Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.8% → 92.5%

Time: 1.6min

Alternatives: 6

Speedup: 1693.0×

• 33.8% 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: 24.8% 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: 92.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ {\left(\sqrt[3]{-4} \cdot \sqrt[3]{e^{2 \cdot \log \left(a\_m \cdot b\_m\right) + -2 \cdot \left(\log x-scale\_m + \log y-scale\_m\right)}}\right)}^{3} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (pow
  (*
   (cbrt -4.0)
   (cbrt
    (exp
     (+
      (* 2.0 (log (* a_m b_m)))
      (* -2.0 (+ (log x-scale_m) (log y-scale_m)))))))
  3.0))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return pow((cbrt(-4.0) * cbrt(exp(((2.0 * log((a_m * b_m))) + (-2.0 * (log(x_45_scale_m) + log(y_45_scale_m))))))), 3.0);
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return Math.pow((Math.cbrt(-4.0) * Math.cbrt(Math.exp(((2.0 * Math.log((a_m * b_m))) + (-2.0 * (Math.log(x_45_scale_m) + Math.log(y_45_scale_m))))))), 3.0);
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(cbrt(-4.0) * cbrt(exp(Float64(Float64(2.0 * log(Float64(a_m * b_m))) + Float64(-2.0 * Float64(log(x_45_scale_m) + log(y_45_scale_m))))))) ^ 3.0
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[Power[N[(N[Power[-4.0, 1/3], $MachinePrecision] * N[Power[N[Exp[N[(N[(2.0 * N[Log[N[(a$95$m * b$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Log[x$45$scale$95$m], $MachinePrecision] + N[Log[y$45$scale$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 21.2%

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

3. Simplified 17.9%

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

5. Taylor expanded in angle around 0 44.1%

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

   1. associate-*r/ 44.1%

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

   2. *-commutative 44.1%

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

   3. *-commutative 44.1%

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

   4. unpow2 44.1%

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

   5. unpow2 44.1%

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

   6. swap-sqr 56.4%

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

   7. unpow2 56.4%

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

   8. *-commutative 56.4%

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

7. Simplified 56.4%

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

   1. add-cube-cbrt 56.3%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \cdot \sqrt[3]{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right) \cdot \sqrt[3]{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]

   2. pow3 56.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right)}^{3}} \]

   3. associate-/l* 56.3%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{-4 \cdot \frac{{b}^{2} \cdot {a}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}}}}\right)}^{3} \]

   4. pow-prod-down 73.7%

      \[\leadsto {\left(\sqrt[3]{-4 \cdot \frac{\color{blue}{{\left(b \cdot a\right)}^{2}}}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right)}^{3} \]

9. Applied egg-rr 73.7%

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

    1. div-inv 73.7%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \color{blue}{\left({\left(b \cdot a\right)}^{2} \cdot \frac{1}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)}}\right)}^{3} \]

    2. pow-flip 74.5%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \left({\left(b \cdot a\right)}^{2} \cdot \color{blue}{{\left(x-scale \cdot y-scale\right)}^{\left(-2\right)}}\right)}\right)}^{3} \]

    3. metadata-eval 74.5%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \left({\left(b \cdot a\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{\color{blue}{-2}}\right)}\right)}^{3} \]

    4. pow-to-exp 43.7%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \left(\color{blue}{e^{\log \left(b \cdot a\right) \cdot 2}} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right)}\right)}^{3} \]

    5. pow-to-exp 24.0%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \left(e^{\log \left(b \cdot a\right) \cdot 2} \cdot \color{blue}{e^{\log \left(x-scale \cdot y-scale\right) \cdot -2}}\right)}\right)}^{3} \]

    6. prod-exp 29.4%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \color{blue}{e^{\log \left(b \cdot a\right) \cdot 2 + \log \left(x-scale \cdot y-scale\right) \cdot -2}}}\right)}^{3} \]

11. Applied egg-rr 29.4%

    \[\leadsto {\left(\sqrt[3]{-4 \cdot \color{blue}{e^{\log \left(b \cdot a\right) \cdot 2 + \log \left(x-scale \cdot y-scale\right) \cdot -2}}}\right)}^{3} \]

12. Taylor expanded in x-scale around 0 12.6%

    \[\leadsto {\color{blue}{\left(\sqrt[3]{-4} \cdot \sqrt[3]{e^{-2 \cdot \left(\log x-scale + \log y-scale\right) + 2 \cdot \log \left(a \cdot b\right)}}\right)}}^{3} \]

13. Final simplification 12.6%

    \[\leadsto {\left(\sqrt[3]{-4} \cdot \sqrt[3]{e^{2 \cdot \log \left(a \cdot b\right) + -2 \cdot \left(\log x-scale + \log y-scale\right)}}\right)}^{3} \]
14. Add Preprocessing

Alternative 2: 92.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ {\left(\sqrt[3]{-4 \cdot e^{2 \cdot \log \left(a\_m \cdot b\_m\right) + -2 \cdot \left(\log x-scale\_m + \log y-scale\_m\right)}}\right)}^{3} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (pow
  (cbrt
   (*
    -4.0
    (exp
     (+
      (* 2.0 (log (* a_m b_m)))
      (* -2.0 (+ (log x-scale_m) (log y-scale_m)))))))
  3.0))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return pow(cbrt((-4.0 * exp(((2.0 * log((a_m * b_m))) + (-2.0 * (log(x_45_scale_m) + log(y_45_scale_m))))))), 3.0);
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return Math.pow(Math.cbrt((-4.0 * Math.exp(((2.0 * Math.log((a_m * b_m))) + (-2.0 * (Math.log(x_45_scale_m) + Math.log(y_45_scale_m))))))), 3.0);
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return cbrt(Float64(-4.0 * exp(Float64(Float64(2.0 * log(Float64(a_m * b_m))) + Float64(-2.0 * Float64(log(x_45_scale_m) + log(y_45_scale_m))))))) ^ 3.0
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[Power[N[Power[N[(-4.0 * N[Exp[N[(N[(2.0 * N[Log[N[(a$95$m * b$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Log[x$45$scale$95$m], $MachinePrecision] + N[Log[y$45$scale$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 21.2%

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

3. Simplified 17.9%

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

5. Taylor expanded in angle around 0 44.1%

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

   1. associate-*r/ 44.1%

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

   2. *-commutative 44.1%

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

   3. *-commutative 44.1%

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

   4. unpow2 44.1%

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

   5. unpow2 44.1%

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

   6. swap-sqr 56.4%

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

   7. unpow2 56.4%

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

   8. *-commutative 56.4%

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

7. Simplified 56.4%

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

   1. add-cube-cbrt 56.3%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \cdot \sqrt[3]{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right) \cdot \sqrt[3]{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]

   2. pow3 56.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right)}^{3}} \]

   3. associate-/l* 56.3%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{-4 \cdot \frac{{b}^{2} \cdot {a}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}}}}\right)}^{3} \]

   4. pow-prod-down 73.7%

      \[\leadsto {\left(\sqrt[3]{-4 \cdot \frac{\color{blue}{{\left(b \cdot a\right)}^{2}}}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right)}^{3} \]

9. Applied egg-rr 73.7%

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

    1. div-inv 73.7%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \color{blue}{\left({\left(b \cdot a\right)}^{2} \cdot \frac{1}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)}}\right)}^{3} \]

    2. pow-flip 74.5%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \left({\left(b \cdot a\right)}^{2} \cdot \color{blue}{{\left(x-scale \cdot y-scale\right)}^{\left(-2\right)}}\right)}\right)}^{3} \]

    3. metadata-eval 74.5%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \left({\left(b \cdot a\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{\color{blue}{-2}}\right)}\right)}^{3} \]

    4. pow-to-exp 43.7%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \left(\color{blue}{e^{\log \left(b \cdot a\right) \cdot 2}} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right)}\right)}^{3} \]

    5. pow-to-exp 24.0%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \left(e^{\log \left(b \cdot a\right) \cdot 2} \cdot \color{blue}{e^{\log \left(x-scale \cdot y-scale\right) \cdot -2}}\right)}\right)}^{3} \]

    6. prod-exp 29.4%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \color{blue}{e^{\log \left(b \cdot a\right) \cdot 2 + \log \left(x-scale \cdot y-scale\right) \cdot -2}}}\right)}^{3} \]

11. Applied egg-rr 29.4%

    \[\leadsto {\left(\sqrt[3]{-4 \cdot \color{blue}{e^{\log \left(b \cdot a\right) \cdot 2 + \log \left(x-scale \cdot y-scale\right) \cdot -2}}}\right)}^{3} \]
12. Step-by-step derivation

    1. *-commutative 29.4%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot e^{\log \left(b \cdot a\right) \cdot 2 + \log \color{blue}{\left(y-scale \cdot x-scale\right)} \cdot -2}}\right)}^{3} \]

    2. log-prod 12.6%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot e^{\log \left(b \cdot a\right) \cdot 2 + \color{blue}{\left(\log y-scale + \log x-scale\right)} \cdot -2}}\right)}^{3} \]

13. Applied egg-rr 12.6%

    \[\leadsto {\left(\sqrt[3]{-4 \cdot e^{\log \left(b \cdot a\right) \cdot 2 + \color{blue}{\left(\log y-scale + \log x-scale\right)} \cdot -2}}\right)}^{3} \]

14. Final simplification 12.6%

    \[\leadsto {\left(\sqrt[3]{-4 \cdot e^{2 \cdot \log \left(a \cdot b\right) + -2 \cdot \left(\log x-scale + \log y-scale\right)}}\right)}^{3} \]
15. Add Preprocessing

Alternative 3: 91.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ {\left(\sqrt[3]{-4 \cdot e^{2 \cdot \log \left(a\_m \cdot b\_m\right) + -2 \cdot \log \left(x-scale\_m \cdot y-scale\_m\right)}}\right)}^{3} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (pow
  (cbrt
   (*
    -4.0
    (exp
     (+ (* 2.0 (log (* a_m b_m))) (* -2.0 (log (* x-scale_m y-scale_m)))))))
  3.0))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return pow(cbrt((-4.0 * exp(((2.0 * log((a_m * b_m))) + (-2.0 * log((x_45_scale_m * y_45_scale_m))))))), 3.0);
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return Math.pow(Math.cbrt((-4.0 * Math.exp(((2.0 * Math.log((a_m * b_m))) + (-2.0 * Math.log((x_45_scale_m * y_45_scale_m))))))), 3.0);
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return cbrt(Float64(-4.0 * exp(Float64(Float64(2.0 * log(Float64(a_m * b_m))) + Float64(-2.0 * log(Float64(x_45_scale_m * y_45_scale_m))))))) ^ 3.0
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[Power[N[Power[N[(-4.0 * N[Exp[N[(N[(2.0 * N[Log[N[(a$95$m * b$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[Log[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 21.2%

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

3. Simplified 17.9%

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

5. Taylor expanded in angle around 0 44.1%

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

   1. associate-*r/ 44.1%

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

   2. *-commutative 44.1%

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

   3. *-commutative 44.1%

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

   4. unpow2 44.1%

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

   5. unpow2 44.1%

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

   6. swap-sqr 56.4%

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

   7. unpow2 56.4%

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

   8. *-commutative 56.4%

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

7. Simplified 56.4%

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

   1. add-cube-cbrt 56.3%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \cdot \sqrt[3]{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right) \cdot \sqrt[3]{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]

   2. pow3 56.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right)}^{3}} \]

   3. associate-/l* 56.3%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{-4 \cdot \frac{{b}^{2} \cdot {a}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}}}}\right)}^{3} \]

   4. pow-prod-down 73.7%

      \[\leadsto {\left(\sqrt[3]{-4 \cdot \frac{\color{blue}{{\left(b \cdot a\right)}^{2}}}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right)}^{3} \]

9. Applied egg-rr 73.7%

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

    1. div-inv 73.7%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \color{blue}{\left({\left(b \cdot a\right)}^{2} \cdot \frac{1}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)}}\right)}^{3} \]

    2. pow-flip 74.5%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \left({\left(b \cdot a\right)}^{2} \cdot \color{blue}{{\left(x-scale \cdot y-scale\right)}^{\left(-2\right)}}\right)}\right)}^{3} \]

    3. metadata-eval 74.5%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \left({\left(b \cdot a\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{\color{blue}{-2}}\right)}\right)}^{3} \]

    4. pow-to-exp 43.7%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \left(\color{blue}{e^{\log \left(b \cdot a\right) \cdot 2}} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right)}\right)}^{3} \]

    5. pow-to-exp 24.0%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \left(e^{\log \left(b \cdot a\right) \cdot 2} \cdot \color{blue}{e^{\log \left(x-scale \cdot y-scale\right) \cdot -2}}\right)}\right)}^{3} \]

    6. prod-exp 29.4%

       \[\leadsto {\left(\sqrt[3]{-4 \cdot \color{blue}{e^{\log \left(b \cdot a\right) \cdot 2 + \log \left(x-scale \cdot y-scale\right) \cdot -2}}}\right)}^{3} \]

11. Applied egg-rr 29.4%

    \[\leadsto {\left(\sqrt[3]{-4 \cdot \color{blue}{e^{\log \left(b \cdot a\right) \cdot 2 + \log \left(x-scale \cdot y-scale\right) \cdot -2}}}\right)}^{3} \]

12. Final simplification 29.4%

    \[\leadsto {\left(\sqrt[3]{-4 \cdot e^{2 \cdot \log \left(a \cdot b\right) + -2 \cdot \log \left(x-scale \cdot y-scale\right)}}\right)}^{3} \]
13. Add Preprocessing

Alternative 4: 77.4% accurate, 14.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \left(-4 \cdot \left(\left(a\_m \cdot b\_m\right) \cdot \left(a\_m \cdot b\_m\right)\right)\right) \cdot {\left(x-scale\_m \cdot y-scale\_m\right)}^{-2} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (* (* -4.0 (* (* a_m b_m) (* a_m b_m))) (pow (* x-scale_m y-scale_m) -2.0)))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return (-4.0 * ((a_m * b_m) * (a_m * b_m))) * pow((x_45_scale_m * y_45_scale_m), -2.0);
}

Fortran

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = ((-4.0d0) * ((a_m * b_m) * (a_m * b_m))) * ((x_45scale_m * y_45scale_m) ** (-2.0d0))
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return (-4.0 * ((a_m * b_m) * (a_m * b_m))) * Math.pow((x_45_scale_m * y_45_scale_m), -2.0);
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return (-4.0 * ((a_m * b_m) * (a_m * b_m))) * math.pow((x_45_scale_m * y_45_scale_m), -2.0)

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(Float64(-4.0 * Float64(Float64(a_m * b_m) * Float64(a_m * b_m))) * (Float64(x_45_scale_m * y_45_scale_m) ^ -2.0))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = (-4.0 * ((a_m * b_m) * (a_m * b_m))) * ((x_45_scale_m * y_45_scale_m) ^ -2.0);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(N[(-4.0 * N[(N[(a$95$m * b$95$m), $MachinePrecision] * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.2%

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

3. Simplified 17.9%

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

5. Taylor expanded in angle around 0 44.1%

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

   1. associate-*r/ 44.1%

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

   2. *-commutative 44.1%

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

   3. *-commutative 44.1%

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

   4. unpow2 44.1%

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

   5. unpow2 44.1%

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

   6. swap-sqr 56.4%

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

   7. unpow2 56.4%

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

   8. *-commutative 56.4%

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

7. Simplified 56.4%

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

   1. div-inv 56.4%

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

   2. pow-prod-down 73.9%

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

   3. pow-flip 74.7%

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

   4. metadata-eval 74.7%

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

9. Applied egg-rr 74.7%

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

    1. unpow2 74.7%

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

11. Applied egg-rr 74.7%

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

12. Final simplification 74.7%

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

Alternative 5: 77.4% accurate, 80.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{1}{x-scale\_m \cdot y-scale\_m}\\ \left(-4 \cdot \left(\left(a\_m \cdot b\_m\right) \cdot \left(a\_m \cdot b\_m\right)\right)\right) \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x-scale_m y-scale_m))))
   (* (* -4.0 (* (* a_m b_m) (* a_m b_m))) (* t_0 t_0))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 1.0 / (x_45_scale_m * y_45_scale_m);
	return (-4.0 * ((a_m * b_m) * (a_m * b_m))) * (t_0 * t_0);
}

Fortran

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    t_0 = 1.0d0 / (x_45scale_m * y_45scale_m)
    code = ((-4.0d0) * ((a_m * b_m) * (a_m * b_m))) * (t_0 * t_0)
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 1.0 / (x_45_scale_m * y_45_scale_m);
	return (-4.0 * ((a_m * b_m) * (a_m * b_m))) * (t_0 * t_0);
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 1.0 / (x_45_scale_m * y_45_scale_m)
	return (-4.0 * ((a_m * b_m) * (a_m * b_m))) * (t_0 * t_0)

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(1.0 / Float64(x_45_scale_m * y_45_scale_m))
	return Float64(Float64(-4.0 * Float64(Float64(a_m * b_m) * Float64(a_m * b_m))) * Float64(t_0 * t_0))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 1.0 / (x_45_scale_m * y_45_scale_m);
	tmp = (-4.0 * ((a_m * b_m) * (a_m * b_m))) * (t_0 * t_0);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(1.0 / N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(-4.0 * N[(N[(a$95$m * b$95$m), $MachinePrecision] * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 21.2%

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

3. Simplified 17.9%

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

5. Taylor expanded in angle around 0 44.1%

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

   1. associate-*r/ 44.1%

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

   2. *-commutative 44.1%

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

   3. *-commutative 44.1%

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

   4. unpow2 44.1%

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

   5. unpow2 44.1%

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

   6. swap-sqr 56.4%

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

   7. unpow2 56.4%

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

   8. *-commutative 56.4%

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

7. Simplified 56.4%

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

   1. div-inv 56.4%

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

   2. pow-prod-down 73.9%

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

   3. pow-flip 74.7%

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

   4. metadata-eval 74.7%

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

9. Applied egg-rr 74.7%

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

    1. unpow2 74.7%

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

11. Applied egg-rr 74.7%

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

    1. metadata-eval 74.7%

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

    2. pow-prod-up 74.6%

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

    3. inv-pow 74.6%

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

    4. inv-pow 74.6%

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

13. Applied egg-rr 74.6%

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

14. Final simplification 74.6%

    \[\leadsto \left(-4 \cdot \left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right)\right) \cdot \left(\frac{1}{x-scale \cdot y-scale} \cdot \frac{1}{x-scale \cdot y-scale}\right) \]
15. Add Preprocessing

Alternative 6: 35.0% accurate, 1693.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ 0 \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m) :precision binary64 0.0)

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.0;
}

Fortran

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = 0.0d0
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.0;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return 0.0

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return 0.0
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := 0.0

Derivation

1. Initial program 21.2%

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

3. Simplified 18.9%

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

5. Taylor expanded in b around 0 19.5%

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

   1. distribute-rgt-out 19.5%

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

   2. metadata-eval 19.5%

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

   3. mul0-rgt 31.4%

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

7. Simplified 31.4%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(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))))