Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.7% → 93.7%

Time: 36.5s

Alternatives: 6

Speedup: 1693.0×

• 33.6% 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: 24.7% 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: 93.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 8.5 \cdot 10^{-151}:\\ \;\;\;\;-4 \cdot {\left(\frac{a\_m \cdot \frac{b\_m}{x-scale\_m}}{y-scale\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left({\left(\sqrt{\frac{b\_m}{y-scale\_m} \cdot \frac{a\_m}{x-scale\_m}}\right)}^{2}\right)}^{2}\\ \end{array} \end{array} \]

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
 (if (<= b_m 8.5e-151)
   (* -4.0 (pow (/ (* a_m (/ b_m x-scale_m)) y-scale_m) 2.0))
   (*
    -4.0
    (pow (pow (sqrt (* (/ b_m y-scale_m) (/ a_m x-scale_m))) 2.0) 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) {
	double tmp;
	if (b_m <= 8.5e-151) {
		tmp = -4.0 * pow(((a_m * (b_m / x_45_scale_m)) / y_45_scale_m), 2.0);
	} else {
		tmp = -4.0 * pow(pow(sqrt(((b_m / y_45_scale_m) * (a_m / x_45_scale_m))), 2.0), 2.0);
	}
	return tmp;
}

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) :: tmp
    if (b_m <= 8.5d-151) then
        tmp = (-4.0d0) * (((a_m * (b_m / x_45scale_m)) / y_45scale_m) ** 2.0d0)
    else
        tmp = (-4.0d0) * ((sqrt(((b_m / y_45scale_m) * (a_m / x_45scale_m))) ** 2.0d0) ** 2.0d0)
    end if
    code = tmp
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 tmp;
	if (b_m <= 8.5e-151) {
		tmp = -4.0 * Math.pow(((a_m * (b_m / x_45_scale_m)) / y_45_scale_m), 2.0);
	} else {
		tmp = -4.0 * Math.pow(Math.pow(Math.sqrt(((b_m / y_45_scale_m) * (a_m / x_45_scale_m))), 2.0), 2.0);
	}
	return tmp;
}

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):
	tmp = 0
	if b_m <= 8.5e-151:
		tmp = -4.0 * math.pow(((a_m * (b_m / x_45_scale_m)) / y_45_scale_m), 2.0)
	else:
		tmp = -4.0 * math.pow(math.pow(math.sqrt(((b_m / y_45_scale_m) * (a_m / x_45_scale_m))), 2.0), 2.0)
	return tmp

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)
	tmp = 0.0
	if (b_m <= 8.5e-151)
		tmp = Float64(-4.0 * (Float64(Float64(a_m * Float64(b_m / x_45_scale_m)) / y_45_scale_m) ^ 2.0));
	else
		tmp = Float64(-4.0 * ((sqrt(Float64(Float64(b_m / y_45_scale_m) * Float64(a_m / x_45_scale_m))) ^ 2.0) ^ 2.0));
	end
	return tmp
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_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b_m <= 8.5e-151)
		tmp = -4.0 * (((a_m * (b_m / x_45_scale_m)) / y_45_scale_m) ^ 2.0);
	else
		tmp = -4.0 * ((sqrt(((b_m / y_45_scale_m) * (a_m / x_45_scale_m))) ^ 2.0) ^ 2.0);
	end
	tmp_2 = tmp;
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_] := If[LessEqual[b$95$m, 8.5e-151], N[(-4.0 * N[Power[N[(N[(a$95$m * N[(b$95$m / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[Power[N[Sqrt[N[(N[(b$95$m / y$45$scale$95$m), $MachinePrecision] * N[(a$95$m / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.49999999999999999e-151

   1. Initial program 30.2%

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

   3. Simplified 26.3%

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

   5. Taylor expanded in angle around 0 41.8%

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

      1. *-commutative 41.8%

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

      2. unpow2 41.8%

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

      3. unpow2 41.8%

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

      4. swap-sqr 59.8%

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

      5. unpow2 59.8%

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

      6. *-commutative 59.8%

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

      7. unpow2 59.8%

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

      8. unpow2 59.8%

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

      9. swap-sqr 77.0%

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

      10. unpow2 77.0%

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

   7. Simplified 77.0%

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

      1. pow1 77.0%

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

      2. div-inv 77.0%

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

      3. *-commutative 77.0%

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

      4. pow-flip 77.0%

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

      5. *-commutative 77.0%

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

      6. metadata-eval 77.0%

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

   9. Applied egg-rr 77.0%

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

       1. unpow1 77.0%

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

   11. Simplified 77.0%

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

   12. Taylor expanded in a around 0 41.8%

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

       1. times-frac 42.9%

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

       2. times-frac 41.8%

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

       3. unpow2 41.8%

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

       4. unpow2 41.8%

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

       5. swap-sqr 59.8%

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

       6. unpow2 59.8%

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

       7. unpow2 59.8%

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

       8. swap-sqr 77.0%

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

       9. times-frac 93.5%

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

       10. unpow2 93.5%

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

       11. associate-/r* 95.9%

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

       12. associate-/l* 93.9%

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

   14. Simplified 93.9%

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

   if 8.49999999999999999e-151 < b

   1. Initial program 12.6%

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

   3. Simplified 18.3%

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

   5. Taylor expanded in angle around 0 49.8%

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

      1. *-commutative 49.8%

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

      2. unpow2 49.8%

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

      3. unpow2 49.8%

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

      4. swap-sqr 62.4%

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

      5. unpow2 62.4%

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

      6. *-commutative 62.4%

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

      7. unpow2 62.4%

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

      8. unpow2 62.4%

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

      9. swap-sqr 83.4%

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

      10. unpow2 83.4%

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

   7. Simplified 83.4%

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

      1. add-sqr-sqrt 83.3%

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

      2. pow2 83.3%

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

      3. div-inv 83.3%

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

      4. *-commutative 83.3%

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

      5. pow-flip 83.6%

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

      6. *-commutative 83.6%

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

      7. metadata-eval 83.6%

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

   9. Applied egg-rr 83.6%

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

       1. add-sqr-sqrt 83.5%

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

       2. pow2 83.5%

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

       3. metadata-eval 83.5%

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

       4. pow-flip 83.2%

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

       5. div-inv 83.3%

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

       6. sqrt-div 83.3%

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

       7. sqrt-pow1 50.7%

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

       8. metadata-eval 50.7%

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

       9. pow1 50.7%

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

       10. sqrt-pow1 53.9%

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

       11. metadata-eval 53.9%

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

       12. pow1 53.9%

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

   11. Applied egg-rr 53.9%

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

       1. times-frac 55.9%

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

       2. *-commutative 55.9%

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

   13. Applied egg-rr 55.9%

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

Alternative 2: 93.7% accurate, 14.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{x-scale\_m}{a\_m} \cdot \frac{y-scale\_m}{b\_m}\\ \mathbf{if}\;b\_m \leq 3.4 \cdot 10^{-150}:\\ \;\;\;\;-4 \cdot {\left(\frac{a\_m \cdot \frac{b\_m}{x-scale\_m}}{y-scale\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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 (* (/ x-scale_m a_m) (/ y-scale_m b_m))))
   (if (<= b_m 3.4e-150)
     (* -4.0 (pow (/ (* a_m (/ b_m x-scale_m)) y-scale_m) 2.0))
     (* -4.0 (/ 1.0 (* 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 = (x_45_scale_m / a_m) * (y_45_scale_m / b_m);
	double tmp;
	if (b_m <= 3.4e-150) {
		tmp = -4.0 * pow(((a_m * (b_m / x_45_scale_m)) / y_45_scale_m), 2.0);
	} else {
		tmp = -4.0 * (1.0 / (t_0 * t_0));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = (x_45scale_m / a_m) * (y_45scale_m / b_m)
    if (b_m <= 3.4d-150) then
        tmp = (-4.0d0) * (((a_m * (b_m / x_45scale_m)) / y_45scale_m) ** 2.0d0)
    else
        tmp = (-4.0d0) * (1.0d0 / (t_0 * t_0))
    end if
    code = tmp
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 = (x_45_scale_m / a_m) * (y_45_scale_m / b_m);
	double tmp;
	if (b_m <= 3.4e-150) {
		tmp = -4.0 * Math.pow(((a_m * (b_m / x_45_scale_m)) / y_45_scale_m), 2.0);
	} else {
		tmp = -4.0 * (1.0 / (t_0 * t_0));
	}
	return tmp;
}

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 = (x_45_scale_m / a_m) * (y_45_scale_m / b_m)
	tmp = 0
	if b_m <= 3.4e-150:
		tmp = -4.0 * math.pow(((a_m * (b_m / x_45_scale_m)) / y_45_scale_m), 2.0)
	else:
		tmp = -4.0 * (1.0 / (t_0 * t_0))
	return tmp

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(Float64(x_45_scale_m / a_m) * Float64(y_45_scale_m / b_m))
	tmp = 0.0
	if (b_m <= 3.4e-150)
		tmp = Float64(-4.0 * (Float64(Float64(a_m * Float64(b_m / x_45_scale_m)) / y_45_scale_m) ^ 2.0));
	else
		tmp = Float64(-4.0 * Float64(1.0 / Float64(t_0 * t_0)));
	end
	return tmp
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_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = (x_45_scale_m / a_m) * (y_45_scale_m / b_m);
	tmp = 0.0;
	if (b_m <= 3.4e-150)
		tmp = -4.0 * (((a_m * (b_m / x_45_scale_m)) / y_45_scale_m) ^ 2.0);
	else
		tmp = -4.0 * (1.0 / (t_0 * t_0));
	end
	tmp_2 = tmp;
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[(N[(x$45$scale$95$m / a$95$m), $MachinePrecision] * N[(y$45$scale$95$m / b$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 3.4e-150], N[(-4.0 * N[Power[N[(N[(a$95$m * N[(b$95$m / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.39999999999999999e-150

   1. Initial program 30.2%

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

   3. Simplified 26.3%

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

   5. Taylor expanded in angle around 0 41.8%

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

      1. *-commutative 41.8%

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

      2. unpow2 41.8%

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

      3. unpow2 41.8%

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

      4. swap-sqr 59.8%

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

      5. unpow2 59.8%

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

      6. *-commutative 59.8%

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

      7. unpow2 59.8%

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

      8. unpow2 59.8%

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

      9. swap-sqr 77.0%

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

      10. unpow2 77.0%

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

   7. Simplified 77.0%

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

      1. pow1 77.0%

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

      2. div-inv 77.0%

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

      3. *-commutative 77.0%

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

      4. pow-flip 77.0%

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

      5. *-commutative 77.0%

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

      6. metadata-eval 77.0%

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

   9. Applied egg-rr 77.0%

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

       1. unpow1 77.0%

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

   11. Simplified 77.0%

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

   12. Taylor expanded in a around 0 41.8%

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

       1. times-frac 42.9%

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

       2. times-frac 41.8%

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

       3. unpow2 41.8%

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

       4. unpow2 41.8%

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

       5. swap-sqr 59.8%

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

       6. unpow2 59.8%

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

       7. unpow2 59.8%

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

       8. swap-sqr 77.0%

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

       9. times-frac 93.5%

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

       10. unpow2 93.5%

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

       11. associate-/r* 95.9%

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

       12. associate-/l* 93.9%

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

   14. Simplified 93.9%

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

   if 3.39999999999999999e-150 < b

   1. Initial program 12.6%

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

   3. Simplified 18.3%

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

   5. Taylor expanded in angle around 0 49.8%

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

      1. *-commutative 49.8%

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

      2. unpow2 49.8%

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

      3. unpow2 49.8%

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

      4. swap-sqr 62.4%

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

      5. unpow2 62.4%

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

      6. *-commutative 62.4%

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

      7. unpow2 62.4%

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

      8. unpow2 62.4%

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

      9. swap-sqr 83.4%

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

      10. unpow2 83.4%

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

   7. Simplified 83.4%

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

      1. clear-num 83.3%

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

      2. inv-pow 83.3%

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

   9. Applied egg-rr 83.3%

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

       1. unpow-1 83.3%

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

   11. Simplified 83.3%

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

       1. pow2 83.3%

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

       2. pow2 83.3%

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

       3. times-frac 95.3%

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

       4. times-frac 93.1%

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

       5. times-frac 97.4%

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

   13. Applied egg-rr 97.4%

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

Alternative 3: 93.7% accurate, 70.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{x-scale\_m}{a\_m} \cdot \frac{y-scale\_m}{b\_m}\\ t_1 := \frac{b\_m \cdot a\_m}{x-scale\_m \cdot y-scale\_m}\\ \mathbf{if}\;x-scale\_m \leq 2.8 \cdot 10^{-202}:\\ \;\;\;\;-4 \cdot \left(t\_1 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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 (* (/ x-scale_m a_m) (/ y-scale_m b_m)))
        (t_1 (/ (* b_m a_m) (* x-scale_m y-scale_m))))
   (if (<= x-scale_m 2.8e-202)
     (* -4.0 (* t_1 t_1))
     (* -4.0 (/ 1.0 (* 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 = (x_45_scale_m / a_m) * (y_45_scale_m / b_m);
	double t_1 = (b_m * a_m) / (x_45_scale_m * y_45_scale_m);
	double tmp;
	if (x_45_scale_m <= 2.8e-202) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = -4.0 * (1.0 / (t_0 * t_0));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_45scale_m / a_m) * (y_45scale_m / b_m)
    t_1 = (b_m * a_m) / (x_45scale_m * y_45scale_m)
    if (x_45scale_m <= 2.8d-202) then
        tmp = (-4.0d0) * (t_1 * t_1)
    else
        tmp = (-4.0d0) * (1.0d0 / (t_0 * t_0))
    end if
    code = tmp
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 = (x_45_scale_m / a_m) * (y_45_scale_m / b_m);
	double t_1 = (b_m * a_m) / (x_45_scale_m * y_45_scale_m);
	double tmp;
	if (x_45_scale_m <= 2.8e-202) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = -4.0 * (1.0 / (t_0 * t_0));
	}
	return tmp;
}

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 = (x_45_scale_m / a_m) * (y_45_scale_m / b_m)
	t_1 = (b_m * a_m) / (x_45_scale_m * y_45_scale_m)
	tmp = 0
	if x_45_scale_m <= 2.8e-202:
		tmp = -4.0 * (t_1 * t_1)
	else:
		tmp = -4.0 * (1.0 / (t_0 * t_0))
	return tmp

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(Float64(x_45_scale_m / a_m) * Float64(y_45_scale_m / b_m))
	t_1 = Float64(Float64(b_m * a_m) / Float64(x_45_scale_m * y_45_scale_m))
	tmp = 0.0
	if (x_45_scale_m <= 2.8e-202)
		tmp = Float64(-4.0 * Float64(t_1 * t_1));
	else
		tmp = Float64(-4.0 * Float64(1.0 / Float64(t_0 * t_0)));
	end
	return tmp
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_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = (x_45_scale_m / a_m) * (y_45_scale_m / b_m);
	t_1 = (b_m * a_m) / (x_45_scale_m * y_45_scale_m);
	tmp = 0.0;
	if (x_45_scale_m <= 2.8e-202)
		tmp = -4.0 * (t_1 * t_1);
	else
		tmp = -4.0 * (1.0 / (t_0 * t_0));
	end
	tmp_2 = tmp;
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[(N[(x$45$scale$95$m / a$95$m), $MachinePrecision] * N[(y$45$scale$95$m / b$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m * a$95$m), $MachinePrecision] / N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 2.8e-202], N[(-4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.8000000000000001e-202

   1. Initial program 25.5%

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

   3. Simplified 23.5%

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

   5. Taylor expanded in angle around 0 47.1%

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

      1. *-commutative 47.1%

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

      2. unpow2 47.1%

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

      3. unpow2 47.1%

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

      4. swap-sqr 64.0%

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

      5. unpow2 64.0%

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

      6. *-commutative 64.0%

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

      7. unpow2 64.0%

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

      8. unpow2 64.0%

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

      9. swap-sqr 82.8%

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

      10. unpow2 82.8%

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

   7. Simplified 82.8%

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

      1. add-sqr-sqrt 82.8%

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

      2. pow2 82.8%

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

      3. div-inv 82.7%

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

      4. *-commutative 82.7%

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

      5. pow-flip 82.9%

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

      6. *-commutative 82.9%

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

      7. metadata-eval 82.9%

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

   9. Applied egg-rr 82.9%

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

       1. unpow2 82.9%

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

       2. add-sqr-sqrt 83.0%

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

       3. metadata-eval 83.0%

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

       4. pow-flip 82.8%

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

       5. div-inv 82.8%

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

       6. add-sqr-sqrt 82.8%

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

       7. sqrt-div 82.8%

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

       8. sqrt-pow1 50.7%

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

       9. metadata-eval 50.7%

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

       10. pow1 50.7%

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

       11. sqrt-pow1 51.9%

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

       12. metadata-eval 51.9%

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

       13. pow1 51.9%

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

       14. sqrt-div 51.9%

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

       15. sqrt-pow1 62.4%

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

       16. metadata-eval 62.4%

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

       17. pow1 62.4%

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

       18. sqrt-pow1 96.1%

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

       19. metadata-eval 96.1%

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

       20. pow1 96.1%

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

   11. Applied egg-rr 96.1%

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

   if 2.8000000000000001e-202 < x-scale

   1. Initial program 21.5%

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

   3. Simplified 23.5%

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

   5. Taylor expanded in angle around 0 40.3%

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

      1. *-commutative 40.3%

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

      2. unpow2 40.3%

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

      3. unpow2 40.3%

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

      4. swap-sqr 55.1%

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

      5. unpow2 55.1%

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

      6. *-commutative 55.1%

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

      7. unpow2 55.1%

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

      8. unpow2 55.1%

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

      9. swap-sqr 73.2%

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

      10. unpow2 73.2%

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

   7. Simplified 73.2%

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

      1. clear-num 73.2%

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

      2. inv-pow 73.2%

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

   9. Applied egg-rr 73.2%

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

       1. unpow-1 73.2%

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

   11. Simplified 73.2%

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

       1. pow2 73.2%

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

       2. pow2 73.2%

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

       3. times-frac 90.6%

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

       4. times-frac 87.7%

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

       5. times-frac 96.6%

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

   13. Applied egg-rr 96.6%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 2.8 \cdot 10^{-202}:\\ \;\;\;\;-4 \cdot \left(\frac{b \cdot a}{x-scale \cdot y-scale} \cdot \frac{b \cdot a}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{1}{\left(\frac{x-scale}{a} \cdot \frac{y-scale}{b}\right) \cdot \left(\frac{x-scale}{a} \cdot \frac{y-scale}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 99.6× speedup

Math

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

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 (/ (* b_m a_m) (* x-scale_m y-scale_m)))) (* -4.0 (* 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 = (b_m * a_m) / (x_45_scale_m * y_45_scale_m);
	return -4.0 * (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 = (b_m * a_m) / (x_45scale_m * y_45scale_m)
    code = (-4.0d0) * (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 = (b_m * a_m) / (x_45_scale_m * y_45_scale_m);
	return -4.0 * (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 = (b_m * a_m) / (x_45_scale_m * y_45_scale_m)
	return -4.0 * (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(Float64(b_m * a_m) / Float64(x_45_scale_m * y_45_scale_m))
	return Float64(-4.0 * 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 = (b_m * a_m) / (x_45_scale_m * y_45_scale_m);
	tmp = -4.0 * (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[(N[(b$95$m * a$95$m), $MachinePrecision] / N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 24.0%

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

3. Simplified 23.5%

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

5. Taylor expanded in angle around 0 44.6%

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

   1. *-commutative 44.6%

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

   2. unpow2 44.6%

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

   3. unpow2 44.6%

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

   4. swap-sqr 60.7%

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

   5. unpow2 60.7%

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

   6. *-commutative 60.7%

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

   7. unpow2 60.7%

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

   8. unpow2 60.7%

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

   9. swap-sqr 79.2%

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

   10. unpow2 79.2%

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

7. Simplified 79.2%

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

   1. add-sqr-sqrt 79.2%

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

   2. pow2 79.2%

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

   3. div-inv 79.2%

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

   4. *-commutative 79.2%

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

   5. pow-flip 79.3%

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

   6. *-commutative 79.3%

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

   7. metadata-eval 79.3%

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

9. Applied egg-rr 79.3%

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

    1. unpow2 79.3%

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

    2. add-sqr-sqrt 79.3%

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

    3. metadata-eval 79.3%

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

    4. pow-flip 79.2%

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

    5. div-inv 79.2%

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

    6. add-sqr-sqrt 79.2%

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

    7. sqrt-div 79.2%

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

    8. sqrt-pow1 50.8%

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

    9. metadata-eval 50.8%

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

    10. pow1 50.8%

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

    11. sqrt-pow1 50.5%

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

    12. metadata-eval 50.5%

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

    13. pow1 50.5%

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

    14. sqrt-div 50.5%

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

    15. sqrt-pow1 59.8%

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

    16. metadata-eval 59.8%

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

    17. pow1 59.8%

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

    18. sqrt-pow1 94.1%

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

    19. metadata-eval 94.1%

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

    20. pow1 94.1%

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

11. Applied egg-rr 94.1%

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

12. Final simplification 94.1%

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

Alternative 5: 84.5% accurate, 99.6× speedup

Math

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

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 x-scale_m)
    (* (/ y-scale_m b_m) (* y-scale_m (/ x-scale_m a_m)))))))

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 / x_45_scale_m) / ((y_45_scale_m / b_m) * (y_45_scale_m * (x_45_scale_m / a_m)))));
}

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 / x_45scale_m) / ((y_45scale_m / b_m) * (y_45scale_m * (x_45scale_m / a_m)))))
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 / x_45_scale_m) / ((y_45_scale_m / b_m) * (y_45_scale_m * (x_45_scale_m / a_m)))));
}

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 / x_45_scale_m) / ((y_45_scale_m / b_m) * (y_45_scale_m * (x_45_scale_m / a_m)))))

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(-4.0 * Float64(a_m * Float64(Float64(b_m / x_45_scale_m) / Float64(Float64(y_45_scale_m / b_m) * Float64(y_45_scale_m * Float64(x_45_scale_m / a_m))))))
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 / x_45_scale_m) / ((y_45_scale_m / b_m) * (y_45_scale_m * (x_45_scale_m / a_m)))));
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[(-4.0 * N[(a$95$m * N[(N[(b$95$m / x$45$scale$95$m), $MachinePrecision] / N[(N[(y$45$scale$95$m / b$95$m), $MachinePrecision] * N[(y$45$scale$95$m * N[(x$45$scale$95$m / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.0%

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

3. Simplified 23.5%

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

5. Taylor expanded in angle around 0 44.6%

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

   1. *-commutative 44.6%

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

   2. unpow2 44.6%

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

   3. unpow2 44.6%

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

   4. swap-sqr 60.7%

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

   5. unpow2 60.7%

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

   6. *-commutative 60.7%

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

   7. unpow2 60.7%

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

   8. unpow2 60.7%

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

   9. swap-sqr 79.2%

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

   10. unpow2 79.2%

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

7. Simplified 79.2%

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

   1. add-sqr-sqrt 79.2%

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

   2. pow2 79.2%

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

   3. div-inv 79.2%

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

   4. *-commutative 79.2%

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

   5. pow-flip 79.3%

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

   6. *-commutative 79.3%

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

   7. metadata-eval 79.3%

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

9. Applied egg-rr 79.3%

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

    1. add-sqr-sqrt 79.2%

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

    2. pow2 79.2%

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

    3. metadata-eval 79.2%

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

    4. pow-flip 79.1%

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

    5. div-inv 79.1%

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

    6. sqrt-div 79.1%

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

    7. sqrt-pow1 52.9%

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

    8. metadata-eval 52.9%

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

    9. pow1 52.9%

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

    10. sqrt-pow1 53.2%

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

    11. metadata-eval 53.2%

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

    12. pow1 53.2%

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

11. Applied egg-rr 53.2%

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

    1. add-sqr-sqrt 53.2%

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

    2. pow-pow 53.2%

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

    3. sqrt-pow1 53.2%

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

    4. metadata-eval 53.2%

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

    5. metadata-eval 53.2%

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

    6. unpow2 53.2%

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

    7. add-sqr-sqrt 53.2%

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

    8. unpow2 53.2%

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

    9. add-sqr-sqrt 58.3%

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

    10. pow2 58.3%

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

    11. frac-times 50.5%

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

    12. pow2 50.5%

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

    13. clear-num 50.5%

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

    14. sqrt-div 50.5%

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

13. Applied egg-rr 85.2%

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

    1. associate-*l* 85.1%

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

    2. associate-/r* 87.5%

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

    3. associate-*r/ 85.4%

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

    4. *-commutative 85.4%

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

    5. associate-/l* 85.0%

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

    6. *-commutative 85.0%

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

    7. associate-*r* 83.4%

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

15. Simplified 83.4%

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

16. Final simplification 83.4%

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

Alternative 6: 35.9% 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 24.0%

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

3. Simplified 23.5%

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

5. Taylor expanded in b around 0 21.8%

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

   1. distribute-rgt-out 21.8%

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

   2. metadata-eval 21.8%

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

   3. mul0-rgt 34.6%

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

7. Simplified 34.6%

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

Reproduce

herbie shell --seed 2024157 
(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))))