Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.4% → 83.5%

Time: 1.6min

Alternatives: 6

Speedup: 2485.0×

• 33.5% 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: 24.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.5% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+113}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{-{\left(\frac{a \cdot b}{y-scale}\right)}^{2}}{x-scale}}{x-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a -1.3e+113)
   (* -4.0 (* (* (/ b x-scale) (/ b x-scale)) (* (/ a y-scale) (/ a y-scale))))
   (/ (* 4.0 (/ (- (pow (/ (* a b) y-scale) 2.0)) x-scale)) x-scale)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= -1.3e+113) {
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)));
	} else {
		tmp = (4.0 * (-pow(((a * b) / y_45_scale), 2.0) / x_45_scale)) / x_45_scale;
	}
	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) :: tmp
    if (a <= (-1.3d+113)) then
        tmp = (-4.0d0) * (((b / x_45scale) * (b / x_45scale)) * ((a / y_45scale) * (a / y_45scale)))
    else
        tmp = (4.0d0 * (-(((a * b) / y_45scale) ** 2.0d0) / x_45scale)) / x_45scale
    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 tmp;
	if (a <= -1.3e+113) {
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)));
	} else {
		tmp = (4.0 * (-Math.pow(((a * b) / y_45_scale), 2.0) / x_45_scale)) / x_45_scale;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= -1.3e+113:
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)))
	else:
		tmp = (4.0 * (-math.pow(((a * b) / y_45_scale), 2.0) / x_45_scale)) / x_45_scale
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= -1.3e+113)
		tmp = Float64(-4.0 * Float64(Float64(Float64(b / x_45_scale) * Float64(b / x_45_scale)) * Float64(Float64(a / y_45_scale) * Float64(a / y_45_scale))));
	else
		tmp = Float64(Float64(4.0 * Float64(Float64(-(Float64(Float64(a * b) / y_45_scale) ^ 2.0)) / x_45_scale)) / x_45_scale);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a <= -1.3e+113)
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)));
	else
		tmp = (4.0 * (-(((a * b) / y_45_scale) ^ 2.0) / x_45_scale)) / x_45_scale;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, -1.3e+113], N[(-4.0 * N[(N[(N[(b / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a / y$45$scale), $MachinePrecision] * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 * N[((-N[Power[N[(N[(a * b), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]) / x$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3e113

   1. Initial program 9.2%

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

   2. Taylor expanded in angle around 0 51.4%

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

      1. *-commutative 51.4%

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

      2. *-commutative 51.4%

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

      3. times-frac 53.7%

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

      4. unpow2 53.7%

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

      5. unpow2 53.7%

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

      6. times-frac 67.1%

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

      7. unpow2 67.1%

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

      8. unpow2 67.1%

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

      9. times-frac 93.4%

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

   4. Simplified 93.4%

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

   if -1.3e113 < a

   1. Initial program 29.3%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in x-scale around 0 19.0%

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

   6. Simplified 25.3%

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

   7. Taylor expanded in angle around 0 55.5%

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

      1. mul-1-neg 55.5%

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

      2. *-commutative 55.5%

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

      3. associate-*r/ 54.9%

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

      4. unpow2 54.9%

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

      5. associate-/r* 59.3%

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

      6. unpow2 59.3%

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

      7. unpow2 59.3%

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

      8. associate-*l/ 66.4%

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

      9. associate-*r/ 69.7%

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

      10. unpow2 69.7%

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

   9. Simplified 69.7%

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

       1. associate-*l/ 70.2%

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

       2. pow2 70.2%

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

       3. pow-prod-down 84.9%

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

   11. Applied egg-rr 84.9%

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

       1. associate-*r/ 85.1%

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

   13. Applied egg-rr 85.1%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+113}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{-{\left(\frac{a \cdot b}{y-scale}\right)}^{2}}{x-scale}}{x-scale}\\ \end{array} \]

Alternative 2: 76.9% accurate, 112.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 5.2 \cdot 10^{-165} \lor \neg \left(y-scale \leq 5 \cdot 10^{+87}\right):\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{x-scale} \cdot \frac{-a}{x-scale \cdot \frac{y-scale \cdot y-scale}{a \cdot \left(b \cdot b\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (or (<= y-scale 5.2e-165) (not (<= y-scale 5e+87)))
   (* -4.0 (* (* (/ b x-scale) (/ b x-scale)) (* (/ a y-scale) (/ a y-scale))))
   (*
    (/ 4.0 x-scale)
    (/ (- a) (* x-scale (/ (* y-scale y-scale) (* a (* b b))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if ((y_45_scale <= 5.2e-165) || !(y_45_scale <= 5e+87)) {
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)));
	} else {
		tmp = (4.0 / x_45_scale) * (-a / (x_45_scale * ((y_45_scale * y_45_scale) / (a * (b * b)))));
	}
	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) :: tmp
    if ((y_45scale <= 5.2d-165) .or. (.not. (y_45scale <= 5d+87))) then
        tmp = (-4.0d0) * (((b / x_45scale) * (b / x_45scale)) * ((a / y_45scale) * (a / y_45scale)))
    else
        tmp = (4.0d0 / x_45scale) * (-a / (x_45scale * ((y_45scale * y_45scale) / (a * (b * b)))))
    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 tmp;
	if ((y_45_scale <= 5.2e-165) || !(y_45_scale <= 5e+87)) {
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)));
	} else {
		tmp = (4.0 / x_45_scale) * (-a / (x_45_scale * ((y_45_scale * y_45_scale) / (a * (b * b)))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if (y_45_scale <= 5.2e-165) or not (y_45_scale <= 5e+87):
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)))
	else:
		tmp = (4.0 / x_45_scale) * (-a / (x_45_scale * ((y_45_scale * y_45_scale) / (a * (b * b)))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if ((y_45_scale <= 5.2e-165) || !(y_45_scale <= 5e+87))
		tmp = Float64(-4.0 * Float64(Float64(Float64(b / x_45_scale) * Float64(b / x_45_scale)) * Float64(Float64(a / y_45_scale) * Float64(a / y_45_scale))));
	else
		tmp = Float64(Float64(4.0 / x_45_scale) * Float64(Float64(-a) / Float64(x_45_scale * Float64(Float64(y_45_scale * y_45_scale) / Float64(a * Float64(b * b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if ((y_45_scale <= 5.2e-165) || ~((y_45_scale <= 5e+87)))
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)));
	else
		tmp = (4.0 / x_45_scale) * (-a / (x_45_scale * ((y_45_scale * y_45_scale) / (a * (b * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[Or[LessEqual[y$45$scale, 5.2e-165], N[Not[LessEqual[y$45$scale, 5e+87]], $MachinePrecision]], N[(-4.0 * N[(N[(N[(b / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a / y$45$scale), $MachinePrecision] * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 / x$45$scale), $MachinePrecision] * N[((-a) / N[(x$45$scale * N[(N[(y$45$scale * y$45$scale), $MachinePrecision] / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 5.20000000000000015e-165 or 4.9999999999999998e87 < y-scale

   1. Initial program 27.0%

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

   2. Taylor expanded in angle around 0 47.8%

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

      1. *-commutative 47.8%

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

      2. *-commutative 47.8%

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

      3. times-frac 47.5%

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

      4. unpow2 47.5%

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

      5. unpow2 47.5%

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

      6. times-frac 57.9%

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

      7. unpow2 57.9%

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

      8. unpow2 57.9%

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

      9. times-frac 80.4%

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

   4. Simplified 80.4%

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

   if 5.20000000000000015e-165 < y-scale < 4.9999999999999998e87

   1. Initial program 21.5%

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

   3. Simplified 21.5%

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

   4. Taylor expanded in x-scale around 0 17.4%

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

   6. Simplified 17.8%

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

   7. Taylor expanded in angle around 0 67.0%

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

      1. mul-1-neg 67.0%

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

      2. associate-/l* 66.7%

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

      3. unpow2 66.7%

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

      4. unpow2 66.7%

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

      5. unpow2 66.7%

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

   9. Simplified 66.7%

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

       1. distribute-frac-neg 66.7%

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

       2. associate-/l* 72.5%

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

       3. clear-num 72.2%

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

       4. frac-times 73.8%

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

       5. pow2 73.8%

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

       6. pow-flip 74.2%

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

       7. metadata-eval 74.2%

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

   11. Applied egg-rr 74.2%

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

       1. associate-/l/ 83.6%

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

       2. distribute-neg-frac 83.6%

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

   13. Simplified 83.6%

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

   14. Taylor expanded in b around 0 86.5%

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

       1. unpow2 86.5%

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

       2. unpow2 86.5%

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

   16. Simplified 86.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 5.2 \cdot 10^{-165} \lor \neg \left(y-scale \leq 5 \cdot 10^{+87}\right):\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{x-scale} \cdot \frac{-a}{x-scale \cdot \frac{y-scale \cdot y-scale}{a \cdot \left(b \cdot b\right)}}\\ \end{array} \]

Alternative 3: 83.7% accurate, 123.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{\frac{y-scale}{b}}\\ \mathbf{if}\;a \leq -6 \cdot 10^{+112}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{x-scale} \cdot \frac{-t_0 \cdot t_0}{x-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (/ y-scale b))))
   (if (<= a -6e+112)
     (*
      -4.0
      (* (* (/ b x-scale) (/ b x-scale)) (* (/ a y-scale) (/ a y-scale))))
     (* (/ 4.0 x-scale) (/ (- (* t_0 t_0)) x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale / b);
	double tmp;
	if (a <= -6e+112) {
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)));
	} else {
		tmp = (4.0 / x_45_scale) * (-(t_0 * t_0) / x_45_scale);
	}
	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 / (y_45scale / b)
    if (a <= (-6d+112)) then
        tmp = (-4.0d0) * (((b / x_45scale) * (b / x_45scale)) * ((a / y_45scale) * (a / y_45scale)))
    else
        tmp = (4.0d0 / x_45scale) * (-(t_0 * t_0) / x_45scale)
    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 / (y_45_scale / b);
	double tmp;
	if (a <= -6e+112) {
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)));
	} else {
		tmp = (4.0 / x_45_scale) * (-(t_0 * t_0) / x_45_scale);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a / (y_45_scale / b)
	tmp = 0
	if a <= -6e+112:
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)))
	else:
		tmp = (4.0 / x_45_scale) * (-(t_0 * t_0) / x_45_scale)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(y_45_scale / b))
	tmp = 0.0
	if (a <= -6e+112)
		tmp = Float64(-4.0 * Float64(Float64(Float64(b / x_45_scale) * Float64(b / x_45_scale)) * Float64(Float64(a / y_45_scale) * Float64(a / y_45_scale))));
	else
		tmp = Float64(Float64(4.0 / x_45_scale) * Float64(Float64(-Float64(t_0 * t_0)) / x_45_scale));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a / (y_45_scale / b);
	tmp = 0.0;
	if (a <= -6e+112)
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)));
	else
		tmp = (4.0 / x_45_scale) * (-(t_0 * t_0) / x_45_scale);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6e+112], N[(-4.0 * N[(N[(N[(b / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a / y$45$scale), $MachinePrecision] * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 / x$45$scale), $MachinePrecision] * N[((-N[(t$95$0 * t$95$0), $MachinePrecision]) / x$45$scale), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.99999999999999958e112

   1. Initial program 9.2%

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

   2. Taylor expanded in angle around 0 51.4%

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

      1. *-commutative 51.4%

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

      2. *-commutative 51.4%

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

      3. times-frac 53.7%

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

      4. unpow2 53.7%

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

      5. unpow2 53.7%

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

      6. times-frac 67.1%

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

      7. unpow2 67.1%

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

      8. unpow2 67.1%

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

      9. times-frac 93.4%

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

   4. Simplified 93.4%

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

   if -5.99999999999999958e112 < a

   1. Initial program 29.3%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in x-scale around 0 19.0%

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

   6. Simplified 25.3%

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

   7. Taylor expanded in angle around 0 55.5%

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

      1. mul-1-neg 55.5%

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

      2. associate-/l* 56.3%

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

      3. unpow2 56.3%

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

      4. unpow2 56.3%

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

      5. unpow2 56.3%

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

   9. Simplified 56.3%

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

       1. times-frac 66.7%

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

   11. Applied egg-rr 66.7%

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

       1. times-frac 84.2%

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

   13. Applied egg-rr 84.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{+112}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{x-scale} \cdot \frac{-\frac{a}{\frac{y-scale}{b}} \cdot \frac{a}{\frac{y-scale}{b}}}{x-scale}\\ \end{array} \]

Alternative 4: 83.5% accurate, 123.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{+112}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{\left(b \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \left(-b\right)\right)}{x-scale}}{x-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a -3.2e+112)
   (* -4.0 (* (* (/ b x-scale) (/ b x-scale)) (* (/ a y-scale) (/ a y-scale))))
   (/
    (* 4.0 (/ (* (* b (/ a y-scale)) (* (/ a y-scale) (- b))) x-scale))
    x-scale)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= -3.2e+112) {
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)));
	} else {
		tmp = (4.0 * (((b * (a / y_45_scale)) * ((a / y_45_scale) * -b)) / x_45_scale)) / x_45_scale;
	}
	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) :: tmp
    if (a <= (-3.2d+112)) then
        tmp = (-4.0d0) * (((b / x_45scale) * (b / x_45scale)) * ((a / y_45scale) * (a / y_45scale)))
    else
        tmp = (4.0d0 * (((b * (a / y_45scale)) * ((a / y_45scale) * -b)) / x_45scale)) / x_45scale
    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 tmp;
	if (a <= -3.2e+112) {
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)));
	} else {
		tmp = (4.0 * (((b * (a / y_45_scale)) * ((a / y_45_scale) * -b)) / x_45_scale)) / x_45_scale;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= -3.2e+112:
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)))
	else:
		tmp = (4.0 * (((b * (a / y_45_scale)) * ((a / y_45_scale) * -b)) / x_45_scale)) / x_45_scale
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= -3.2e+112)
		tmp = Float64(-4.0 * Float64(Float64(Float64(b / x_45_scale) * Float64(b / x_45_scale)) * Float64(Float64(a / y_45_scale) * Float64(a / y_45_scale))));
	else
		tmp = Float64(Float64(4.0 * Float64(Float64(Float64(b * Float64(a / y_45_scale)) * Float64(Float64(a / y_45_scale) * Float64(-b))) / x_45_scale)) / x_45_scale);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a <= -3.2e+112)
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)));
	else
		tmp = (4.0 * (((b * (a / y_45_scale)) * ((a / y_45_scale) * -b)) / x_45_scale)) / x_45_scale;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, -3.2e+112], N[(-4.0 * N[(N[(N[(b / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a / y$45$scale), $MachinePrecision] * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 * N[(N[(N[(b * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a / y$45$scale), $MachinePrecision] * (-b)), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.19999999999999986e112

   1. Initial program 9.2%

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

   2. Taylor expanded in angle around 0 51.4%

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

      1. *-commutative 51.4%

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

      2. *-commutative 51.4%

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

      3. times-frac 53.7%

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

      4. unpow2 53.7%

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

      5. unpow2 53.7%

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

      6. times-frac 67.1%

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

      7. unpow2 67.1%

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

      8. unpow2 67.1%

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

      9. times-frac 93.4%

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

   4. Simplified 93.4%

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

   if -3.19999999999999986e112 < a

   1. Initial program 29.3%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in x-scale around 0 19.0%

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

   6. Simplified 25.3%

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

   7. Taylor expanded in angle around 0 55.5%

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

      1. mul-1-neg 55.5%

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

      2. *-commutative 55.5%

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

      3. associate-*r/ 54.9%

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

      4. unpow2 54.9%

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

      5. associate-/r* 59.3%

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

      6. unpow2 59.3%

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

      7. unpow2 59.3%

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

      8. associate-*l/ 66.4%

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

      9. associate-*r/ 69.7%

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

      10. unpow2 69.7%

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

   9. Simplified 69.7%

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

       1. associate-*l/ 70.2%

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

       2. pow2 70.2%

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

       3. pow-prod-down 84.9%

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

   11. Applied egg-rr 84.9%

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

       1. unpow2 84.9%

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

   13. Applied egg-rr 84.9%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{+112}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{\left(b \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \left(-b\right)\right)}{x-scale}}{x-scale}\\ \end{array} \]

Alternative 5: 77.5% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.8%

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

2. Taylor expanded in angle around 0 49.3%

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

   1. *-commutative 49.3%

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

   2. *-commutative 49.3%

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

   3. times-frac 48.7%

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

   4. unpow2 48.7%

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

   5. unpow2 48.7%

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

   6. times-frac 59.8%

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

   7. unpow2 59.8%

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

   8. unpow2 59.8%

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

   9. times-frac 78.5%

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

4. Simplified 78.5%

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

5. Final simplification 78.5%

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

Alternative 6: 35.3% 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 25.8%

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

   1. fma-neg 25.2%

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

3. Simplified 20.4%

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

4. Taylor expanded in b around 0 18.1%

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

   1. *-commutative 18.1%

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

   2. *-commutative 18.1%

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

   3. *-commutative 18.1%

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

   4. distribute-lft-out 18.1%

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

6. Simplified 28.8%

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

7. Final simplification 28.8%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023194 
(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))))