Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.3% → 94.6%

Time: 1.8min

Alternatives: 6

Speedup: 2485.0×

• 33.9% 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: 25.3% 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: 94.6% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (-4.0d0) * (((a / x_45scale) * (b / y_45scale)) ** 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.6%

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

3. Simplified 20.5%

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

4. Taylor expanded in angle around 0 53.5%

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

   1. add-sqr-sqrt 53.5%

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

   2. pow2 53.5%

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

   3. div-inv 53.5%

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

   4. pow-prod-down 66.4%

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

   5. pow-prod-down 82.6%

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

   6. pow-flip 83.0%

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

   7. metadata-eval 83.0%

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

6. Applied egg-rr 83.0%

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

7. Taylor expanded in a around 0 92.8%

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

   1. times-frac 95.3%

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

9. Simplified 95.3%

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

10. Final simplification 95.3%

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

Alternative 2: 90.2% accurate, 130.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{y-scale} \cdot \frac{b}{x-scale}\\ \mathbf{if}\;a \leq 1.45 \cdot 10^{-229}:\\ \;\;\;\;-4 \cdot \frac{t_0 \cdot \frac{a}{\frac{x-scale}{b}}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t_0}{\frac{y-scale}{\frac{b}{x-scale}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ a y-scale) (/ b x-scale))))
   (if (<= a 1.45e-229)
     (* -4.0 (/ (* t_0 (/ a (/ x-scale b))) y-scale))
     (* -4.0 (/ (* a t_0) (/ y-scale (/ b x-scale)))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a / y_45_scale) * (b / x_45_scale);
	double tmp;
	if (a <= 1.45e-229) {
		tmp = -4.0 * ((t_0 * (a / (x_45_scale / b))) / y_45_scale);
	} else {
		tmp = -4.0 * ((a * t_0) / (y_45_scale / (b / x_45_scale)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a / y_45scale) * (b / x_45scale)
    if (a <= 1.45d-229) then
        tmp = (-4.0d0) * ((t_0 * (a / (x_45scale / b))) / y_45scale)
    else
        tmp = (-4.0d0) * ((a * t_0) / (y_45scale / (b / x_45scale)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a / y_45_scale) * (b / x_45_scale);
	double tmp;
	if (a <= 1.45e-229) {
		tmp = -4.0 * ((t_0 * (a / (x_45_scale / b))) / y_45_scale);
	} else {
		tmp = -4.0 * ((a * t_0) / (y_45_scale / (b / x_45_scale)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (a / y_45_scale) * (b / x_45_scale)
	tmp = 0
	if a <= 1.45e-229:
		tmp = -4.0 * ((t_0 * (a / (x_45_scale / b))) / y_45_scale)
	else:
		tmp = -4.0 * ((a * t_0) / (y_45_scale / (b / x_45_scale)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale))
	tmp = 0.0
	if (a <= 1.45e-229)
		tmp = Float64(-4.0 * Float64(Float64(t_0 * Float64(a / Float64(x_45_scale / b))) / y_45_scale));
	else
		tmp = Float64(-4.0 * Float64(Float64(a * t_0) / Float64(y_45_scale / Float64(b / x_45_scale))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (a / y_45_scale) * (b / x_45_scale);
	tmp = 0.0;
	if (a <= 1.45e-229)
		tmp = -4.0 * ((t_0 * (a / (x_45_scale / b))) / y_45_scale);
	else
		tmp = -4.0 * ((a * t_0) / (y_45_scale / (b / x_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.45e-229], N[(-4.0 * N[(N[(t$95$0 * N[(a / N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(a * t$95$0), $MachinePrecision] / N[(y$45$scale / N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.45e-229

   1. Initial program 23.7%

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

   3. Simplified 21.7%

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

   4. Taylor expanded in angle around 0 55.3%

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

      1. add-sqr-sqrt 55.3%

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

      2. pow2 55.3%

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

      3. div-inv 55.3%

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

      4. pow-prod-down 68.0%

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

      5. pow-prod-down 83.3%

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

      6. pow-flip 83.3%

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

      7. metadata-eval 83.3%

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

   6. Applied egg-rr 83.3%

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

   7. Taylor expanded in a around 0 93.5%

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

      1. times-frac 95.2%

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

   9. Simplified 95.2%

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

       1. unpow2 95.2%

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

       2. associate-*r/ 95.2%

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

       3. associate-*l/ 93.8%

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

       4. associate-*l/ 89.4%

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

       5. associate-/l* 90.0%

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

       6. frac-times 87.1%

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

       7. *-commutative 87.1%

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

       8. times-frac 92.6%

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

   11. Applied egg-rr 92.6%

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

   if 1.45e-229 < a

   1. Initial program 21.0%

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

   3. Simplified 18.9%

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

   4. Taylor expanded in angle around 0 51.0%

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

      1. add-sqr-sqrt 51.0%

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

      2. pow2 51.0%

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

      3. div-inv 51.0%

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

      4. pow-prod-down 64.0%

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

      5. pow-prod-down 81.5%

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

      6. pow-flip 82.5%

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

      7. metadata-eval 82.5%

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

   6. Applied egg-rr 82.5%

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

   7. Taylor expanded in a around 0 91.8%

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

      1. times-frac 95.4%

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

   9. Simplified 95.4%

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

       1. unpow2 95.4%

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

       2. frac-times 87.7%

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

       3. associate-/l* 88.7%

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

       4. associate-*r/ 87.7%

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

       5. frac-times 89.9%

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

       6. *-commutative 89.9%

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

       7. times-frac 86.5%

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

       8. *-commutative 86.5%

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

       9. associate-/l* 92.3%

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

   11. Applied egg-rr 92.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.45 \cdot 10^{-229}:\\ \;\;\;\;-4 \cdot \frac{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \frac{a}{\frac{x-scale}{b}}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}{\frac{y-scale}{\frac{b}{x-scale}}}\\ \end{array} \]

Alternative 3: 86.2% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.6%

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

3. Simplified 20.5%

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

4. Taylor expanded in angle around 0 53.5%

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

   1. add-sqr-sqrt 53.5%

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

   2. pow2 53.5%

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

   3. div-inv 53.5%

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

   4. pow-prod-down 66.4%

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

   5. pow-prod-down 82.6%

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

   6. pow-flip 83.0%

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

   7. metadata-eval 83.0%

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

6. Applied egg-rr 83.0%

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

7. Taylor expanded in a around 0 92.8%

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

   1. times-frac 95.3%

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

9. Simplified 95.3%

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

    1. frac-times 92.8%

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

    2. unpow2 92.8%

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

    3. clear-num 92.8%

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

    4. *-commutative 92.8%

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

    5. frac-times 90.8%

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

    6. *-un-lft-identity 90.8%

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

    7. times-frac 87.3%

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

    8. *-commutative 87.3%

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

11. Applied egg-rr 87.3%

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

12. Final simplification 87.3%

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

Alternative 4: 89.9% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.6%

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

3. Simplified 20.5%

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

4. Taylor expanded in angle around 0 53.5%

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

   1. add-sqr-sqrt 53.5%

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

   2. pow2 53.5%

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

   3. div-inv 53.5%

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

   4. pow-prod-down 66.4%

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

   5. pow-prod-down 82.6%

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

   6. pow-flip 83.0%

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

   7. metadata-eval 83.0%

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

6. Applied egg-rr 83.0%

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

7. Taylor expanded in a around 0 92.8%

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

   1. times-frac 95.3%

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

9. Simplified 95.3%

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

    1. unpow2 95.3%

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

    2. frac-times 88.5%

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

    3. associate-/l* 89.0%

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

    4. associate-*r/ 87.6%

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

    5. frac-times 89.6%

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

    6. *-commutative 89.6%

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

    7. times-frac 87.1%

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

    8. *-commutative 87.1%

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

    9. associate-/l* 91.0%

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

11. Applied egg-rr 91.0%

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

12. Final simplification 91.0%

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

Alternative 5: 94.1% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.6%

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

3. Simplified 20.5%

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

4. Taylor expanded in angle around 0 53.5%

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

   1. associate-*r/ 53.5%

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

   2. *-commutative 53.5%

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

   3. *-commutative 53.5%

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

   4. pow-prod-down 66.3%

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

   5. *-commutative 66.3%

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

   6. pow-prod-down 82.9%

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

6. Applied egg-rr 82.9%

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

   1. *-commutative 82.9%

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

   2. pow-prod-down 66.4%

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

8. Applied egg-rr 66.4%

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

   1. *-commutative 66.4%

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

   2. unpow-prod-down 82.9%

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

   3. *-un-lft-identity 82.9%

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

   4. times-frac 82.9%

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

   5. metadata-eval 82.9%

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

   6. unpow2 82.9%

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

   7. unpow2 82.9%

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

   8. frac-times 92.8%

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

   9. *-commutative 92.8%

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

   10. *-commutative 92.8%

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

   11. unpow2 92.8%

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

   12. *-commutative 92.8%

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

   13. frac-times 95.3%

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

   14. unpow2 95.3%

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

   15. associate-*l* 95.3%

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

10. Applied egg-rr 94.7%

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

11. Final simplification 94.7%

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

Alternative 6: 35.1% 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 22.6%

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

3. Simplified 22.7%

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

4. Taylor expanded in b around 0 22.7%

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

   1. distribute-rgt-out 22.7%

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

   2. metadata-eval 22.7%

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

   3. mul0-rgt 30.3%

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

6. Simplified 30.3%

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

7. Final simplification 30.3%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023318 
(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))))