Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.4% → 95.4%

Time: 1.5min

Alternatives: 4

Speedup: 2485.0×

• 34.1% 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: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{x-scale} \cdot \frac{a}{y-scale}\\ t_1 := \frac{angle}{180} \cdot \pi\\ t_2 := \sin t_1\\ t_3 := \frac{x-scale \cdot y-scale}{b \cdot a}\\ t_4 := \cos t_1\\ t_5 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t_2\right) \cdot t_4}{x-scale}}{y-scale}\\ \mathbf{if}\;t_5 \cdot t_5 - \left(4 \cdot \frac{\frac{{\left(a \cdot t_2\right)}^{2} + {\left(b \cdot t_4\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t_4\right)}^{2} + {\left(b \cdot t_2\right)}^{2}}{y-scale}}{y-scale} \leq 2 \cdot 10^{-123}:\\ \;\;\;\;t_0 \cdot \left(-4 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4}{t_3}}{t_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ b x-scale) (/ a y-scale)))
        (t_1 (* (/ angle 180.0) PI))
        (t_2 (sin t_1))
        (t_3 (/ (* x-scale y-scale) (* b a)))
        (t_4 (cos t_1))
        (t_5
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_2) t_4) x-scale)
          y-scale)))
   (if (<=
        (-
         (* t_5 t_5)
         (*
          (*
           4.0
           (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_4) 2.0)) x-scale) x-scale))
          (/ (/ (+ (pow (* a t_4) 2.0) (pow (* b t_2) 2.0)) y-scale) y-scale)))
        2e-123)
     (* t_0 (* -4.0 t_0))
     (/ (/ -4.0 t_3) t_3))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b / x_45_scale) * (a / y_45_scale);
	double t_1 = (angle / 180.0) * ((double) M_PI);
	double t_2 = sin(t_1);
	double t_3 = (x_45_scale * y_45_scale) / (b * a);
	double t_4 = cos(t_1);
	double t_5 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale;
	double tmp;
	if (((t_5 * t_5) - ((4.0 * (((pow((a * t_2), 2.0) + pow((b * t_4), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_4), 2.0) + pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale))) <= 2e-123) {
		tmp = t_0 * (-4.0 * t_0);
	} else {
		tmp = (-4.0 / t_3) / t_3;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b / x_45_scale) * (a / y_45_scale);
	double t_1 = (angle / 180.0) * Math.PI;
	double t_2 = Math.sin(t_1);
	double t_3 = (x_45_scale * y_45_scale) / (b * a);
	double t_4 = Math.cos(t_1);
	double t_5 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale;
	double tmp;
	if (((t_5 * t_5) - ((4.0 * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_4), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_4), 2.0) + Math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale))) <= 2e-123) {
		tmp = t_0 * (-4.0 * t_0);
	} else {
		tmp = (-4.0 / t_3) / t_3;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b / x_45_scale) * (a / y_45_scale)
	t_1 = (angle / 180.0) * math.pi
	t_2 = math.sin(t_1)
	t_3 = (x_45_scale * y_45_scale) / (b * a)
	t_4 = math.cos(t_1)
	t_5 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale
	tmp = 0
	if ((t_5 * t_5) - ((4.0 * (((math.pow((a * t_2), 2.0) + math.pow((b * t_4), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_4), 2.0) + math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale))) <= 2e-123:
		tmp = t_0 * (-4.0 * t_0)
	else:
		tmp = (-4.0 / t_3) / t_3
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b / x_45_scale) * Float64(a / y_45_scale))
	t_1 = Float64(Float64(angle / 180.0) * pi)
	t_2 = sin(t_1)
	t_3 = Float64(Float64(x_45_scale * y_45_scale) / Float64(b * a))
	t_4 = cos(t_1)
	t_5 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale)
	tmp = 0.0
	if (Float64(Float64(t_5 * t_5) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_4) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_4) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale))) <= 2e-123)
		tmp = Float64(t_0 * Float64(-4.0 * t_0));
	else
		tmp = Float64(Float64(-4.0 / t_3) / t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b / x_45_scale) * (a / y_45_scale);
	t_1 = (angle / 180.0) * pi;
	t_2 = sin(t_1);
	t_3 = (x_45_scale * y_45_scale) / (b * a);
	t_4 = cos(t_1);
	t_5 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale;
	tmp = 0.0;
	if (((t_5 * t_5) - ((4.0 * (((((a * t_2) ^ 2.0) + ((b * t_4) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_4) ^ 2.0) + ((b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale))) <= 2e-123)
		tmp = t_0 * (-4.0 * t_0);
	else
		tmp = (-4.0 / t_3) / t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b / x$45$scale), $MachinePrecision] * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$4), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$5 * t$95$5), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$4), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-123], N[(t$95$0 * N[(-4.0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 / t$95$3), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 4 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale))) < 2.0000000000000001e-123

   1. Initial program 63.8%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in angle around 0 62.8%

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

      1. associate-*r/ 62.8%

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

      2. *-commutative 62.8%

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

      3. unpow2 62.8%

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

      4. unpow2 62.8%

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

      5. swap-sqr 76.3%

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

      6. unpow2 76.3%

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

   6. Simplified 76.3%

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

      1. div-inv 75.8%

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

      2. *-commutative 75.8%

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

      3. pow-prod-down 76.8%

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

      4. pow-flip 76.8%

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

      5. metadata-eval 76.8%

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

   8. Applied egg-rr 76.8%

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

      1. associate-*l* 76.8%

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

      2. add-exp-log 76.0%

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

      3. add-exp-log 75.9%

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

      4. prod-exp 75.8%

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

      5. log-pow 37.3%

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

      6. metadata-eval 37.3%

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

      7. cancel-sign-sub-inv 37.3%

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

      8. *-commutative 37.3%

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

      9. exp-diff 35.7%

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

      10. add-exp-log 35.7%

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

      11. pow-to-exp 77.3%

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

      12. clear-num 77.4%

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

      13. div-inv 77.4%

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

      14. add-sqr-sqrt 77.4%

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

   10. Applied egg-rr 94.3%

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

       1. div-inv 94.2%

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

       2. div-inv 94.2%

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

       3. clear-num 94.3%

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

       4. *-commutative 94.3%

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

       5. times-frac 94.3%

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

       6. clear-num 94.3%

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

       7. *-commutative 94.3%

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

       8. times-frac 99.7%

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

   12. Applied egg-rr 99.7%

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

   if 2.0000000000000001e-123 < (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 4 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in angle around 0 40.2%

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

      1. associate-*r/ 40.2%

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

      2. *-commutative 40.2%

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

      3. unpow2 40.2%

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

      4. unpow2 40.2%

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

      5. swap-sqr 50.7%

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

      6. unpow2 50.7%

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

   6. Simplified 50.7%

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

      1. div-inv 50.7%

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

      2. *-commutative 50.7%

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

      3. pow-prod-down 79.4%

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

      4. pow-flip 80.0%

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

      5. metadata-eval 80.0%

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

   8. Applied egg-rr 80.0%

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

      1. associate-*l* 80.0%

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

      2. add-exp-log 78.5%

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

      3. add-exp-log 78.1%

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

      4. prod-exp 78.1%

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

      5. log-pow 46.8%

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

      6. metadata-eval 46.8%

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

      7. cancel-sign-sub-inv 46.8%

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

      8. *-commutative 46.8%

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

      9. exp-diff 44.2%

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

      10. add-exp-log 44.3%

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

      11. pow-to-exp 79.4%

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

      12. clear-num 79.5%

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

      13. div-inv 79.5%

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

      14. add-sqr-sqrt 79.4%

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

   10. Applied egg-rr 97.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4}{\frac{x-scale \cdot y-scale}{b \cdot a}}}{\frac{x-scale \cdot y-scale}{b \cdot a}}\\ \end{array} \]

Alternative 2: 88.4% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.9%

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

3. Simplified 18.0%

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

4. Taylor expanded in angle around 0 48.0%

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

   1. add-exp-log 47.7%

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

   2. pow-prod-down 59.0%

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

   3. pow-to-exp 32.0%

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

   4. div-exp 33.7%

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

   5. pow-prod-down 43.5%

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

6. Applied egg-rr 43.5%

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

   1. exp-diff 41.2%

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

   2. add-exp-log 41.3%

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

   3. pow-to-exp 78.7%

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

   4. unpow2 78.7%

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

   5. unpow2 78.7%

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

   6. times-frac 96.4%

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

8. Applied egg-rr 96.4%

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

   1. *-commutative 96.4%

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

   2. *-commutative 96.4%

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

   3. frac-times 90.2%

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

10. Applied egg-rr 90.2%

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

11. Final simplification 90.2%

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

Alternative 3: 93.9% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.9%

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

3. Simplified 18.0%

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

4. Taylor expanded in angle around 0 48.0%

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

   1. add-exp-log 47.7%

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

   2. pow-prod-down 59.0%

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

   3. pow-to-exp 32.0%

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

   4. div-exp 33.7%

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

   5. pow-prod-down 43.5%

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

6. Applied egg-rr 43.5%

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

   1. exp-diff 41.2%

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

   2. add-exp-log 41.3%

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

   3. pow-to-exp 78.7%

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

   4. unpow2 78.7%

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

   5. unpow2 78.7%

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

   6. times-frac 96.4%

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

8. Applied egg-rr 96.4%

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

9. Final simplification 96.4%

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

Alternative 4: 36.1% accurate, 2485.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.9%

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

3. Simplified 18.1%

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

4. Taylor expanded in b around 0 20.2%

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

   1. distribute-rgt-out 20.2%

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

   2. metadata-eval 20.2%

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

   3. mul0-rgt 32.0%

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

6. Simplified 32.0%

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

7. Final simplification 32.0%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023338 
(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))))