Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 23.4% → 94.2%

Time: 1.8min

Alternatives: 7

Speedup: 2485.0×

• 35.4% 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: 23.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.2% accurate, 21.4× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := \frac{b \cdot a}{y-scale \cdot x-scale}\\ \mathbf{if}\;b \leq 1.7 \cdot 10^{-5}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{1}{\frac{1}{{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* b a) (* y-scale x-scale))))
   (if (<= b 1.7e-5)
     (* -4.0 (* t_0 t_0))
     (* -4.0 (/ 1.0 (/ 1.0 (pow (* (/ a y-scale) (/ b x-scale)) 2.0)))))))

C

b = abs(b);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * a) / (y_45_scale * x_45_scale);
	double tmp;
	if (b <= 1.7e-5) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * (1.0 / (1.0 / pow(((a / y_45_scale) * (b / x_45_scale)), 2.0)));
	}
	return tmp;
}

Fortran

NOTE: b should be positive before calling this function
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 = (b * a) / (y_45scale * x_45scale)
    if (b <= 1.7d-5) then
        tmp = (-4.0d0) * (t_0 * t_0)
    else
        tmp = (-4.0d0) * (1.0d0 / (1.0d0 / (((a / y_45scale) * (b / x_45scale)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

b = Math.abs(b);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * a) / (y_45_scale * x_45_scale);
	double tmp;
	if (b <= 1.7e-5) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * (1.0 / (1.0 / Math.pow(((a / y_45_scale) * (b / x_45_scale)), 2.0)));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b * a) / (y_45_scale * x_45_scale)
	tmp = 0
	if b <= 1.7e-5:
		tmp = -4.0 * (t_0 * t_0)
	else:
		tmp = -4.0 * (1.0 / (1.0 / math.pow(((a / y_45_scale) * (b / x_45_scale)), 2.0)))
	return tmp

Julia

b = abs(b)
function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b * a) / Float64(y_45_scale * x_45_scale))
	tmp = 0.0
	if (b <= 1.7e-5)
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	else
		tmp = Float64(-4.0 * Float64(1.0 / Float64(1.0 / (Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale)) ^ 2.0))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b * a) / (y_45_scale * x_45_scale);
	tmp = 0.0;
	if (b <= 1.7e-5)
		tmp = -4.0 * (t_0 * t_0);
	else
		tmp = -4.0 * (1.0 / (1.0 / (((a / y_45_scale) * (b / x_45_scale)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.7e-5], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(1.0 / N[(1.0 / N[Power[N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.7e-5

   1. Initial program 33.9%

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

   3. Simplified 27.2%

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

   4. Taylor expanded in angle around 0 51.4%

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

      1. *-commutative 51.4%

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

      2. times-frac 52.4%

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

   6. Simplified 52.4%

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

      1. frac-times 51.4%

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

      2. *-commutative 51.4%

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

      3. clear-num 51.4%

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

      4. pow-prod-down 64.2%

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

      5. *-commutative 64.2%

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

      6. pow-prod-down 79.6%

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

   8. Applied egg-rr 79.6%

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

      1. pow-prod-down 64.2%

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

      2. clear-num 64.2%

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

      3. pow-prod-down 79.6%

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

      4. unpow2 79.6%

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

      5. unpow2 79.6%

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

      6. times-frac 97.2%

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

   10. Applied egg-rr 97.2%

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

   if 1.7e-5 < b

   1. Initial program 4.1%

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

   3. Simplified 2.3%

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

   4. Taylor expanded in angle around 0 39.9%

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

      1. *-commutative 39.9%

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

      2. times-frac 41.8%

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

   6. Simplified 41.8%

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

      1. frac-times 39.9%

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

      2. *-commutative 39.9%

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

      3. clear-num 39.9%

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

      4. pow-prod-down 54.1%

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

      5. *-commutative 54.1%

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

      6. pow-prod-down 79.3%

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

   8. Applied egg-rr 79.3%

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

      1. pow-prod-down 54.1%

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

      2. clear-num 54.1%

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

      3. inv-pow 54.1%

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

      4. add-sqr-sqrt 54.0%

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

      5. pow2 54.0%

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

      6. sqrt-div 54.1%

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

      7. pow-prod-down 79.2%

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

      8. unpow2 79.2%

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

      9. sqrt-prod 46.3%

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

      10. add-sqr-sqrt 81.0%

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

      11. sqrt-pow1 86.2%

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

      12. metadata-eval 86.2%

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

      13. pow1 86.2%

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

   10. Applied egg-rr 86.2%

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

       1. unpow-1 86.2%

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

       2. times-frac 91.1%

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

   12. Simplified 91.1%

       \[\leadsto -4 \cdot \frac{1}{\color{blue}{\frac{1}{{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-5}:\\ \;\;\;\;-4 \cdot \left(\frac{b \cdot a}{y-scale \cdot x-scale} \cdot \frac{b \cdot a}{y-scale \cdot x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{1}{\frac{1}{{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}}}\\ \end{array} \]

Alternative 2: 94.2% accurate, 22.2× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := \frac{b \cdot a}{y-scale \cdot x-scale}\\ \mathbf{if}\;b \leq 10^{-5}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* b a) (* y-scale x-scale))))
   (if (<= b 1e-5)
     (* -4.0 (* t_0 t_0))
     (* -4.0 (pow (* (/ a y-scale) (/ b x-scale)) 2.0)))))

C

b = abs(b);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * a) / (y_45_scale * x_45_scale);
	double tmp;
	if (b <= 1e-5) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * pow(((a / y_45_scale) * (b / x_45_scale)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: b should be positive before calling this function
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 = (b * a) / (y_45scale * x_45scale)
    if (b <= 1d-5) then
        tmp = (-4.0d0) * (t_0 * t_0)
    else
        tmp = (-4.0d0) * (((a / y_45scale) * (b / x_45scale)) ** 2.0d0)
    end if
    code = tmp
end function

Java

b = Math.abs(b);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * a) / (y_45_scale * x_45_scale);
	double tmp;
	if (b <= 1e-5) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * Math.pow(((a / y_45_scale) * (b / x_45_scale)), 2.0);
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b * a) / (y_45_scale * x_45_scale)
	tmp = 0
	if b <= 1e-5:
		tmp = -4.0 * (t_0 * t_0)
	else:
		tmp = -4.0 * math.pow(((a / y_45_scale) * (b / x_45_scale)), 2.0)
	return tmp

Julia

b = abs(b)
function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b * a) / Float64(y_45_scale * x_45_scale))
	tmp = 0.0
	if (b <= 1e-5)
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	else
		tmp = Float64(-4.0 * (Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale)) ^ 2.0));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b * a) / (y_45_scale * x_45_scale);
	tmp = 0.0;
	if (b <= 1e-5)
		tmp = -4.0 * (t_0 * t_0);
	else
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1e-5], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.00000000000000008e-5

   1. Initial program 33.9%

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

   3. Simplified 27.2%

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

   4. Taylor expanded in angle around 0 51.4%

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

      1. *-commutative 51.4%

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

      2. times-frac 52.4%

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

   6. Simplified 52.4%

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

      1. frac-times 51.4%

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

      2. *-commutative 51.4%

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

      3. clear-num 51.4%

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

      4. pow-prod-down 64.2%

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

      5. *-commutative 64.2%

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

      6. pow-prod-down 79.6%

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

   8. Applied egg-rr 79.6%

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

      1. pow-prod-down 64.2%

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

      2. clear-num 64.2%

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

      3. pow-prod-down 79.6%

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

      4. unpow2 79.6%

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

      5. unpow2 79.6%

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

      6. times-frac 97.2%

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

   10. Applied egg-rr 97.2%

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

   if 1.00000000000000008e-5 < b

   1. Initial program 4.1%

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

   3. Simplified 2.3%

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

   4. Taylor expanded in angle around 0 39.9%

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

      1. *-commutative 39.9%

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

      2. times-frac 41.8%

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

   6. Simplified 41.8%

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

      1. frac-times 39.9%

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

      2. *-commutative 39.9%

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

      3. clear-num 39.9%

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

      4. pow-prod-down 54.1%

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

      5. *-commutative 54.1%

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

      6. pow-prod-down 79.3%

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

   8. Applied egg-rr 79.3%

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

      1. pow-prod-down 54.1%

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

      2. clear-num 54.1%

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

      3. div-inv 54.0%

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

      4. pow-flip 54.0%

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

      5. metadata-eval 54.0%

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

      6. expm1-log1p-u 53.7%

         \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({a}^{2} \cdot {b}^{2}\right) \cdot {\left(y-scale \cdot x-scale\right)}^{-2}\right)\right)} \]

      7. pow-prod-down 78.1%

         \[\leadsto -4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(a \cdot b\right)}^{2}} \cdot {\left(y-scale \cdot x-scale\right)}^{-2}\right)\right) \]

      8. expm1-udef 65.2%

         \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(a \cdot b\right)}^{2} \cdot {\left(y-scale \cdot x-scale\right)}^{-2}\right)} - 1\right)} \]

   10. Applied egg-rr 70.3%

       \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{a \cdot b}{y-scale \cdot x-scale}\right)}^{2}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 84.9%

          \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{a \cdot b}{y-scale \cdot x-scale}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 86.2%

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

       3. times-frac 91.1%

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

   12. Simplified 91.1%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-5}:\\ \;\;\;\;-4 \cdot \left(\frac{b \cdot a}{y-scale \cdot x-scale} \cdot \frac{b \cdot a}{y-scale \cdot x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}\\ \end{array} \]

Alternative 3: 94.2% accurate, 22.2× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := \frac{b \cdot a}{y-scale \cdot x-scale}\\ \mathbf{if}\;b \leq 35000:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{y-scale}{a} \cdot \frac{x-scale}{b}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* b a) (* y-scale x-scale))))
   (if (<= b 35000.0)
     (* -4.0 (* t_0 t_0))
     (* -4.0 (pow (* (/ y-scale a) (/ x-scale b)) -2.0)))))

C

b = abs(b);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * a) / (y_45_scale * x_45_scale);
	double tmp;
	if (b <= 35000.0) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * pow(((y_45_scale / a) * (x_45_scale / b)), -2.0);
	}
	return tmp;
}

Fortran

NOTE: b should be positive before calling this function
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 = (b * a) / (y_45scale * x_45scale)
    if (b <= 35000.0d0) then
        tmp = (-4.0d0) * (t_0 * t_0)
    else
        tmp = (-4.0d0) * (((y_45scale / a) * (x_45scale / b)) ** (-2.0d0))
    end if
    code = tmp
end function

Java

b = Math.abs(b);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * a) / (y_45_scale * x_45_scale);
	double tmp;
	if (b <= 35000.0) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * Math.pow(((y_45_scale / a) * (x_45_scale / b)), -2.0);
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b * a) / (y_45_scale * x_45_scale)
	tmp = 0
	if b <= 35000.0:
		tmp = -4.0 * (t_0 * t_0)
	else:
		tmp = -4.0 * math.pow(((y_45_scale / a) * (x_45_scale / b)), -2.0)
	return tmp

Julia

b = abs(b)
function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b * a) / Float64(y_45_scale * x_45_scale))
	tmp = 0.0
	if (b <= 35000.0)
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	else
		tmp = Float64(-4.0 * (Float64(Float64(y_45_scale / a) * Float64(x_45_scale / b)) ^ -2.0));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b * a) / (y_45_scale * x_45_scale);
	tmp = 0.0;
	if (b <= 35000.0)
		tmp = -4.0 * (t_0 * t_0);
	else
		tmp = -4.0 * (((y_45_scale / a) * (x_45_scale / b)) ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 35000.0], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(N[(y$45$scale / a), $MachinePrecision] * N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 35000

   1. Initial program 34.2%

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

   3. Simplified 27.5%

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

   4. Taylor expanded in angle around 0 51.6%

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

      1. *-commutative 51.6%

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

      2. times-frac 52.7%

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

   6. Simplified 52.7%

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

      1. frac-times 51.6%

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

      2. *-commutative 51.6%

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

      3. clear-num 51.6%

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

      4. pow-prod-down 64.4%

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

      5. *-commutative 64.4%

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

      6. pow-prod-down 79.7%

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

   8. Applied egg-rr 79.7%

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

      1. pow-prod-down 64.4%

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

      2. clear-num 64.4%

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

      3. pow-prod-down 79.7%

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

      4. unpow2 79.7%

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

      5. unpow2 79.7%

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

      6. times-frac 97.3%

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

   10. Applied egg-rr 97.3%

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

   if 35000 < b

   1. Initial program 2.4%

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

   3. Simplified 0.6%

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

   4. Taylor expanded in angle around 0 38.9%

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

      1. *-commutative 38.9%

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

      2. times-frac 40.8%

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

   6. Simplified 40.8%

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

      1. frac-times 38.9%

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

      2. *-commutative 38.9%

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

      3. clear-num 38.8%

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

      4. pow-prod-down 53.3%

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

      5. *-commutative 53.3%

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

      6. pow-prod-down 78.9%

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

   8. Applied egg-rr 78.9%

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

      1. inv-pow 78.9%

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

      2. pow-prod-down 53.3%

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

      3. add-sqr-sqrt 53.3%

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

      4. unpow-prod-down 53.3%

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

      5. sqrt-div 53.2%

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

      6. sqrt-pow1 30.5%

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

      7. metadata-eval 30.5%

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

      8. pow1 30.5%

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

      9. pow-prod-down 30.5%

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

      10. unpow2 30.5%

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

      11. sqrt-prod 19.9%

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

      12. add-sqr-sqrt 32.3%

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

   10. Applied egg-rr 85.9%

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

       1. pow-sqr 86.1%

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

       2. times-frac 91.0%

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

       3. metadata-eval 91.0%

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

   12. Simplified 91.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 35000:\\ \;\;\;\;-4 \cdot \left(\frac{b \cdot a}{y-scale \cdot x-scale} \cdot \frac{b \cdot a}{y-scale \cdot x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{y-scale}{a} \cdot \frac{x-scale}{b}\right)}^{-2}\\ \end{array} \]

Alternative 4: 90.3% accurate, 146.2× speedup

Math

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

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* -4.0 (* (/ a y-scale) (* (/ b x-scale) (/ (/ a (/ x-scale b)) y-scale)))))

C

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

Fortran

NOTE: b should be positive before calling this function
real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (-4.0d0) * ((a / y_45scale) * ((b / x_45scale) * ((a / (x_45scale / b)) / y_45scale)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(-4.0 * N[(N[(a / y$45$scale), $MachinePrecision] * N[(N[(b / x$45$scale), $MachinePrecision] * N[(N[(a / N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.1%

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

3. Simplified 21.6%

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

4. Taylor expanded in angle around 0 48.8%

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

   1. *-commutative 48.8%

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

   2. times-frac 50.0%

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

6. Simplified 50.0%

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

   1. frac-times 48.8%

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

   2. *-commutative 48.8%

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

   3. clear-num 48.8%

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

   4. pow-prod-down 61.9%

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

   5. *-commutative 61.9%

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

   6. pow-prod-down 79.5%

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

8. Applied egg-rr 79.5%

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

   1. unpow2 79.5%

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

   2. unpow2 79.5%

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

   3. times-frac 94.7%

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

10. Applied egg-rr 94.7%

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

    1. metadata-eval 94.7%

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

    2. frac-times 94.7%

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

    3. clear-num 94.8%

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

    4. frac-times 89.5%

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

    5. clear-num 89.5%

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

    6. frac-times 94.3%

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

    7. associate-*l* 91.8%

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

    8. associate-*l/ 90.0%

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

    9. clear-num 90.0%

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

    10. un-div-inv 90.0%

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

12. Applied egg-rr 90.0%

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

13. Final simplification 90.0%

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

Alternative 5: 92.5% accurate, 146.2× speedup

Math

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

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* -4.0 (* (/ (* b a) (* y-scale x-scale)) (* a (/ b (* y-scale x-scale))))))

C

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

Fortran

NOTE: b should be positive before calling this function
real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (-4.0d0) * (((b * a) / (y_45scale * x_45scale)) * (a * (b / (y_45scale * x_45scale))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(-4.0 * N[(N[(N[(b * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.1%

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

3. Simplified 21.6%

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

4. Taylor expanded in angle around 0 48.8%

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

   1. *-commutative 48.8%

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

   2. times-frac 50.0%

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

6. Simplified 50.0%

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

   1. frac-times 48.8%

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

   2. *-commutative 48.8%

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

   3. clear-num 48.8%

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

   4. pow-prod-down 61.9%

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

   5. *-commutative 61.9%

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

   6. pow-prod-down 79.5%

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

8. Applied egg-rr 79.5%

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

   1. pow-prod-down 61.9%

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

   2. clear-num 61.9%

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

   3. pow-prod-down 79.5%

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

   4. unpow2 79.5%

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

   5. unpow2 79.5%

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

   6. times-frac 94.7%

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

10. Applied egg-rr 94.7%

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

11. Taylor expanded in a around 0 94.7%

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

    1. *-commutative 94.7%

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

    2. associate-*r/ 92.9%

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

    3. *-commutative 92.9%

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

13. Simplified 92.9%

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

14. Final simplification 92.9%

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

Alternative 6: 94.0% accurate, 146.2× speedup

Math

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

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* b a) (* y-scale x-scale)))) (* -4.0 (* t_0 t_0))))

C

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

Fortran

NOTE: b should be positive before calling this function
real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = (b * a) / (y_45scale * x_45scale)
    code = (-4.0d0) * (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 27.1%

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

3. Simplified 21.6%

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

4. Taylor expanded in angle around 0 48.8%

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

   1. *-commutative 48.8%

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

   2. times-frac 50.0%

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

6. Simplified 50.0%

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

   1. frac-times 48.8%

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

   2. *-commutative 48.8%

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

   3. clear-num 48.8%

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

   4. pow-prod-down 61.9%

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

   5. *-commutative 61.9%

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

   6. pow-prod-down 79.5%

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

8. Applied egg-rr 79.5%

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

   1. pow-prod-down 61.9%

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

   2. clear-num 61.9%

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

   3. pow-prod-down 79.5%

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

   4. unpow2 79.5%

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

   5. unpow2 79.5%

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

   6. times-frac 94.7%

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

10. Applied egg-rr 94.7%

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

11. Final simplification 94.7%

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

Alternative 7: 34.1% accurate, 2485.0× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ 0 \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle x-scale y-scale) :precision binary64 0.0)

C

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

Fortran

NOTE: b should be positive before calling this function
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 27.1%

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

3. Simplified 24.8%

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

4. Taylor expanded in b around 0 23.8%

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

   1. distribute-rgt-out 23.8%

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

   2. metadata-eval 23.8%

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

   3. mul0-rgt 35.4%

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

6. Simplified 35.4%

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

7. Final simplification 35.4%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023308 
(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))))