Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.5% → 93.9%

Time: 1.8min

Alternatives: 7

Speedup: 1693.0×

• 34.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.5% 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: 93.9% accurate, 15.4× 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 25.6%

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

3. Simplified 22.5%

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

5. Taylor expanded in angle around 0 52.8%

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

   1. *-commutative 52.8%

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

7. Simplified 52.8%

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

   1. pow-prod-down 62.3%

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

   2. add-sqr-sqrt 62.3%

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

   3. pow2 62.3%

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

   4. *-commutative 62.3%

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

   5. pow-prod-down 78.4%

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

9. Applied egg-rr 78.4%

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

10. Taylor expanded in a around 0 94.9%

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

    1. times-frac 96.2%

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

12. Simplified 96.2%

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

13. Final simplification 96.2%

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

Alternative 2: 89.7% accurate, 76.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 1.05 \cdot 10^{+188}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \frac{b}{y-scale}}{x-scale \cdot \left(\frac{x-scale}{a} \cdot \frac{y-scale}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{\frac{x-scale}{a}} \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}{y-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale 1.05e+188)
   (* -4.0 (/ (* a (/ b y-scale)) (* x-scale (* (/ x-scale a) (/ y-scale b)))))
   (*
    -4.0
    (/ (* (/ b (/ x-scale a)) (* (/ 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) {
	double tmp;
	if (x_45_scale <= 1.05e+188) {
		tmp = -4.0 * ((a * (b / y_45_scale)) / (x_45_scale * ((x_45_scale / a) * (y_45_scale / b))));
	} else {
		tmp = -4.0 * (((b / (x_45_scale / a)) * ((a / y_45_scale) * (b / x_45_scale))) / y_45_scale);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (x_45scale <= 1.05d+188) then
        tmp = (-4.0d0) * ((a * (b / y_45scale)) / (x_45scale * ((x_45scale / a) * (y_45scale / b))))
    else
        tmp = (-4.0d0) * (((b / (x_45scale / a)) * ((a / y_45scale) * (b / x_45scale))) / y_45scale)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 1.05e+188) {
		tmp = -4.0 * ((a * (b / y_45_scale)) / (x_45_scale * ((x_45_scale / a) * (y_45_scale / b))));
	} else {
		tmp = -4.0 * (((b / (x_45_scale / a)) * ((a / y_45_scale) * (b / x_45_scale))) / y_45_scale);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if x_45_scale <= 1.05e+188:
		tmp = -4.0 * ((a * (b / y_45_scale)) / (x_45_scale * ((x_45_scale / a) * (y_45_scale / b))))
	else:
		tmp = -4.0 * (((b / (x_45_scale / a)) * ((a / y_45_scale) * (b / x_45_scale))) / y_45_scale)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (x_45_scale <= 1.05e+188)
		tmp = Float64(-4.0 * Float64(Float64(a * Float64(b / y_45_scale)) / Float64(x_45_scale * Float64(Float64(x_45_scale / a) * Float64(y_45_scale / b)))));
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(b / Float64(x_45_scale / a)) * Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale))) / y_45_scale));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (x_45_scale <= 1.05e+188)
		tmp = -4.0 * ((a * (b / y_45_scale)) / (x_45_scale * ((x_45_scale / a) * (y_45_scale / b))));
	else
		tmp = -4.0 * (((b / (x_45_scale / a)) * ((a / y_45_scale) * (b / x_45_scale))) / y_45_scale);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[x$45$scale, 1.05e+188], N[(-4.0 * N[(N[(a * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(N[(x$45$scale / a), $MachinePrecision] * N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[(b / N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision] * N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.04999999999999993e188

   1. Initial program 25.0%

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

   3. Simplified 20.9%

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

   5. Taylor expanded in angle around 0 54.6%

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

      1. *-commutative 54.6%

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

   7. Simplified 54.6%

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

      1. pow-prod-down 63.0%

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

      2. add-sqr-sqrt 62.9%

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

      3. pow2 62.9%

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

      4. *-commutative 62.9%

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

      5. pow-prod-down 79.0%

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

   9. Applied egg-rr 79.0%

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

   10. Taylor expanded in a around 0 95.9%

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

       1. times-frac 96.7%

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

   12. Simplified 96.7%

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

       1. unpow2 96.7%

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

       2. frac-times 93.1%

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

       3. clear-num 93.1%

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

       4. associate-*l/ 93.2%

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

       5. frac-times 91.2%

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

       6. *-un-lft-identity 91.2%

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

       7. times-frac 93.5%

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

   14. Applied egg-rr 93.5%

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

   if 1.04999999999999993e188 < x-scale

   1. Initial program 31.8%

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

   3. Simplified 36.8%

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

   5. Taylor expanded in angle around 0 36.0%

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

      1. *-commutative 36.0%

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

   7. Simplified 36.0%

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

      1. pow-prod-down 56.0%

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

      2. add-sqr-sqrt 56.0%

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

      3. pow2 56.0%

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

      4. *-commutative 56.0%

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

      5. pow-prod-down 72.3%

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

   9. Applied egg-rr 72.3%

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

   10. Taylor expanded in a around 0 85.3%

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

       1. times-frac 91.7%

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

   12. Simplified 91.7%

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

       1. unpow2 91.7%

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

       2. associate-*r/ 91.7%

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

       3. associate-*l/ 87.9%

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

       4. clear-num 87.9%

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

       5. associate-*l/ 88.1%

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

       6. *-un-lft-identity 88.1%

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

       7. frac-times 81.6%

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

       8. *-commutative 81.6%

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

       9. times-frac 96.3%

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

   14. Applied egg-rr 96.3%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.05 \cdot 10^{+188}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \frac{b}{y-scale}}{x-scale \cdot \left(\frac{x-scale}{a} \cdot \frac{y-scale}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{\frac{x-scale}{a}} \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}{y-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 99.6× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* -4.0 (* (/ b y-scale) (* (/ a x-scale) (* (/ a 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 * ((b / y_45_scale) * ((a / x_45_scale) * ((a / 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) * ((b / y_45scale) * ((a / x_45scale) * ((a / 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 * ((b / y_45_scale) * ((a / x_45_scale) * ((a / y_45_scale) * (b / x_45_scale))));
}

Python

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

Wolfram

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

Derivation

1. Initial program 25.6%

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

3. Simplified 22.5%

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

5. Taylor expanded in angle around 0 52.8%

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

   1. *-commutative 52.8%

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

7. Simplified 52.8%

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

   1. pow-prod-down 62.3%

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

   2. add-sqr-sqrt 62.3%

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

   3. pow2 62.3%

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

   4. *-commutative 62.3%

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

   5. pow-prod-down 78.4%

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

9. Applied egg-rr 78.4%

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

10. Taylor expanded in a around 0 94.9%

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

    1. times-frac 96.2%

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

12. Simplified 96.2%

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

    1. unpow2 96.2%

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

    2. *-commutative 96.2%

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

    3. associate-*l* 93.1%

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

    4. frac-times 88.5%

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

    5. *-commutative 88.5%

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

    6. times-frac 88.7%

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

14. Applied egg-rr 88.7%

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

15. Final simplification 88.7%

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

Alternative 4: 87.1% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.6%

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

3. Simplified 22.5%

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

5. Taylor expanded in angle around 0 52.8%

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

   1. *-commutative 52.8%

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

7. Simplified 52.8%

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

   1. pow-prod-down 62.3%

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

   2. add-sqr-sqrt 62.3%

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

   3. pow2 62.3%

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

   4. *-commutative 62.3%

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

   5. pow-prod-down 78.4%

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

9. Applied egg-rr 78.4%

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

10. Taylor expanded in a around 0 94.9%

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

    1. times-frac 96.2%

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

12. Simplified 96.2%

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

    1. frac-times 94.9%

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

    2. unpow2 94.9%

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

    3. clear-num 94.9%

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

    4. clear-num 94.9%

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

    5. clear-num 94.9%

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

    6. associate-/l* 92.9%

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

    7. add-sqr-sqrt 38.3%

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

    8. sqrt-prod 52.5%

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

    9. unpow2 52.5%

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

    10. add-sqr-sqrt 23.6%

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

    11. sqrt-prod 45.9%

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

    12. unpow2 45.9%

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

    13. sqrt-div 45.2%

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

    14. frac-times 45.2%

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

    15. *-un-lft-identity 45.2%

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

    16. times-frac 45.1%

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

    17. sqrt-div 45.5%

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

14. Applied egg-rr 90.1%

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

15. Final simplification 90.1%

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

Alternative 5: 89.4% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

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 / y_45scale)) / (x_45scale * ((x_45scale / a) * (y_45scale / b))))
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 / y_45_scale)) / (x_45_scale * ((x_45_scale / a) * (y_45_scale / b))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.6%

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

3. Simplified 22.5%

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

5. Taylor expanded in angle around 0 52.8%

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

   1. *-commutative 52.8%

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

7. Simplified 52.8%

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

   1. pow-prod-down 62.3%

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

   2. add-sqr-sqrt 62.3%

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

   3. pow2 62.3%

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

   4. *-commutative 62.3%

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

   5. pow-prod-down 78.4%

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

9. Applied egg-rr 78.4%

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

10. Taylor expanded in a around 0 94.9%

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

    1. times-frac 96.2%

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

12. Simplified 96.2%

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

    1. unpow2 96.2%

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

    2. frac-times 91.6%

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

    3. clear-num 91.5%

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

    4. associate-*l/ 91.7%

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

    5. frac-times 89.5%

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

    6. *-un-lft-identity 89.5%

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

    7. times-frac 91.9%

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

14. Applied egg-rr 91.9%

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

15. Final simplification 91.9%

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

Alternative 6: 89.4% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.6%

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

3. Simplified 22.5%

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

5. Taylor expanded in angle around 0 52.8%

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

   1. *-commutative 52.8%

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

7. Simplified 52.8%

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

   1. pow-prod-down 62.3%

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

   2. add-sqr-sqrt 62.3%

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

   3. pow2 62.3%

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

   4. *-commutative 62.3%

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

   5. pow-prod-down 78.4%

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

9. Applied egg-rr 78.4%

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

10. Taylor expanded in a around 0 94.9%

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

    1. times-frac 96.2%

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

12. Simplified 96.2%

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

    1. unpow2 96.2%

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

    2. frac-times 91.6%

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

    3. clear-num 91.5%

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

    4. associate-*r/ 90.9%

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

    5. frac-times 90.5%

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

    6. *-un-lft-identity 90.5%

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

    7. clear-num 90.5%

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

    8. associate-*l/ 91.0%

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

    9. *-un-lft-identity 91.0%

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

    10. times-frac 94.5%

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

14. Applied egg-rr 94.5%

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

15. Final simplification 94.5%

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

Alternative 7: 34.8% accurate, 1693.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.6%

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

3. Simplified 21.9%

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

5. Taylor expanded in b around 0 23.9%

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

   1. distribute-rgt-out 23.9%

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

   2. metadata-eval 23.9%

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

   3. mul0-rgt 34.5%

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

7. Simplified 34.5%

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

8. Final simplification 34.5%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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))))