Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.0% → 92.0%

Time: 1.6min

Alternatives: 4

Speedup: 2485.0×

• 34.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.0% 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: 92.0% accurate, 118.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{x-scale \cdot y-scale}\\ t_1 := \frac{y-scale}{b} \cdot \frac{x-scale}{a}\\ \mathbf{if}\;y-scale \leq 7.6 \cdot 10^{-272}:\\ \;\;\;\;-4 \cdot \frac{1}{t_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(b \cdot \left(t_0 \cdot \left(b \cdot t_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* x-scale y-scale))) (t_1 (* (/ y-scale b) (/ x-scale a))))
   (if (<= y-scale 7.6e-272)
     (* -4.0 (/ 1.0 (* t_1 t_1)))
     (* -4.0 (* b (* t_0 (* b t_0)))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (x_45_scale * y_45_scale);
	double t_1 = (y_45_scale / b) * (x_45_scale / a);
	double tmp;
	if (y_45_scale <= 7.6e-272) {
		tmp = -4.0 * (1.0 / (t_1 * t_1));
	} else {
		tmp = -4.0 * (b * (t_0 * (b * t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a / (x_45scale * y_45scale)
    t_1 = (y_45scale / b) * (x_45scale / a)
    if (y_45scale <= 7.6d-272) then
        tmp = (-4.0d0) * (1.0d0 / (t_1 * t_1))
    else
        tmp = (-4.0d0) * (b * (t_0 * (b * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (x_45_scale * y_45_scale);
	double t_1 = (y_45_scale / b) * (x_45_scale / a);
	double tmp;
	if (y_45_scale <= 7.6e-272) {
		tmp = -4.0 * (1.0 / (t_1 * t_1));
	} else {
		tmp = -4.0 * (b * (t_0 * (b * t_0)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a / (x_45_scale * y_45_scale)
	t_1 = (y_45_scale / b) * (x_45_scale / a)
	tmp = 0
	if y_45_scale <= 7.6e-272:
		tmp = -4.0 * (1.0 / (t_1 * t_1))
	else:
		tmp = -4.0 * (b * (t_0 * (b * t_0)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(x_45_scale * y_45_scale))
	t_1 = Float64(Float64(y_45_scale / b) * Float64(x_45_scale / a))
	tmp = 0.0
	if (y_45_scale <= 7.6e-272)
		tmp = Float64(-4.0 * Float64(1.0 / Float64(t_1 * t_1)));
	else
		tmp = Float64(-4.0 * Float64(b * Float64(t_0 * Float64(b * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a / (x_45_scale * y_45_scale);
	t_1 = (y_45_scale / b) * (x_45_scale / a);
	tmp = 0.0;
	if (y_45_scale <= 7.6e-272)
		tmp = -4.0 * (1.0 / (t_1 * t_1));
	else
		tmp = -4.0 * (b * (t_0 * (b * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$45$scale / b), $MachinePrecision] * N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, 7.6e-272], N[(-4.0 * N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(b * N[(t$95$0 * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.5999999999999994e-272

   1. Initial program 27.1%

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

   3. Simplified 22.1%

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

   4. Taylor expanded in angle around 0 48.9%

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

      1. times-frac 48.2%

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

   6. Simplified 48.2%

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

      1. add-sqr-sqrt 48.2%

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

      2. pow2 48.2%

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

      3. frac-times 48.9%

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

      4. *-commutative 48.9%

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

      5. pow-prod-down 64.7%

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

      6. pow-prod-down 80.9%

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

   8. Applied egg-rr 80.9%

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

   9. Taylor expanded in b around 0 93.7%

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

       1. *-commutative 93.7%

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

       2. associate-/l* 94.6%

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

   11. Simplified 94.6%

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

       1. unpow2 94.6%

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

       2. clear-num 94.6%

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

       3. clear-num 94.5%

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

       4. frac-times 94.6%

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

       5. metadata-eval 94.6%

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

       6. associate-/r* 93.0%

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

       7. associate-/r* 93.6%

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

       8. *-commutative 93.6%

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

       9. *-commutative 93.6%

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

       10. times-frac 90.6%

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

       11. *-commutative 90.6%

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

       12. *-commutative 90.6%

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

       13. times-frac 96.5%

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

   13. Applied egg-rr 96.5%

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

   if 7.5999999999999994e-272 < y-scale

   1. Initial program 28.3%

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

   3. Simplified 24.7%

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

   4. Taylor expanded in angle around 0 49.2%

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

      1. times-frac 49.3%

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

   6. Simplified 49.3%

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

      1. add-sqr-sqrt 49.3%

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

      2. pow2 49.3%

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

      3. frac-times 49.2%

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

      4. *-commutative 49.2%

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

      5. pow-prod-down 60.3%

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

      6. pow-prod-down 78.1%

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

   8. Applied egg-rr 78.1%

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

   9. Taylor expanded in b around 0 96.2%

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

       1. *-commutative 96.2%

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

       2. associate-/l* 95.4%

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

   11. Simplified 95.4%

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

       1. unpow2 95.4%

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

       2. div-inv 95.4%

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

       3. clear-num 95.4%

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

       4. associate-*l* 95.4%

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

       5. div-inv 95.4%

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

       6. clear-num 95.4%

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

   13. Applied egg-rr 95.4%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 7.6 \cdot 10^{-272}:\\ \;\;\;\;-4 \cdot \frac{1}{\left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right) \cdot \left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(b \cdot \left(\frac{a}{x-scale \cdot y-scale} \cdot \left(b \cdot \frac{a}{x-scale \cdot y-scale}\right)\right)\right)\\ \end{array} \]

Alternative 2: 93.5% accurate, 22.6× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* -4.0 (pow (/ b (/ (* x-scale y-scale) a)) 2.0)))

C

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

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 = (-4.0d0) * ((b / ((x_45scale * y_45scale) / a)) ** 2.0d0)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * Math.pow((b / ((x_45_scale * y_45_scale) / a)), 2.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(-4.0 * N[Power[N[(b / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.6%

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

3. Simplified 23.2%

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

4. Taylor expanded in angle around 0 49.1%

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

   1. times-frac 48.6%

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

6. Simplified 48.6%

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

   1. add-sqr-sqrt 48.6%

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

   2. pow2 48.6%

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

   3. frac-times 49.0%

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

   4. *-commutative 49.0%

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

   5. pow-prod-down 62.8%

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

   6. pow-prod-down 79.7%

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

8. Applied egg-rr 79.7%

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

9. Taylor expanded in b around 0 94.8%

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

    1. *-commutative 94.8%

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

    2. associate-/l* 94.9%

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

11. Simplified 94.9%

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

12. Final simplification 94.9%

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

Alternative 3: 90.6% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.6%

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

3. Simplified 23.2%

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

4. Taylor expanded in angle around 0 49.1%

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

   1. times-frac 48.6%

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

6. Simplified 48.6%

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

   1. add-sqr-sqrt 48.6%

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

   2. pow2 48.6%

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

   3. frac-times 49.0%

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

   4. *-commutative 49.0%

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

   5. pow-prod-down 62.8%

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

   6. pow-prod-down 79.7%

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

8. Applied egg-rr 79.7%

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

9. Taylor expanded in b around 0 94.8%

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

    1. *-commutative 94.8%

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

    2. associate-/l* 94.9%

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

11. Simplified 94.9%

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

    1. unpow2 94.9%

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

    2. div-inv 94.9%

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

    3. clear-num 94.9%

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

    4. associate-*l* 93.9%

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

    5. div-inv 93.9%

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

    6. clear-num 93.8%

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

13. Applied egg-rr 93.8%

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

14. Final simplification 93.8%

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

Alternative 4: 34.0% 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 27.6%

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

3. Simplified 24.1%

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

4. Taylor expanded in b around 0 29.1%

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

   1. distribute-rgt-out 29.1%

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

   2. metadata-eval 29.1%

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

   3. mul0-rgt 40.4%

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

6. Simplified 40.4%

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

7. Final simplification 40.4%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023333 
(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))))