Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.0% → 79.8%

Time: 1.5min

Alternatives: 7

Speedup: 2485.0×

• 33.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4}{x-scale} \cdot \frac{b}{x-scale}\\ t_1 := \sqrt{\frac{{\left(b \cdot a\right)}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}}}\\ \mathbf{if}\;x-scale \leq -1.6 \cdot 10^{-86}:\\ \;\;\;\;t_0 \cdot \frac{a \cdot \frac{b}{y-scale}}{\frac{y-scale}{a}}\\ \mathbf{elif}\;x-scale \leq 5.8 \cdot 10^{-157}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{a \cdot \frac{b \cdot a}{y-scale}}{y-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ -4.0 x-scale) (/ b x-scale)))
        (t_1 (sqrt (/ (pow (* b a) 2.0) (pow (* x-scale y-scale) 2.0)))))
   (if (<= x-scale -1.6e-86)
     (* t_0 (/ (* a (/ b y-scale)) (/ y-scale a)))
     (if (<= x-scale 5.8e-157)
       (* -4.0 (* t_1 t_1))
       (* t_0 (/ (* a (/ (* b a) y-scale)) y-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (-4.0 / x_45_scale) * (b / x_45_scale);
	double t_1 = sqrt((pow((b * a), 2.0) / pow((x_45_scale * y_45_scale), 2.0)));
	double tmp;
	if (x_45_scale <= -1.6e-86) {
		tmp = t_0 * ((a * (b / y_45_scale)) / (y_45_scale / a));
	} else if (x_45_scale <= 5.8e-157) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = t_0 * ((a * ((b * a) / y_45_scale)) / y_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) :: t_1
    real(8) :: tmp
    t_0 = ((-4.0d0) / x_45scale) * (b / x_45scale)
    t_1 = sqrt((((b * a) ** 2.0d0) / ((x_45scale * y_45scale) ** 2.0d0)))
    if (x_45scale <= (-1.6d-86)) then
        tmp = t_0 * ((a * (b / y_45scale)) / (y_45scale / a))
    else if (x_45scale <= 5.8d-157) then
        tmp = (-4.0d0) * (t_1 * t_1)
    else
        tmp = t_0 * ((a * ((b * a) / y_45scale)) / y_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 = (-4.0 / x_45_scale) * (b / x_45_scale);
	double t_1 = Math.sqrt((Math.pow((b * a), 2.0) / Math.pow((x_45_scale * y_45_scale), 2.0)));
	double tmp;
	if (x_45_scale <= -1.6e-86) {
		tmp = t_0 * ((a * (b / y_45_scale)) / (y_45_scale / a));
	} else if (x_45_scale <= 5.8e-157) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = t_0 * ((a * ((b * a) / y_45_scale)) / y_45_scale);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (-4.0 / x_45_scale) * (b / x_45_scale)
	t_1 = math.sqrt((math.pow((b * a), 2.0) / math.pow((x_45_scale * y_45_scale), 2.0)))
	tmp = 0
	if x_45_scale <= -1.6e-86:
		tmp = t_0 * ((a * (b / y_45_scale)) / (y_45_scale / a))
	elif x_45_scale <= 5.8e-157:
		tmp = -4.0 * (t_1 * t_1)
	else:
		tmp = t_0 * ((a * ((b * a) / y_45_scale)) / y_45_scale)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(-4.0 / x_45_scale) * Float64(b / x_45_scale))
	t_1 = sqrt(Float64((Float64(b * a) ^ 2.0) / (Float64(x_45_scale * y_45_scale) ^ 2.0)))
	tmp = 0.0
	if (x_45_scale <= -1.6e-86)
		tmp = Float64(t_0 * Float64(Float64(a * Float64(b / y_45_scale)) / Float64(y_45_scale / a)));
	elseif (x_45_scale <= 5.8e-157)
		tmp = Float64(-4.0 * Float64(t_1 * t_1));
	else
		tmp = Float64(t_0 * Float64(Float64(a * Float64(Float64(b * a) / y_45_scale)) / y_45_scale));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (-4.0 / x_45_scale) * (b / x_45_scale);
	t_1 = sqrt((((b * a) ^ 2.0) / ((x_45_scale * y_45_scale) ^ 2.0)));
	tmp = 0.0;
	if (x_45_scale <= -1.6e-86)
		tmp = t_0 * ((a * (b / y_45_scale)) / (y_45_scale / a));
	elseif (x_45_scale <= 5.8e-157)
		tmp = -4.0 * (t_1 * t_1);
	else
		tmp = t_0 * ((a * ((b * a) / y_45_scale)) / y_45_scale);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(-4.0 / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[Power[N[(b * a), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$45$scale, -1.6e-86], N[(t$95$0 * N[(N[(a * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 5.8e-157], N[(-4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(a * N[(N[(b * a), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -1.60000000000000003e-86

   1. Initial program 38.5%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in angle around 0 51.8%

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

      1. associate-/l* 53.8%

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

      2. associate-*r/ 53.8%

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

      3. unpow2 53.8%

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

      4. unpow2 53.8%

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

      5. times-frac 55.1%

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

      6. unpow2 55.1%

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

      7. unpow2 55.1%

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

   6. Simplified 55.1%

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

      1. times-frac 56.2%

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

      2. associate-/l* 58.3%

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

      3. associate-/l* 65.6%

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

   8. Applied egg-rr 65.6%

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

      1. unpow2 65.6%

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

      2. associate-/r/ 65.6%

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

      3. associate-/r/ 72.8%

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

      4. unpow2 72.8%

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

   10. Simplified 72.8%

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

   11. Taylor expanded in a around 0 72.8%

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

       1. unpow2 72.8%

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

       2. associate-*l/ 81.1%

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

       3. *-commutative 81.1%

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

   13. Simplified 81.1%

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

       1. *-commutative 81.1%

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

       2. associate-/r/ 81.1%

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

       3. associate-*l/ 84.5%

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

   15. Applied egg-rr 84.5%

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

   if -1.60000000000000003e-86 < x-scale < 5.79999999999999977e-157

   1. Initial program 17.8%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in angle around 0 47.7%

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

      1. associate-/l* 47.7%

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

      2. associate-*r/ 47.7%

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

      3. unpow2 47.7%

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

      4. unpow2 47.7%

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

      5. times-frac 51.4%

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

      6. unpow2 51.4%

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

      7. unpow2 51.4%

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

   6. Simplified 51.4%

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

   7. Taylor expanded in a around 0 47.7%

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

      1. unpow2 47.7%

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

      2. unpow2 47.7%

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

      3. swap-sqr 54.5%

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

      4. unpow2 54.5%

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

      5. unpow2 54.5%

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

      6. unpow2 54.5%

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

      7. swap-sqr 91.6%

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

      8. unpow2 91.6%

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

   9. Simplified 91.6%

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

       1. add-sqr-sqrt 91.6%

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

   11. Applied egg-rr 91.6%

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

   if 5.79999999999999977e-157 < x-scale

   1. Initial program 32.4%

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

   3. Simplified 31.4%

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

   4. Taylor expanded in angle around 0 57.0%

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

      1. associate-/l* 57.3%

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

      2. associate-*r/ 57.3%

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

      3. unpow2 57.3%

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

      4. unpow2 57.3%

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

      5. times-frac 61.7%

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

      6. unpow2 61.7%

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

      7. unpow2 61.7%

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

   6. Simplified 61.7%

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

      1. times-frac 61.7%

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

      2. associate-/l* 65.8%

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

      3. associate-/l* 68.8%

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

   8. Applied egg-rr 68.8%

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

      1. unpow2 68.8%

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

      2. associate-/r/ 68.8%

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

      3. associate-/r/ 72.0%

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

      4. unpow2 72.0%

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

   10. Simplified 72.0%

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

       1. associate-*r/ 72.1%

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

       2. associate-/l* 81.8%

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

   12. Applied egg-rr 81.8%

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

   13. Taylor expanded in a around 0 70.6%

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

       1. *-commutative 70.6%

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

       2. associate-/l* 73.0%

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

       3. associate-/r/ 73.8%

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

       4. unpow2 73.8%

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

       5. associate-*l* 82.2%

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

       6. *-commutative 82.2%

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

       7. *-commutative 82.2%

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

       8. associate-*r/ 86.0%

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

   15. Simplified 86.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -1.6 \cdot 10^{-86}:\\ \;\;\;\;\left(\frac{-4}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \frac{a \cdot \frac{b}{y-scale}}{\frac{y-scale}{a}}\\ \mathbf{elif}\;x-scale \leq 5.8 \cdot 10^{-157}:\\ \;\;\;\;-4 \cdot \left(\sqrt{\frac{{\left(b \cdot a\right)}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}}} \cdot \sqrt{\frac{{\left(b \cdot a\right)}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \frac{a \cdot \frac{b \cdot a}{y-scale}}{y-scale}\\ \end{array} \]

Alternative 2: 79.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4}{x-scale} \cdot \frac{b}{x-scale}\\ \mathbf{if}\;x-scale \leq -1.32 \cdot 10^{-86}:\\ \;\;\;\;t_0 \cdot \frac{a \cdot \frac{b}{y-scale}}{\frac{y-scale}{a}}\\ \mathbf{elif}\;x-scale \leq 2.25 \cdot 10^{-157}:\\ \;\;\;\;-4 \cdot \frac{{\left(b \cdot a\right)}^{2}}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{a \cdot \frac{b \cdot a}{y-scale}}{y-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ -4.0 x-scale) (/ b x-scale))))
   (if (<= x-scale -1.32e-86)
     (* t_0 (/ (* a (/ b y-scale)) (/ y-scale a)))
     (if (<= x-scale 2.25e-157)
       (*
        -4.0
        (/ (pow (* b a) 2.0) (* (* x-scale y-scale) (* x-scale y-scale))))
       (* t_0 (/ (* a (/ (* b a) y-scale)) y-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (-4.0 / x_45_scale) * (b / x_45_scale);
	double tmp;
	if (x_45_scale <= -1.32e-86) {
		tmp = t_0 * ((a * (b / y_45_scale)) / (y_45_scale / a));
	} else if (x_45_scale <= 2.25e-157) {
		tmp = -4.0 * (pow((b * a), 2.0) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
	} else {
		tmp = t_0 * ((a * ((b * a) / y_45_scale)) / y_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 = ((-4.0d0) / x_45scale) * (b / x_45scale)
    if (x_45scale <= (-1.32d-86)) then
        tmp = t_0 * ((a * (b / y_45scale)) / (y_45scale / a))
    else if (x_45scale <= 2.25d-157) then
        tmp = (-4.0d0) * (((b * a) ** 2.0d0) / ((x_45scale * y_45scale) * (x_45scale * y_45scale)))
    else
        tmp = t_0 * ((a * ((b * a) / y_45scale)) / y_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 = (-4.0 / x_45_scale) * (b / x_45_scale);
	double tmp;
	if (x_45_scale <= -1.32e-86) {
		tmp = t_0 * ((a * (b / y_45_scale)) / (y_45_scale / a));
	} else if (x_45_scale <= 2.25e-157) {
		tmp = -4.0 * (Math.pow((b * a), 2.0) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
	} else {
		tmp = t_0 * ((a * ((b * a) / y_45_scale)) / y_45_scale);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (-4.0 / x_45_scale) * (b / x_45_scale)
	tmp = 0
	if x_45_scale <= -1.32e-86:
		tmp = t_0 * ((a * (b / y_45_scale)) / (y_45_scale / a))
	elif x_45_scale <= 2.25e-157:
		tmp = -4.0 * (math.pow((b * a), 2.0) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))
	else:
		tmp = t_0 * ((a * ((b * a) / y_45_scale)) / y_45_scale)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(-4.0 / x_45_scale) * Float64(b / x_45_scale))
	tmp = 0.0
	if (x_45_scale <= -1.32e-86)
		tmp = Float64(t_0 * Float64(Float64(a * Float64(b / y_45_scale)) / Float64(y_45_scale / a)));
	elseif (x_45_scale <= 2.25e-157)
		tmp = Float64(-4.0 * Float64((Float64(b * a) ^ 2.0) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))));
	else
		tmp = Float64(t_0 * Float64(Float64(a * Float64(Float64(b * a) / y_45_scale)) / y_45_scale));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (-4.0 / x_45_scale) * (b / x_45_scale);
	tmp = 0.0;
	if (x_45_scale <= -1.32e-86)
		tmp = t_0 * ((a * (b / y_45_scale)) / (y_45_scale / a));
	elseif (x_45_scale <= 2.25e-157)
		tmp = -4.0 * (((b * a) ^ 2.0) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
	else
		tmp = t_0 * ((a * ((b * a) / y_45_scale)) / y_45_scale);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(-4.0 / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -1.32e-86], N[(t$95$0 * N[(N[(a * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 2.25e-157], N[(-4.0 * N[(N[Power[N[(b * a), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(a * N[(N[(b * a), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -1.32e-86

   1. Initial program 38.5%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in angle around 0 51.8%

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

      1. associate-/l* 53.8%

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

      2. associate-*r/ 53.8%

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

      3. unpow2 53.8%

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

      4. unpow2 53.8%

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

      5. times-frac 55.1%

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

      6. unpow2 55.1%

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

      7. unpow2 55.1%

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

   6. Simplified 55.1%

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

      1. times-frac 56.2%

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

      2. associate-/l* 58.3%

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

      3. associate-/l* 65.6%

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

   8. Applied egg-rr 65.6%

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

      1. unpow2 65.6%

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

      2. associate-/r/ 65.6%

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

      3. associate-/r/ 72.8%

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

      4. unpow2 72.8%

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

   10. Simplified 72.8%

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

   11. Taylor expanded in a around 0 72.8%

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

       1. unpow2 72.8%

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

       2. associate-*l/ 81.1%

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

       3. *-commutative 81.1%

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

   13. Simplified 81.1%

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

       1. *-commutative 81.1%

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

       2. associate-/r/ 81.1%

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

       3. associate-*l/ 84.5%

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

   15. Applied egg-rr 84.5%

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

   if -1.32e-86 < x-scale < 2.24999999999999999e-157

   1. Initial program 17.8%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in angle around 0 47.7%

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

      1. associate-/l* 47.7%

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

      2. associate-*r/ 47.7%

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

      3. unpow2 47.7%

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

      4. unpow2 47.7%

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

      5. times-frac 51.4%

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

      6. unpow2 51.4%

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

      7. unpow2 51.4%

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

   6. Simplified 51.4%

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

   7. Taylor expanded in a around 0 47.7%

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

      1. unpow2 47.7%

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

      2. unpow2 47.7%

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

      3. swap-sqr 54.5%

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

      4. unpow2 54.5%

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

      5. unpow2 54.5%

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

      6. unpow2 54.5%

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

      7. swap-sqr 91.6%

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

      8. unpow2 91.6%

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

   9. Simplified 91.6%

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

       1. unpow2 91.6%

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

   11. Applied egg-rr 91.6%

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

   if 2.24999999999999999e-157 < x-scale

   1. Initial program 32.4%

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

   3. Simplified 31.4%

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

   4. Taylor expanded in angle around 0 57.0%

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

      1. associate-/l* 57.3%

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

      2. associate-*r/ 57.3%

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

      3. unpow2 57.3%

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

      4. unpow2 57.3%

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

      5. times-frac 61.7%

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

      6. unpow2 61.7%

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

      7. unpow2 61.7%

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

   6. Simplified 61.7%

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

      1. times-frac 61.7%

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

      2. associate-/l* 65.8%

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

      3. associate-/l* 68.8%

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

   8. Applied egg-rr 68.8%

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

      1. unpow2 68.8%

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

      2. associate-/r/ 68.8%

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

      3. associate-/r/ 72.0%

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

      4. unpow2 72.0%

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

   10. Simplified 72.0%

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

       1. associate-*r/ 72.1%

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

       2. associate-/l* 81.8%

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

   12. Applied egg-rr 81.8%

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

   13. Taylor expanded in a around 0 70.6%

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

       1. *-commutative 70.6%

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

       2. associate-/l* 73.0%

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

       3. associate-/r/ 73.8%

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

       4. unpow2 73.8%

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

       5. associate-*l* 82.2%

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

       6. *-commutative 82.2%

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

       7. *-commutative 82.2%

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

       8. associate-*r/ 86.0%

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

   15. Simplified 86.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -1.32 \cdot 10^{-86}:\\ \;\;\;\;\left(\frac{-4}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \frac{a \cdot \frac{b}{y-scale}}{\frac{y-scale}{a}}\\ \mathbf{elif}\;x-scale \leq 2.25 \cdot 10^{-157}:\\ \;\;\;\;-4 \cdot \frac{{\left(b \cdot a\right)}^{2}}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \frac{a \cdot \frac{b \cdot a}{y-scale}}{y-scale}\\ \end{array} \]

Alternative 3: 77.9% accurate, 146.2× speedup

Math

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

FPCore

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.2%

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

3. Simplified 26.6%

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

4. Taylor expanded in angle around 0 52.9%

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

   1. associate-/l* 53.7%

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

   2. associate-*r/ 53.7%

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

   3. unpow2 53.7%

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

   4. unpow2 53.7%

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

   5. times-frac 56.8%

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

   6. unpow2 56.8%

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

   7. unpow2 56.8%

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

6. Simplified 56.8%

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

   1. times-frac 57.2%

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

   2. associate-/l* 61.7%

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

   3. associate-/l* 65.6%

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

8. Applied egg-rr 65.6%

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

   1. unpow2 65.6%

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

   2. associate-/r/ 65.6%

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

   3. associate-/r/ 70.7%

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

   4. unpow2 70.7%

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

10. Simplified 70.7%

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

    1. associate-*r/ 71.3%

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

    2. associate-/l* 80.0%

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

12. Applied egg-rr 80.0%

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

    1. associate-/r/ 80.0%

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

    2. *-commutative 80.0%

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

    3. associate-*r/ 77.9%

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

    4. associate-*l* 79.9%

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

14. Applied egg-rr 79.9%

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

15. Final simplification 79.9%

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

Alternative 4: 78.8% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

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) / x_45scale) * (b / x_45scale)) * ((a * (b / y_45scale)) / (y_45scale / a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.2%

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

3. Simplified 26.6%

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

4. Taylor expanded in angle around 0 52.9%

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

   1. associate-/l* 53.7%

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

   2. associate-*r/ 53.7%

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

   3. unpow2 53.7%

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

   4. unpow2 53.7%

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

   5. times-frac 56.8%

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

   6. unpow2 56.8%

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

   7. unpow2 56.8%

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

6. Simplified 56.8%

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

   1. times-frac 57.2%

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

   2. associate-/l* 61.7%

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

   3. associate-/l* 65.6%

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

8. Applied egg-rr 65.6%

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

   1. unpow2 65.6%

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

   2. associate-/r/ 65.6%

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

   3. associate-/r/ 70.7%

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

   4. unpow2 70.7%

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

10. Simplified 70.7%

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

11. Taylor expanded in a around 0 70.7%

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

    1. unpow2 70.7%

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

    2. associate-*l/ 77.9%

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

    3. *-commutative 77.9%

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

13. Simplified 77.9%

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

    1. *-commutative 77.9%

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

    2. associate-/r/ 77.9%

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

    3. associate-*l/ 81.8%

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

15. Applied egg-rr 81.8%

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

16. Final simplification 81.8%

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

Alternative 5: 77.7% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.2%

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

3. Simplified 26.6%

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

4. Taylor expanded in angle around 0 52.9%

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

   1. associate-/l* 53.7%

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

   2. associate-*r/ 53.7%

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

   3. unpow2 53.7%

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

   4. unpow2 53.7%

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

   5. times-frac 56.8%

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

   6. unpow2 56.8%

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

   7. unpow2 56.8%

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

6. Simplified 56.8%

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

   1. times-frac 57.2%

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

   2. associate-/l* 61.7%

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

   3. associate-/l* 65.6%

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

8. Applied egg-rr 65.6%

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

   1. unpow2 65.6%

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

   2. associate-/r/ 65.6%

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

   3. associate-/r/ 70.7%

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

   4. unpow2 70.7%

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

10. Simplified 70.7%

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

    1. associate-*r/ 71.3%

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

    2. associate-/l* 80.0%

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

12. Applied egg-rr 80.0%

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

13. Taylor expanded in a around 0 68.6%

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

    1. *-commutative 68.6%

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

    2. associate-/l* 71.3%

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

    3. associate-/r/ 71.5%

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

    4. unpow2 71.5%

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

    5. associate-*l* 78.8%

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

    6. *-commutative 78.8%

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

    7. *-commutative 78.8%

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

    8. associate-*r/ 82.7%

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

15. Simplified 82.7%

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

16. Final simplification 82.7%

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

Alternative 6: 77.7% accurate, 146.2× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.2%

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

3. Simplified 26.6%

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

4. Taylor expanded in angle around 0 52.9%

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

   1. associate-/l* 53.7%

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

   2. associate-*r/ 53.7%

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

   3. unpow2 53.7%

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

   4. unpow2 53.7%

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

   5. times-frac 56.8%

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

   6. unpow2 56.8%

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

   7. unpow2 56.8%

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

6. Simplified 56.8%

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

   1. times-frac 57.2%

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

   2. associate-/l* 61.7%

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

   3. associate-/l* 65.6%

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

8. Applied egg-rr 65.6%

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

   1. unpow2 65.6%

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

   2. associate-/r/ 65.6%

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

   3. associate-/r/ 70.7%

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

   4. unpow2 70.7%

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

10. Simplified 70.7%

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

    1. associate-*r/ 71.3%

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

    2. associate-/l* 80.0%

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

12. Applied egg-rr 80.0%

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

    1. associate-*l/ 82.8%

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

    2. *-commutative 82.8%

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

14. Applied egg-rr 82.8%

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

15. Final simplification 82.8%

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

Alternative 7: 35.0% accurate, 2485.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.2%

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

3. Simplified 27.8%

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

4. Taylor expanded in b around 0 27.6%

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

   1. distribute-rgt-out 27.6%

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

   2. metadata-eval 27.6%

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

   3. mul0-rgt 41.0%

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

6. Simplified 41.0%

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

7. Final simplification 41.0%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023293 
(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))))