Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.5% → 95.0%

Time: 32.5s

Alternatives: 3

Speedup: 1693.0×

• 33.9% 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: 25.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \frac{b \cdot a}{x-scale \cdot y-scale}\\ t_3 := \cos t\_0\\ t_4 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_3}{x-scale}}{y-scale}\\ \mathbf{if}\;t\_4 \cdot t\_4 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_3\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_3\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \leq 0:\\ \;\;\;\;-4 \cdot {\left(b \cdot \frac{\frac{a}{x-scale}}{y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(-4 \cdot t\_2\right)\\ \end{array} \end{array} \]

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 (/ (* b a) (* x-scale y-scale)))
        (t_3 (cos t_0))
        (t_4
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_3) x-scale)
          y-scale)))
   (if (<=
        (-
         (* t_4 t_4)
         (*
          (*
           4.0
           (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_3) 2.0)) x-scale) x-scale))
          (/ (/ (+ (pow (* a t_3) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))
        0.0)
     (* -4.0 (pow (* b (/ (/ a x-scale) y-scale)) 2.0))
     (* t_2 (* -4.0 t_2)))))

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 = (b * a) / (x_45_scale * y_45_scale);
	double t_3 = cos(t_0);
	double t_4 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_3) / x_45_scale) / y_45_scale;
	double tmp;
	if (((t_4 * t_4) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_3), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_3), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))) <= 0.0) {
		tmp = -4.0 * pow((b * ((a / x_45_scale) / y_45_scale)), 2.0);
	} else {
		tmp = t_2 * (-4.0 * t_2);
	}
	return tmp;
}

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 = (b * a) / (x_45_scale * y_45_scale);
	double t_3 = Math.cos(t_0);
	double t_4 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_3) / x_45_scale) / y_45_scale;
	double tmp;
	if (((t_4 * t_4) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_3), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_3), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))) <= 0.0) {
		tmp = -4.0 * Math.pow((b * ((a / x_45_scale) / y_45_scale)), 2.0);
	} else {
		tmp = t_2 * (-4.0 * t_2);
	}
	return tmp;
}

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 = (b * a) / (x_45_scale * y_45_scale)
	t_3 = math.cos(t_0)
	t_4 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_3) / x_45_scale) / y_45_scale
	tmp = 0
	if ((t_4 * t_4) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_3), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_3), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))) <= 0.0:
		tmp = -4.0 * math.pow((b * ((a / x_45_scale) / y_45_scale)), 2.0)
	else:
		tmp = t_2 * (-4.0 * t_2)
	return tmp

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 = Float64(Float64(b * a) / Float64(x_45_scale * y_45_scale))
	t_3 = cos(t_0)
	t_4 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_3) / x_45_scale) / y_45_scale)
	tmp = 0.0
	if (Float64(Float64(t_4 * t_4) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_3) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_3) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale))) <= 0.0)
		tmp = Float64(-4.0 * (Float64(b * Float64(Float64(a / x_45_scale) / y_45_scale)) ^ 2.0));
	else
		tmp = Float64(t_2 * Float64(-4.0 * t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = (b * a) / (x_45_scale * y_45_scale);
	t_3 = cos(t_0);
	t_4 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_3) / x_45_scale) / y_45_scale;
	tmp = 0.0;
	if (((t_4 * t_4) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_3) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_3) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale))) <= 0.0)
		tmp = -4.0 * ((b * ((a / x_45_scale) / y_45_scale)) ^ 2.0);
	else
		tmp = t_2 * (-4.0 * t_2);
	end
	tmp_2 = tmp;
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[(N[(b * a), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$4 = 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$3), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$4 * t$95$4), $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$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$3), $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], 0.0], N[(-4.0 * N[Power[N[(b * N[(N[(a / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(-4.0 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 #s(literal 4 binary64) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale))) < 0.0

   1. Initial program 76.6%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in angle around 0 74.5%

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

      1. times-frac 75.8%

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

   7. Simplified 75.8%

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

      1. frac-times 74.5%

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

      2. unpow-prod-down 75.7%

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

      3. *-commutative 75.7%

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

      4. unpow-prod-down 86.6%

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

      5. pow2 86.6%

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

      6. pow-to-exp 67.4%

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

      7. add-exp-log 67.3%

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

      8. div-exp 71.0%

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

      9. pow2 71.0%

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

      10. log-pow 32.1%

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

   9. Applied egg-rr 32.1%

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

       1. *-un-lft-identity 32.1%

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

       2. add-sqr-sqrt 32.1%

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

       3. pow2 32.1%

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

       4. exp-diff 29.6%

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

       5. pow-to-exp 38.8%

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

       6. pow2 38.8%

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

       7. *-commutative 38.8%

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

       8. pow-to-exp 86.6%

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

       9. pow2 86.6%

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

       10. sqrt-div 86.6%

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

       11. sqrt-prod 71.5%

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

       12. add-sqr-sqrt 90.5%

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

       13. sqrt-prod 41.8%

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

       14. add-sqr-sqrt 93.2%

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

   11. Applied egg-rr 93.2%

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

       1. *-lft-identity 93.2%

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

       2. associate-/l* 95.9%

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

       3. associate-/r* 99.8%

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

   13. Simplified 99.8%

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

   if 0.0 < (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 #s(literal 4 binary64) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)))

   1. Initial program 0.7%

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

   3. Simplified 1.2%

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

   5. Taylor expanded in angle around 0 36.0%

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

      1. associate-*r/ 36.0%

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

      2. *-commutative 36.0%

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

      3. unpow2 36.0%

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

      4. unpow2 36.0%

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

      5. swap-sqr 55.6%

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

      6. unpow2 55.6%

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

      7. unpow2 55.6%

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

      8. unpow2 55.6%

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

      9. swap-sqr 76.3%

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

      10. unpow2 76.3%

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

   7. Simplified 76.3%

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

      1. unpow2 76.3%

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

   9. Applied egg-rr 76.3%

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

       1. unpow2 76.3%

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

   11. Applied egg-rr 76.3%

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

       1. associate-/l* 76.3%

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

       2. pow2 76.3%

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

       3. pow-to-exp 37.2%

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

       4. pow2 37.2%

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

       5. pow-to-exp 19.7%

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

       6. *-commutative 19.7%

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

       7. exp-diff 25.4%

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

       8. *-commutative 25.4%

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

       9. add-sqr-sqrt 25.4%

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

       10. associate-*l* 25.4%

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

   13. Applied egg-rr 94.3%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \leq 0:\\ \;\;\;\;-4 \cdot {\left(b \cdot \frac{\frac{a}{x-scale}}{y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot a}{x-scale \cdot y-scale} \cdot \left(-4 \cdot \frac{b \cdot a}{x-scale \cdot y-scale}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.9% accurate, 99.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot a}{x-scale \cdot y-scale}\\ t\_0 \cdot \left(-4 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* b a) (* x-scale y-scale)))) (* t_0 (* -4.0 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 t_0 * (-4.0 * 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 = t_0 * ((-4.0d0) * 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 t_0 * (-4.0 * 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 t_0 * (-4.0 * 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(t_0 * Float64(-4.0 * 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 = t_0 * (-4.0 * 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[(t$95$0 * N[(-4.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 22.7%

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

3. Simplified 20.2%

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

5. Taylor expanded in angle around 0 47.2%

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

   1. associate-*r/ 47.2%

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

   2. *-commutative 47.2%

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

   3. unpow2 47.2%

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

   4. unpow2 47.2%

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

   5. swap-sqr 61.4%

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

   6. unpow2 61.4%

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

   7. unpow2 61.4%

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

   8. unpow2 61.4%

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

   9. swap-sqr 79.3%

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

   10. unpow2 79.3%

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

7. Simplified 79.3%

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

   1. unpow2 79.3%

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

9. Applied egg-rr 79.3%

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

    1. unpow2 79.3%

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

11. Applied egg-rr 79.3%

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

    1. associate-/l* 79.3%

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

    2. pow2 79.3%

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

    3. pow-to-exp 45.9%

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

    4. pow2 45.9%

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

    5. pow-to-exp 22.6%

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

    6. *-commutative 22.6%

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

    7. exp-diff 27.4%

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

    8. *-commutative 27.4%

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

    9. add-sqr-sqrt 27.3%

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

    10. associate-*l* 27.3%

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

13. Applied egg-rr 94.0%

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

14. Final simplification 94.0%

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

Alternative 3: 35.3% accurate, 1693.0× 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 22.7%

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

3. Simplified 20.6%

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

5. Taylor expanded in b around 0 20.7%

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

   1. distribute-rgt-out 20.7%

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

   2. metadata-eval 20.7%

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

   3. mul0-rgt 33.5%

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

7. Simplified 33.5%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024123 
(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))))