Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.9% → 94.0%

Time: 1.4min

Alternatives: 5

Speedup: 2485.0×

• 32.8% 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: 25.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.0% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ \mathbf{if}\;y-scale \leq -5.5 \cdot 10^{-229}:\\ \;\;\;\;{\left(\frac{a \cdot b}{y-scale \cdot x-scale}\right)}^{2} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ a x-scale) (/ b y-scale))))
   (if (<= y-scale -5.5e-229)
     (* (pow (/ (* a b) (* y-scale x-scale)) 2.0) -4.0)
     (* -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 = (a / x_45_scale) * (b / y_45_scale);
	double tmp;
	if (y_45_scale <= -5.5e-229) {
		tmp = pow(((a * b) / (y_45_scale * x_45_scale)), 2.0) * -4.0;
	} else {
		tmp = -4.0 * (t_0 * 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) :: tmp
    t_0 = (a / x_45scale) * (b / y_45scale)
    if (y_45scale <= (-5.5d-229)) then
        tmp = (((a * b) / (y_45scale * x_45scale)) ** 2.0d0) * (-4.0d0)
    else
        tmp = (-4.0d0) * (t_0 * 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) * (b / y_45_scale);
	double tmp;
	if (y_45_scale <= -5.5e-229) {
		tmp = Math.pow(((a * b) / (y_45_scale * x_45_scale)), 2.0) * -4.0;
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (a / x_45_scale) * (b / y_45_scale)
	tmp = 0
	if y_45_scale <= -5.5e-229:
		tmp = math.pow(((a * b) / (y_45_scale * x_45_scale)), 2.0) * -4.0
	else:
		tmp = -4.0 * (t_0 * t_0)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a / x_45_scale) * Float64(b / y_45_scale))
	tmp = 0.0
	if (y_45_scale <= -5.5e-229)
		tmp = Float64((Float64(Float64(a * b) / Float64(y_45_scale * x_45_scale)) ^ 2.0) * -4.0);
	else
		tmp = Float64(-4.0 * Float64(t_0 * 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) * (b / y_45_scale);
	tmp = 0.0;
	if (y_45_scale <= -5.5e-229)
		tmp = (((a * b) / (y_45_scale * x_45_scale)) ^ 2.0) * -4.0;
	else
		tmp = -4.0 * (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a / x$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, -5.5e-229], N[(N[Power[N[(N[(a * b), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -4.0), $MachinePrecision], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < -5.5000000000000001e-229

   1. Initial program 27.9%

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

   3. Simplified 23.2%

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

   4. Taylor expanded in angle around 0 54.4%

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

      1. associate-*r/ 54.4%

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

      2. pow-prod-down 65.9%

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

      3. pow-prod-down 77.0%

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

   6. Applied egg-rr 77.0%

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

      1. *-un-lft-identity 77.0%

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

      2. times-frac 77.0%

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

      3. metadata-eval 77.0%

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

      4. pow-prod-down 61.3%

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

      5. pow2 61.3%

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

      6. pow2 61.3%

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

      7. *-commutative 61.3%

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

      8. unpow-prod-down 54.4%

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

      9. pow2 54.4%

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

      10. pow2 54.4%

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

      11. frac-times 53.9%

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

      12. frac-times 68.3%

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

      13. *-commutative 68.3%

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

   8. Applied egg-rr 92.1%

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

   9. Taylor expanded in a around 0 97.1%

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

   if -5.5000000000000001e-229 < y-scale

   1. Initial program 22.1%

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

   3. Simplified 20.4%

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

   4. Taylor expanded in angle around 0 48.2%

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

      1. associate-*r/ 48.2%

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

      2. pow-prod-down 62.8%

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

      3. pow-prod-down 74.0%

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

   6. Applied egg-rr 74.0%

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

      1. *-un-lft-identity 74.0%

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

      2. times-frac 74.0%

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

      3. metadata-eval 74.0%

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

      4. pow-prod-down 59.1%

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

      5. pow2 59.1%

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

      6. pow2 59.1%

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

      7. *-commutative 59.1%

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

      8. unpow-prod-down 48.2%

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

      9. pow2 48.2%

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

      10. pow2 48.2%

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

      11. frac-times 47.4%

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

      12. frac-times 58.9%

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

      13. *-commutative 58.9%

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

   8. Applied egg-rr 96.0%

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

      1. unpow2 96.0%

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

   10. Applied egg-rr 96.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -5.5 \cdot 10^{-229}:\\ \;\;\;\;{\left(\frac{a \cdot b}{y-scale \cdot x-scale}\right)}^{2} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \]

Alternative 2: 93.9% accurate, 130.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ t_1 := \frac{y-scale \cdot x-scale}{a \cdot b}\\ \mathbf{if}\;y-scale \leq -1.3 \cdot 10^{-228}:\\ \;\;\;\;\frac{-4}{t_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ a x-scale) (/ b y-scale)))
        (t_1 (/ (* y-scale x-scale) (* a b))))
   (if (<= y-scale -1.3e-228) (/ -4.0 (* t_1 t_1)) (* -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 = (a / x_45_scale) * (b / y_45_scale);
	double t_1 = (y_45_scale * x_45_scale) / (a * b);
	double tmp;
	if (y_45_scale <= -1.3e-228) {
		tmp = -4.0 / (t_1 * t_1);
	} else {
		tmp = -4.0 * (t_0 * 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) * (b / y_45scale)
    t_1 = (y_45scale * x_45scale) / (a * b)
    if (y_45scale <= (-1.3d-228)) then
        tmp = (-4.0d0) / (t_1 * t_1)
    else
        tmp = (-4.0d0) * (t_0 * 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) * (b / y_45_scale);
	double t_1 = (y_45_scale * x_45_scale) / (a * b);
	double tmp;
	if (y_45_scale <= -1.3e-228) {
		tmp = -4.0 / (t_1 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (a / x_45_scale) * (b / y_45_scale)
	t_1 = (y_45_scale * x_45_scale) / (a * b)
	tmp = 0
	if y_45_scale <= -1.3e-228:
		tmp = -4.0 / (t_1 * t_1)
	else:
		tmp = -4.0 * (t_0 * t_0)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a / x_45_scale) * Float64(b / y_45_scale))
	t_1 = Float64(Float64(y_45_scale * x_45_scale) / Float64(a * b))
	tmp = 0.0
	if (y_45_scale <= -1.3e-228)
		tmp = Float64(-4.0 / Float64(t_1 * t_1));
	else
		tmp = Float64(-4.0 * Float64(t_0 * 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) * (b / y_45_scale);
	t_1 = (y_45_scale * x_45_scale) / (a * b);
	tmp = 0.0;
	if (y_45_scale <= -1.3e-228)
		tmp = -4.0 / (t_1 * t_1);
	else
		tmp = -4.0 * (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a / x$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$45$scale * x$45$scale), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, -1.3e-228], N[(-4.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < -1.3e-228

   1. Initial program 27.9%

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

      1. cancel-sign-sub-inv 27.9%

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

      2. fma-def 31.5%

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

   3. Simplified 26.9%

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

   4. Taylor expanded in angle around 0 54.4%

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

      1. times-frac 53.9%

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

      2. unpow2 53.9%

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

      3. unpow2 53.9%

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

      4. unpow2 53.9%

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

      5. unpow2 53.9%

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

   6. Simplified 53.9%

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

   7. Taylor expanded in a around 0 53.9%

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

      1. unpow2 53.9%

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

      2. unpow2 53.9%

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

      3. times-frac 68.3%

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

   9. Simplified 68.3%

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

       1. metadata-eval 68.3%

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

       2. frac-times 53.9%

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

       3. frac-times 54.4%

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

       4. pow2 54.4%

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

       5. pow2 54.4%

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

       6. pow-prod-down 65.9%

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

       7. pow2 65.9%

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

       8. pow2 65.9%

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

       9. unpow-prod-down 77.0%

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

       10. *-commutative 77.0%

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

       11. times-frac 77.0%

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

       12. *-un-lft-identity 77.0%

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

       13. associate-/l* 77.0%

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

   11. Applied egg-rr 77.0%

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

       1. add-sqr-sqrt 77.0%

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

       2. sqrt-div 77.0%

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

       3. unpow2 77.0%

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

       4. sqrt-prod 44.7%

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

       5. add-sqr-sqrt 56.8%

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

       6. *-commutative 56.8%

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

       7. unpow2 56.8%

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

       8. sqrt-prod 33.1%

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

       9. add-sqr-sqrt 53.4%

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

       10. sqrt-div 53.4%

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

       11. unpow2 53.4%

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

       12. sqrt-prod 35.5%

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

       13. add-sqr-sqrt 60.0%

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

       14. *-commutative 60.0%

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

       15. unpow2 60.0%

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

       16. sqrt-prod 55.7%

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

       17. add-sqr-sqrt 96.7%

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

   13. Applied egg-rr 96.7%

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

   if -1.3e-228 < y-scale

   1. Initial program 22.1%

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

   3. Simplified 20.4%

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

   4. Taylor expanded in angle around 0 48.2%

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

      1. associate-*r/ 48.2%

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

      2. pow-prod-down 62.8%

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

      3. pow-prod-down 74.0%

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

   6. Applied egg-rr 74.0%

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

      1. *-un-lft-identity 74.0%

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

      2. times-frac 74.0%

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

      3. metadata-eval 74.0%

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

      4. pow-prod-down 59.1%

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

      5. pow2 59.1%

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

      6. pow2 59.1%

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

      7. *-commutative 59.1%

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

      8. unpow-prod-down 48.2%

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

      9. pow2 48.2%

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

      10. pow2 48.2%

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

      11. frac-times 47.4%

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

      12. frac-times 58.9%

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

      13. *-commutative 58.9%

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

   8. Applied egg-rr 96.0%

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

      1. unpow2 96.0%

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

   10. Applied egg-rr 96.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -1.3 \cdot 10^{-228}:\\ \;\;\;\;\frac{-4}{\frac{y-scale \cdot x-scale}{a \cdot b} \cdot \frac{y-scale \cdot x-scale}{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \]

Alternative 3: 78.2% accurate, 146.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_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) * (((a / x_45scale) * (a / x_45scale)) * ((b / y_45scale) * (b / y_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 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.8%

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

   1. cancel-sign-sub-inv 24.8%

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

   2. fma-def 27.9%

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

3. Simplified 25.2%

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

4. Taylor expanded in angle around 0 51.1%

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

   1. times-frac 50.4%

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

   2. unpow2 50.4%

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

   3. unpow2 50.4%

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

   4. unpow2 50.4%

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

   5. unpow2 50.4%

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

6. Simplified 50.4%

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

7. Taylor expanded in a around 0 50.4%

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

   1. unpow2 50.4%

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

   2. unpow2 50.4%

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

   3. times-frac 63.2%

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

9. Simplified 63.2%

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

10. Taylor expanded in b around 0 63.2%

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

    1. unpow2 63.2%

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

    2. unpow2 63.2%

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

    3. times-frac 78.0%

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

12. Simplified 78.0%

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

13. Final simplification 78.0%

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

Alternative 4: 93.8% accurate, 146.2× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ a x-scale) (/ b 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 = (a / x_45_scale) * (b / 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 = (a / x_45scale) * (b / 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 = (a / x_45_scale) * (b / y_45_scale);
	return -4.0 * (t_0 * t_0);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (a / x_45_scale) * (b / 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(a / x_45_scale) * Float64(b / 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 = (a / x_45_scale) * (b / 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[(a / x$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 24.8%

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

3. Simplified 21.7%

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

4. Taylor expanded in angle around 0 51.1%

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

   1. associate-*r/ 51.1%

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

   2. pow-prod-down 64.2%

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

   3. pow-prod-down 75.4%

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

6. Applied egg-rr 75.4%

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

   1. *-un-lft-identity 75.4%

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

   2. times-frac 75.4%

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

   3. metadata-eval 75.4%

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

   4. pow-prod-down 60.1%

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

   5. pow2 60.1%

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

   6. pow2 60.1%

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

   7. *-commutative 60.1%

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

   8. unpow-prod-down 51.1%

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

   9. pow2 51.1%

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

   10. pow2 51.1%

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

   11. frac-times 50.4%

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

   12. frac-times 63.2%

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

   13. *-commutative 63.2%

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

8. Applied egg-rr 94.2%

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

   1. unpow2 94.2%

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

10. Applied egg-rr 94.2%

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

11. Final simplification 94.2%

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

Alternative 5: 36.6% 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 24.8%

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

   1. cancel-sign-sub-inv 24.8%

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

   2. fma-def 27.9%

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

3. Simplified 25.2%

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

4. Taylor expanded in b around 0 23.6%

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

   1. *-commutative 23.6%

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

   2. *-commutative 23.6%

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

   3. *-commutative 23.6%

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

   4. distribute-lft-out 23.6%

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

6. Simplified 36.1%

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

7. Final simplification 36.1%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023189 
(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))))