Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.0% → 94.3%

Time: 20.7s

Alternatives: 11

Speedup: 40.5×

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

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

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)))))

Initial Program: 25.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

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)))))

Alternative 1: 94.3% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.6%

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

3. Taylor expanded in angle around 0

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

   1. times-frac N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. unpow2 N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 69.4

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

5. Applied rewrites 69.4%

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

7. Applied rewrites 80.6%

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

9. Applied rewrites 95.7%

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

11. Applied rewrites 95.7%

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

Alternative 2: 78.1% accurate, 26.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 1.02 \cdot 10^{-165}:\\ \;\;\;\;\left(\frac{-4 \cdot b}{y-scale \cdot x-scale} \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;x-scale \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\left(-4 \cdot \left(a \cdot \frac{a}{x-scale \cdot x-scale}\right)\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\\ \mathbf{elif}\;x-scale \leq 2.6 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(\frac{b}{y-scale} \cdot a\right) \cdot \left(a \cdot b\right)}{\left(x-scale \cdot x-scale\right) \cdot y-scale} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{a}{x-scale} \cdot b\right) \cdot \left(a \cdot b\right)}{y-scale \cdot \left(y-scale \cdot x-scale\right)} \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale 1.02e-165)
   (* (* (/ (* -4.0 b) (* y-scale x-scale)) (/ b (* y-scale x-scale))) (* a a))
   (if (<= x-scale 2e-43)
     (*
      (* -4.0 (* a (/ a (* x-scale x-scale))))
      (* (/ b y-scale) (/ b y-scale)))
     (if (<= x-scale 2.6e+131)
       (*
        (/ (* (* (/ b y-scale) a) (* a b)) (* (* x-scale x-scale) y-scale))
        -4.0)
       (*
        (/ (* (* (/ a x-scale) b) (* a b)) (* y-scale (* y-scale x-scale)))
        -4.0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 1.02e-165) {
		tmp = (((-4.0 * b) / (y_45_scale * x_45_scale)) * (b / (y_45_scale * x_45_scale))) * (a * a);
	} else if (x_45_scale <= 2e-43) {
		tmp = (-4.0 * (a * (a / (x_45_scale * x_45_scale)))) * ((b / y_45_scale) * (b / y_45_scale));
	} else if (x_45_scale <= 2.6e+131) {
		tmp = ((((b / y_45_scale) * a) * (a * b)) / ((x_45_scale * x_45_scale) * y_45_scale)) * -4.0;
	} else {
		tmp = ((((a / x_45_scale) * b) * (a * b)) / (y_45_scale * (y_45_scale * x_45_scale))) * -4.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (x_45scale <= 1.02d-165) then
        tmp = ((((-4.0d0) * b) / (y_45scale * x_45scale)) * (b / (y_45scale * x_45scale))) * (a * a)
    else if (x_45scale <= 2d-43) then
        tmp = ((-4.0d0) * (a * (a / (x_45scale * x_45scale)))) * ((b / y_45scale) * (b / y_45scale))
    else if (x_45scale <= 2.6d+131) then
        tmp = ((((b / y_45scale) * a) * (a * b)) / ((x_45scale * x_45scale) * y_45scale)) * (-4.0d0)
    else
        tmp = ((((a / x_45scale) * b) * (a * b)) / (y_45scale * (y_45scale * x_45scale))) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 1.02e-165) {
		tmp = (((-4.0 * b) / (y_45_scale * x_45_scale)) * (b / (y_45_scale * x_45_scale))) * (a * a);
	} else if (x_45_scale <= 2e-43) {
		tmp = (-4.0 * (a * (a / (x_45_scale * x_45_scale)))) * ((b / y_45_scale) * (b / y_45_scale));
	} else if (x_45_scale <= 2.6e+131) {
		tmp = ((((b / y_45_scale) * a) * (a * b)) / ((x_45_scale * x_45_scale) * y_45_scale)) * -4.0;
	} else {
		tmp = ((((a / x_45_scale) * b) * (a * b)) / (y_45_scale * (y_45_scale * x_45_scale))) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if x_45_scale <= 1.02e-165:
		tmp = (((-4.0 * b) / (y_45_scale * x_45_scale)) * (b / (y_45_scale * x_45_scale))) * (a * a)
	elif x_45_scale <= 2e-43:
		tmp = (-4.0 * (a * (a / (x_45_scale * x_45_scale)))) * ((b / y_45_scale) * (b / y_45_scale))
	elif x_45_scale <= 2.6e+131:
		tmp = ((((b / y_45_scale) * a) * (a * b)) / ((x_45_scale * x_45_scale) * y_45_scale)) * -4.0
	else:
		tmp = ((((a / x_45_scale) * b) * (a * b)) / (y_45_scale * (y_45_scale * x_45_scale))) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (x_45_scale <= 1.02e-165)
		tmp = Float64(Float64(Float64(Float64(-4.0 * b) / Float64(y_45_scale * x_45_scale)) * Float64(b / Float64(y_45_scale * x_45_scale))) * Float64(a * a));
	elseif (x_45_scale <= 2e-43)
		tmp = Float64(Float64(-4.0 * Float64(a * Float64(a / Float64(x_45_scale * x_45_scale)))) * Float64(Float64(b / y_45_scale) * Float64(b / y_45_scale)));
	elseif (x_45_scale <= 2.6e+131)
		tmp = Float64(Float64(Float64(Float64(Float64(b / y_45_scale) * a) * Float64(a * b)) / Float64(Float64(x_45_scale * x_45_scale) * y_45_scale)) * -4.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a / x_45_scale) * b) * Float64(a * b)) / Float64(y_45_scale * Float64(y_45_scale * x_45_scale))) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (x_45_scale <= 1.02e-165)
		tmp = (((-4.0 * b) / (y_45_scale * x_45_scale)) * (b / (y_45_scale * x_45_scale))) * (a * a);
	elseif (x_45_scale <= 2e-43)
		tmp = (-4.0 * (a * (a / (x_45_scale * x_45_scale)))) * ((b / y_45_scale) * (b / y_45_scale));
	elseif (x_45_scale <= 2.6e+131)
		tmp = ((((b / y_45_scale) * a) * (a * b)) / ((x_45_scale * x_45_scale) * y_45_scale)) * -4.0;
	else
		tmp = ((((a / x_45_scale) * b) * (a * b)) / (y_45_scale * (y_45_scale * x_45_scale))) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[x$45$scale, 1.02e-165], N[(N[(N[(N[(-4.0 * b), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 2e-43], N[(N[(-4.0 * N[(a * N[(a / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b / y$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 2.6e+131], N[(N[(N[(N[(N[(b / y$45$scale), $MachinePrecision] * a), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(N[(N[(a / x$45$scale), $MachinePrecision] * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x-scale < 1.02e-165

   1. Initial program 21.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-8 \cdot \frac{{b}^{2} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

   5. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-8 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right), \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale}, -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale} + \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(a \cdot a\right)} \]

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 63.6%

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

   10. Applied rewrites 78.8%

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

   if 1.02e-165 < x-scale < 2.00000000000000015e-43

   1. Initial program 33.4%

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

   3. Taylor expanded in angle around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 72.3

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

   5. Applied rewrites 72.3%

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

   7. Applied rewrites 89.0%

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

   if 2.00000000000000015e-43 < x-scale < 2.6e131

   1. Initial program 37.1%

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

   3. Taylor expanded in angle around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 99.7%

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

   11. Applied rewrites 96.5%

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

   if 2.6e131 < x-scale

   1. Initial program 29.8%

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

   3. Taylor expanded in angle around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 73.2%

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

   9. Applied rewrites 97.1%

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

   11. Applied rewrites 82.9%

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

Alternative 3: 77.7% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 9.2 \cdot 10^{-188} \lor \neg \left(y-scale \leq 2.3 \cdot 10^{+159}\right):\\ \;\;\;\;\left(\frac{-4 \cdot b}{y-scale \cdot x-scale} \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{a}{x-scale} \cdot b\right) \cdot \left(a \cdot b\right)}{y-scale \cdot \left(y-scale \cdot x-scale\right)} \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (or (<= y-scale 9.2e-188) (not (<= y-scale 2.3e+159)))
   (* (* (/ (* -4.0 b) (* y-scale x-scale)) (/ b (* y-scale x-scale))) (* a a))
   (*
    (/ (* (* (/ a x-scale) b) (* a b)) (* y-scale (* y-scale x-scale)))
    -4.0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if ((y_45_scale <= 9.2e-188) || !(y_45_scale <= 2.3e+159)) {
		tmp = (((-4.0 * b) / (y_45_scale * x_45_scale)) * (b / (y_45_scale * x_45_scale))) * (a * a);
	} else {
		tmp = ((((a / x_45_scale) * b) * (a * b)) / (y_45_scale * (y_45_scale * x_45_scale))) * -4.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if ((y_45scale <= 9.2d-188) .or. (.not. (y_45scale <= 2.3d+159))) then
        tmp = ((((-4.0d0) * b) / (y_45scale * x_45scale)) * (b / (y_45scale * x_45scale))) * (a * a)
    else
        tmp = ((((a / x_45scale) * b) * (a * b)) / (y_45scale * (y_45scale * x_45scale))) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if ((y_45_scale <= 9.2e-188) || !(y_45_scale <= 2.3e+159)) {
		tmp = (((-4.0 * b) / (y_45_scale * x_45_scale)) * (b / (y_45_scale * x_45_scale))) * (a * a);
	} else {
		tmp = ((((a / x_45_scale) * b) * (a * b)) / (y_45_scale * (y_45_scale * x_45_scale))) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if (y_45_scale <= 9.2e-188) or not (y_45_scale <= 2.3e+159):
		tmp = (((-4.0 * b) / (y_45_scale * x_45_scale)) * (b / (y_45_scale * x_45_scale))) * (a * a)
	else:
		tmp = ((((a / x_45_scale) * b) * (a * b)) / (y_45_scale * (y_45_scale * x_45_scale))) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if ((y_45_scale <= 9.2e-188) || !(y_45_scale <= 2.3e+159))
		tmp = Float64(Float64(Float64(Float64(-4.0 * b) / Float64(y_45_scale * x_45_scale)) * Float64(b / Float64(y_45_scale * x_45_scale))) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a / x_45_scale) * b) * Float64(a * b)) / Float64(y_45_scale * Float64(y_45_scale * x_45_scale))) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if ((y_45_scale <= 9.2e-188) || ~((y_45_scale <= 2.3e+159)))
		tmp = (((-4.0 * b) / (y_45_scale * x_45_scale)) * (b / (y_45_scale * x_45_scale))) * (a * a);
	else
		tmp = ((((a / x_45_scale) * b) * (a * b)) / (y_45_scale * (y_45_scale * x_45_scale))) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[Or[LessEqual[y$45$scale, 9.2e-188], N[Not[LessEqual[y$45$scale, 2.3e+159]], $MachinePrecision]], N[(N[(N[(N[(-4.0 * b), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a / x$45$scale), $MachinePrecision] * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 9.1999999999999999e-188 or 2.29999999999999995e159 < y-scale

   1. Initial program 25.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-8 \cdot \frac{{b}^{2} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

   5. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-8 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right), \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale}, -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale} + \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(a \cdot a\right)} \]

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 58.5%

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

   10. Applied rewrites 74.7%

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

   if 9.1999999999999999e-188 < y-scale < 2.29999999999999995e159

   1. Initial program 26.9%

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

   3. Taylor expanded in angle around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 81.7%

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

   9. Applied rewrites 99.6%

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

   11. Applied rewrites 90.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 9.2 \cdot 10^{-188} \lor \neg \left(y-scale \leq 2.3 \cdot 10^{+159}\right):\\ \;\;\;\;\left(\frac{-4 \cdot b}{y-scale \cdot x-scale} \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{a}{x-scale} \cdot b\right) \cdot \left(a \cdot b\right)}{y-scale \cdot \left(y-scale \cdot x-scale\right)} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{-210} \lor \neg \left(a \leq 1.8 \cdot 10^{-149}\right):\\ \;\;\;\;\left(\frac{-4 \cdot b}{y-scale \cdot x-scale} \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \frac{-4 \cdot a}{x-scale \cdot x-scale}\right) \cdot \left(b \cdot \frac{b}{y-scale \cdot y-scale}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (or (<= a 4.8e-210) (not (<= a 1.8e-149)))
   (* (* (/ (* -4.0 b) (* y-scale x-scale)) (/ b (* y-scale x-scale))) (* a a))
   (*
    (* a (/ (* -4.0 a) (* x-scale x-scale)))
    (* b (/ b (* y-scale y-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if ((a <= 4.8e-210) || !(a <= 1.8e-149)) {
		tmp = (((-4.0 * b) / (y_45_scale * x_45_scale)) * (b / (y_45_scale * x_45_scale))) * (a * a);
	} else {
		tmp = (a * ((-4.0 * a) / (x_45_scale * x_45_scale))) * (b * (b / (y_45_scale * y_45_scale)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if ((a <= 4.8d-210) .or. (.not. (a <= 1.8d-149))) then
        tmp = ((((-4.0d0) * b) / (y_45scale * x_45scale)) * (b / (y_45scale * x_45scale))) * (a * a)
    else
        tmp = (a * (((-4.0d0) * a) / (x_45scale * x_45scale))) * (b * (b / (y_45scale * y_45scale)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if ((a <= 4.8e-210) || !(a <= 1.8e-149)) {
		tmp = (((-4.0 * b) / (y_45_scale * x_45_scale)) * (b / (y_45_scale * x_45_scale))) * (a * a);
	} else {
		tmp = (a * ((-4.0 * a) / (x_45_scale * x_45_scale))) * (b * (b / (y_45_scale * y_45_scale)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if (a <= 4.8e-210) or not (a <= 1.8e-149):
		tmp = (((-4.0 * b) / (y_45_scale * x_45_scale)) * (b / (y_45_scale * x_45_scale))) * (a * a)
	else:
		tmp = (a * ((-4.0 * a) / (x_45_scale * x_45_scale))) * (b * (b / (y_45_scale * y_45_scale)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if ((a <= 4.8e-210) || !(a <= 1.8e-149))
		tmp = Float64(Float64(Float64(Float64(-4.0 * b) / Float64(y_45_scale * x_45_scale)) * Float64(b / Float64(y_45_scale * x_45_scale))) * Float64(a * a));
	else
		tmp = Float64(Float64(a * Float64(Float64(-4.0 * a) / Float64(x_45_scale * x_45_scale))) * Float64(b * Float64(b / Float64(y_45_scale * y_45_scale))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if ((a <= 4.8e-210) || ~((a <= 1.8e-149)))
		tmp = (((-4.0 * b) / (y_45_scale * x_45_scale)) * (b / (y_45_scale * x_45_scale))) * (a * a);
	else
		tmp = (a * ((-4.0 * a) / (x_45_scale * x_45_scale))) * (b * (b / (y_45_scale * y_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[Or[LessEqual[a, 4.8e-210], N[Not[LessEqual[a, 1.8e-149]], $MachinePrecision]], N[(N[(N[(N[(-4.0 * b), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-4.0 * a), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * N[(b / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.80000000000000008e-210 or 1.8000000000000001e-149 < a

   1. Initial program 24.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-8 \cdot \frac{{b}^{2} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

   5. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-8 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right), \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale}, -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale} + \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(a \cdot a\right)} \]

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 63.7%

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

   10. Applied rewrites 79.5%

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

   if 4.80000000000000008e-210 < a < 1.8000000000000001e-149

   1. Initial program 36.7%

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

   3. Taylor expanded in angle around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 49.7

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

   5. Applied rewrites 49.7%

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

   7. Applied rewrites 48.4%

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

   9. Applied rewrites 48.3%

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

   11. Applied rewrites 77.7%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{-210} \lor \neg \left(a \leq 1.8 \cdot 10^{-149}\right):\\ \;\;\;\;\left(\frac{-4 \cdot b}{y-scale \cdot x-scale} \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \frac{-4 \cdot a}{x-scale \cdot x-scale}\right) \cdot \left(b \cdot \frac{b}{y-scale \cdot y-scale}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.5% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 9.5 \cdot 10^{-200} \lor \neg \left(x-scale \leq 1.12 \cdot 10^{+120}\right):\\ \;\;\;\;\left(\frac{-4}{\left(y-scale \cdot x-scale\right) \cdot y-scale} \cdot \left(\frac{b}{x-scale} \cdot b\right)\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{-4}{\left(x-scale \cdot x-scale\right) \cdot y-scale} \cdot \frac{b}{y-scale}\right) \cdot b\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (or (<= x-scale 9.5e-200) (not (<= x-scale 1.12e+120)))
   (* (* (/ -4.0 (* (* y-scale x-scale) y-scale)) (* (/ b x-scale) b)) (* a a))
   (*
    (* (* (/ -4.0 (* (* x-scale x-scale) y-scale)) (/ b y-scale)) b)
    (* a a))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if ((x_45_scale <= 9.5e-200) || !(x_45_scale <= 1.12e+120)) {
		tmp = ((-4.0 / ((y_45_scale * x_45_scale) * y_45_scale)) * ((b / x_45_scale) * b)) * (a * a);
	} else {
		tmp = (((-4.0 / ((x_45_scale * x_45_scale) * y_45_scale)) * (b / y_45_scale)) * b) * (a * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if ((x_45scale <= 9.5d-200) .or. (.not. (x_45scale <= 1.12d+120))) then
        tmp = (((-4.0d0) / ((y_45scale * x_45scale) * y_45scale)) * ((b / x_45scale) * b)) * (a * a)
    else
        tmp = ((((-4.0d0) / ((x_45scale * x_45scale) * y_45scale)) * (b / y_45scale)) * b) * (a * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if ((x_45_scale <= 9.5e-200) || !(x_45_scale <= 1.12e+120)) {
		tmp = ((-4.0 / ((y_45_scale * x_45_scale) * y_45_scale)) * ((b / x_45_scale) * b)) * (a * a);
	} else {
		tmp = (((-4.0 / ((x_45_scale * x_45_scale) * y_45_scale)) * (b / y_45_scale)) * b) * (a * a);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if (x_45_scale <= 9.5e-200) or not (x_45_scale <= 1.12e+120):
		tmp = ((-4.0 / ((y_45_scale * x_45_scale) * y_45_scale)) * ((b / x_45_scale) * b)) * (a * a)
	else:
		tmp = (((-4.0 / ((x_45_scale * x_45_scale) * y_45_scale)) * (b / y_45_scale)) * b) * (a * a)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if ((x_45_scale <= 9.5e-200) || !(x_45_scale <= 1.12e+120))
		tmp = Float64(Float64(Float64(-4.0 / Float64(Float64(y_45_scale * x_45_scale) * y_45_scale)) * Float64(Float64(b / x_45_scale) * b)) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(Float64(-4.0 / Float64(Float64(x_45_scale * x_45_scale) * y_45_scale)) * Float64(b / y_45_scale)) * b) * Float64(a * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if ((x_45_scale <= 9.5e-200) || ~((x_45_scale <= 1.12e+120)))
		tmp = ((-4.0 / ((y_45_scale * x_45_scale) * y_45_scale)) * ((b / x_45_scale) * b)) * (a * a);
	else
		tmp = (((-4.0 / ((x_45_scale * x_45_scale) * y_45_scale)) * (b / y_45_scale)) * b) * (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[Or[LessEqual[x$45$scale, 9.5e-200], N[Not[LessEqual[x$45$scale, 1.12e+120]], $MachinePrecision]], N[(N[(N[(-4.0 / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b / x$45$scale), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-4.0 / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 9.4999999999999995e-200 or 1.12000000000000005e120 < x-scale

   1. Initial program 23.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-8 \cdot \frac{{b}^{2} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-8 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right), \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale}, -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale} + \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(a \cdot a\right)} \]

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 62.3%

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

   10. Applied rewrites 68.3%

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

   if 9.4999999999999995e-200 < x-scale < 1.12000000000000005e120

   1. Initial program 34.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-8 \cdot \frac{{b}^{2} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-8 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right), \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale}, -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale} + \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(a \cdot a\right)} \]

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 59.5%

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

   10. Applied rewrites 56.4%

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

   12. Applied rewrites 72.7%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 9.5 \cdot 10^{-200} \lor \neg \left(x-scale \leq 1.12 \cdot 10^{+120}\right):\\ \;\;\;\;\left(\frac{-4}{\left(y-scale \cdot x-scale\right) \cdot y-scale} \cdot \left(\frac{b}{x-scale} \cdot b\right)\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{-4}{\left(x-scale \cdot x-scale\right) \cdot y-scale} \cdot \frac{b}{y-scale}\right) \cdot b\right) \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.4% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 3.3 \cdot 10^{+141}:\\ \;\;\;\;\left(\left(\frac{-4}{\left(x-scale \cdot x-scale\right) \cdot y-scale} \cdot \frac{b}{y-scale}\right) \cdot b\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(b \cdot b\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale 3.3e+141)
   (* (* (* (/ -4.0 (* (* x-scale x-scale) y-scale)) (/ b y-scale)) b) (* a a))
   (*
    (/ (* -4.0 (* b b)) (* (* y-scale x-scale) (* y-scale x-scale)))
    (* a a))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 3.3e+141) {
		tmp = (((-4.0 / ((x_45_scale * x_45_scale) * y_45_scale)) * (b / y_45_scale)) * b) * (a * a);
	} else {
		tmp = ((-4.0 * (b * b)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (a * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (x_45scale <= 3.3d+141) then
        tmp = ((((-4.0d0) / ((x_45scale * x_45scale) * y_45scale)) * (b / y_45scale)) * b) * (a * a)
    else
        tmp = (((-4.0d0) * (b * b)) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * (a * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 3.3e+141) {
		tmp = (((-4.0 / ((x_45_scale * x_45_scale) * y_45_scale)) * (b / y_45_scale)) * b) * (a * a);
	} else {
		tmp = ((-4.0 * (b * b)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (a * a);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if x_45_scale <= 3.3e+141:
		tmp = (((-4.0 / ((x_45_scale * x_45_scale) * y_45_scale)) * (b / y_45_scale)) * b) * (a * a)
	else:
		tmp = ((-4.0 * (b * b)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (a * a)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (x_45_scale <= 3.3e+141)
		tmp = Float64(Float64(Float64(Float64(-4.0 / Float64(Float64(x_45_scale * x_45_scale) * y_45_scale)) * Float64(b / y_45_scale)) * b) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(-4.0 * Float64(b * b)) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * Float64(a * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (x_45_scale <= 3.3e+141)
		tmp = (((-4.0 / ((x_45_scale * x_45_scale) * y_45_scale)) * (b / y_45_scale)) * b) * (a * a);
	else
		tmp = ((-4.0 * (b * b)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[x$45$scale, 3.3e+141], N[(N[(N[(N[(-4.0 / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 3.2999999999999997e141

   1. Initial program 25.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-8 \cdot \frac{{b}^{2} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

   5. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-8 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right), \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale}, -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale} + \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(a \cdot a\right)} \]

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 62.4%

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

   10. Applied rewrites 56.0%

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

   12. Applied rewrites 68.7%

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

   if 3.2999999999999997e141 < x-scale

   1. Initial program 28.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-8 \cdot \frac{{b}^{2} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

   5. Applied rewrites 57.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-8 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right), \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale}, -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale} + \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(a \cdot a\right)} \]

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 57.1%

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

Alternative 7: 86.5% accurate, 32.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.6%

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

3. Taylor expanded in angle around 0

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

   1. times-frac N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. unpow2 N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 69.4

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

5. Applied rewrites 69.4%

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

7. Applied rewrites 80.6%

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

9. Applied rewrites 95.7%

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

11. Applied rewrites 87.1%

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

Alternative 8: 85.5% accurate, 32.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.6%

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

3. Taylor expanded in angle around 0

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

   1. times-frac N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. unpow2 N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 69.4

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

5. Applied rewrites 69.4%

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

7. Applied rewrites 80.6%

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

9. Applied rewrites 95.7%

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

11. Applied rewrites 86.4%

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

Alternative 9: 75.3% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.6%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{{a}^{2} \cdot \left(-8 \cdot \frac{{b}^{2} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

5. Applied rewrites 51.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-8 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right), \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale}, -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale} + \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(a \cdot a\right)} \]

6. Taylor expanded in angle around 0

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

8. Applied rewrites 61.6%

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

10. Applied rewrites 77.5%

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

Alternative 10: 61.1% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.6%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{{a}^{2} \cdot \left(-8 \cdot \frac{{b}^{2} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

5. Applied rewrites 51.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-8 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right), \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale}, -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale} + \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(a \cdot a\right)} \]

6. Taylor expanded in angle around 0

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

8. Applied rewrites 61.6%

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

10. Applied rewrites 61.6%

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

Alternative 11: 61.1% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.6%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{{a}^{2} \cdot \left(-8 \cdot \frac{{b}^{2} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

5. Applied rewrites 51.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-8 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right), \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{y-scale}, -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale} + \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(a \cdot a\right)} \]

6. Taylor expanded in angle around 0

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

8. Applied rewrites 61.6%

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

Reproduce

herbie shell --seed 2024318 
(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))))