94.0%

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

      18. associate-*l* 92.3%

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

      19. associate-*r/ 92.3%

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

      20. *-rgt-identity 92.3%

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

   7. Simplified 92.3%

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

   8. Taylor expanded in a around -inf 94.2%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.76 \cdot 10^{+34}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot a}{y-scale \cdot x-scale}\right)}^{2}\\ \end{array} \]

Alternative 2: 94.1% accurate, 130.4× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := a \cdot \frac{b}{y-scale \cdot x-scale}\\ t_1 := \frac{a}{y-scale} \cdot \frac{b}{x-scale}\\ \mathbf{if}\;b \leq 7 \cdot 10^{-34}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (/ b (* y-scale x-scale))))
        (t_1 (* (/ a y-scale) (/ b x-scale))))
   (if (<= b 7e-34) (* -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 * (b / (y_45_scale * x_45_scale));
	double t_1 = (a / y_45_scale) * (b / x_45_scale);
	double tmp;
	if (b <= 7e-34) {
		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 * (b / (y_45scale * x_45scale))
    t_1 = (a / y_45scale) * (b / x_45scale)
    if (b <= 7d-34) 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 * (b / (y_45_scale * x_45_scale));
	double t_1 = (a / y_45_scale) * (b / x_45_scale);
	double tmp;
	if (b <= 7e-34) {
		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 * (b / (y_45_scale * x_45_scale))
	t_1 = (a / y_45_scale) * (b / x_45_scale)
	tmp = 0
	if b <= 7e-34:
		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(a * Float64(b / Float64(y_45_scale * x_45_scale)))
	t_1 = Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale))
	tmp = 0.0
	if (b <= 7e-34)
		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 * (b / (y_45_scale * x_45_scale));
	t_1 = (a / y_45_scale) * (b / x_45_scale);
	tmp = 0.0;
	if (b <= 7e-34)
		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[(a * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7e-34], 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 < 7e-34

   1. Initial program 31.6%

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

   2. Taylor expanded in angle around 0 46.5%

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

      1. times-frac 46.5%

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

      2. unpow2 46.5%

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

      3. unpow2 46.5%

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

      4. unpow2 46.5%

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

      5. unpow2 46.5%

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

   4. Simplified 46.5%

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

      1. associate-*r/ 46.1%

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

      2. times-frac 52.8%

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

      3. pow2 52.8%

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

   6. Applied egg-rr 52.8%

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

      1. add-sqr-sqrt 52.8%

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

      2. pow2 52.8%

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

      3. associate-/l* 51.9%

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

      4. frac-times 51.9%

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

      5. un-div-inv 51.7%

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

      6. sqrt-prod 51.7%

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

      7. sqrt-prod 30.3%

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

      8. add-sqr-sqrt 42.5%

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

      9. frac-times 42.5%

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

      10. clear-num 42.5%

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

      11. times-frac 42.5%

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

      12. sqrt-prod 28.6%

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

      13. add-sqr-sqrt 45.6%

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

   8. Applied egg-rr 91.8%

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

   if 7e-34 < b

   1. Initial program 9.6%

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

   2. Taylor expanded in angle around 0 47.8%

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

      1. div-inv 47.8%

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

      2. pow-prod-down 59.8%

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

      3. pow-prod-down 76.0%

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

      4. pow-flip 76.9%

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

      5. metadata-eval 76.9%

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

   4. Applied egg-rr 76.9%

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

   5. Taylor expanded in a around 0 47.8%

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

      1. unpow2 47.8%

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

      2. unpow2 47.8%

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

      3. unpow2 47.8%

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

      4. unpow2 47.8%

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

      5. times-frac 49.0%

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

      6. associate-/r* 50.6%

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

      7. times-frac 66.1%

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

      8. associate-*r/ 72.4%

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

      9. associate-*l/ 72.5%

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

      10. swap-sqr 89.9%

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

      11. unpow2 89.9%

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

      12. associate-*l/ 91.5%

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

      13. associate-*r/ 94.2%

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

      14. associate-/r* 94.1%

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

      15. *-rgt-identity 94.1%

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

      16. *-commutative 94.1%

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

      17. associate-*r/ 94.0%

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

      18. associate-*l* 94.0%

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

      19. associate-*r/ 94.1%

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

      20. *-rgt-identity 94.1%

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

   7. Simplified 94.1%

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

      1. unpow2 94.1%

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

   9. Applied egg-rr 94.1%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{-34}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)\right)\\ \end{array} \]

Alternative 3: 94.2% accurate, 146.2× speedup

Math

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

FPCore

NOTE: b should be positive before calling this function
(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

b = abs(b);
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

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 * (b / (y_45scale * 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 * (b / (y_45_scale * 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 * (b / (y_45_scale * 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(a * Float64(b / Float64(y_45_scale * 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 * (b / (y_45_scale * 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[(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 25.6%

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

2. Taylor expanded in angle around 0 46.9%

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

   1. div-inv 46.8%

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

   2. pow-prod-down 59.0%

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

   3. pow-prod-down 76.6%

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

   4. pow-flip 77.3%

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

   5. metadata-eval 77.3%

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

4. Applied egg-rr 77.3%

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

5. Taylor expanded in a around 0 46.9%

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

   1. unpow2 46.9%

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

   2. unpow2 46.9%

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

   3. unpow2 46.9%

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

   4. unpow2 46.9%

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

   5. times-frac 47.2%

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

   6. associate-/r* 51.2%

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

   7. times-frac 61.9%

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

   8. associate-*r/ 68.8%

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

   9. associate-*l/ 71.2%

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

   10. swap-sqr 91.3%

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

   11. unpow2 91.3%

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

   12. associate-*l/ 90.5%

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

   13. associate-*r/ 92.0%

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

   14. associate-/r* 92.2%

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

   15. *-rgt-identity 92.2%

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

   16. *-commutative 92.2%

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

   17. associate-*r/ 92.2%

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

   18. associate-*l* 92.5%

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

   19. associate-*r/ 92.4%

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

   20. *-rgt-identity 92.4%

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

7. Simplified 92.4%

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

   1. unpow2 92.4%

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

9. Applied egg-rr 92.4%

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

10. Final simplification 92.4%

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

Alternative 4: 35.1% 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 25.6%

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

   1. fma-neg 25.9%

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

3. Simplified 22.5%

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

4. Taylor expanded in b around 0 25.2%

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

   1. *-commutative 25.2%

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

   2. *-commutative 25.2%

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

   3. *-commutative 25.2%

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

   4. distribute-lft-out 25.2%

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

6. Simplified 36.6%

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

7. Final simplification 36.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023214 
(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))))