Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.5% → 90.5%

Time: 18.5s

Alternatives: 7

Speedup: 40.5×

• 33.0% 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: 25.5% 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: 90.5% accurate, 13.7× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 8.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{a \cdot b}{y-scale} \cdot \frac{\frac{-4 \cdot \frac{a \cdot b}{x-scale\_m}}{y-scale}}{x-scale\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\frac{b}{y-scale \cdot x-scale\_m}\right)}^{2} \cdot -4\right) \cdot a\right) \cdot a\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (if (<= x-scale_m 8.5e+157)
   (*
    (/ (* a b) y-scale)
    (/ (/ (* -4.0 (/ (* a b) x-scale_m)) y-scale) x-scale_m))
   (* (* (* (pow (/ b (* y-scale x-scale_m)) 2.0) -4.0) a) a)))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (x_45_scale_m <= 8.5e+157) {
		tmp = ((a * b) / y_45_scale) * (((-4.0 * ((a * b) / x_45_scale_m)) / y_45_scale) / x_45_scale_m);
	} else {
		tmp = ((pow((b / (y_45_scale * x_45_scale_m)), 2.0) * -4.0) * a) * a;
	}
	return tmp;
}

Fortran

x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (x_45scale_m <= 8.5d+157) then
        tmp = ((a * b) / y_45scale) * ((((-4.0d0) * ((a * b) / x_45scale_m)) / y_45scale) / x_45scale_m)
    else
        tmp = ((((b / (y_45scale * x_45scale_m)) ** 2.0d0) * (-4.0d0)) * a) * a
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (x_45_scale_m <= 8.5e+157) {
		tmp = ((a * b) / y_45_scale) * (((-4.0 * ((a * b) / x_45_scale_m)) / y_45_scale) / x_45_scale_m);
	} else {
		tmp = ((Math.pow((b / (y_45_scale * x_45_scale_m)), 2.0) * -4.0) * a) * a;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	tmp = 0
	if x_45_scale_m <= 8.5e+157:
		tmp = ((a * b) / y_45_scale) * (((-4.0 * ((a * b) / x_45_scale_m)) / y_45_scale) / x_45_scale_m)
	else:
		tmp = ((math.pow((b / (y_45_scale * x_45_scale_m)), 2.0) * -4.0) * a) * a
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (x_45_scale_m <= 8.5e+157)
		tmp = Float64(Float64(Float64(a * b) / y_45_scale) * Float64(Float64(Float64(-4.0 * Float64(Float64(a * b) / x_45_scale_m)) / y_45_scale) / x_45_scale_m));
	else
		tmp = Float64(Float64(Float64((Float64(b / Float64(y_45_scale * x_45_scale_m)) ^ 2.0) * -4.0) * a) * a);
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0;
	if (x_45_scale_m <= 8.5e+157)
		tmp = ((a * b) / y_45_scale) * (((-4.0 * ((a * b) / x_45_scale_m)) / y_45_scale) / x_45_scale_m);
	else
		tmp = ((((b / (y_45_scale * x_45_scale_m)) ^ 2.0) * -4.0) * a) * a;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[x$45$scale$95$m, 8.5e+157], N[(N[(N[(a * b), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(N[(N[(-4.0 * N[(N[(a * b), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(b / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 8.4999999999999998e157

   1. Initial program 19.3%

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

   5. Applied rewrites 53.3%

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

   7. Applied rewrites 80.4%

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

   9. Applied rewrites 84.8%

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

   11. Applied rewrites 91.4%

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

   if 8.4999999999999998e157 < x-scale

   1. Initial program 46.9%

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

   3. Taylor expanded in 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 56.5%

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

   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 46.5%

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

   10. Applied rewrites 68.3%

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

   12. Applied rewrites 90.3%

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

Alternative 2: 91.4% accurate, 26.8× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{a \cdot b}{x-scale\_m}\\ \mathbf{if}\;y-scale \leq 1.4 \cdot 10^{-228}:\\ \;\;\;\;\frac{\left(\frac{\frac{-4}{y-scale}}{x-scale\_m} \cdot \left(a \cdot b\right)\right) \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{-4 \cdot t\_0}{y-scale}}{y-scale}\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (/ (* a b) x-scale_m)))
   (if (<= y-scale 1.4e-228)
     (/
      (* (* (/ (/ -4.0 y-scale) x-scale_m) (* a b)) (* a b))
      (* y-scale x-scale_m))
     (* t_0 (/ (/ (* -4.0 t_0) y-scale) y-scale)))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (a * b) / x_45_scale_m;
	double tmp;
	if (y_45_scale <= 1.4e-228) {
		tmp = ((((-4.0 / y_45_scale) / x_45_scale_m) * (a * b)) * (a * b)) / (y_45_scale * x_45_scale_m);
	} else {
		tmp = t_0 * (((-4.0 * t_0) / y_45_scale) / y_45_scale);
	}
	return tmp;
}

Fortran

x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * b) / x_45scale_m
    if (y_45scale <= 1.4d-228) then
        tmp = (((((-4.0d0) / y_45scale) / x_45scale_m) * (a * b)) * (a * b)) / (y_45scale * x_45scale_m)
    else
        tmp = t_0 * ((((-4.0d0) * t_0) / y_45scale) / y_45scale)
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (a * b) / x_45_scale_m;
	double tmp;
	if (y_45_scale <= 1.4e-228) {
		tmp = ((((-4.0 / y_45_scale) / x_45_scale_m) * (a * b)) * (a * b)) / (y_45_scale * x_45_scale_m);
	} else {
		tmp = t_0 * (((-4.0 * t_0) / y_45_scale) / y_45_scale);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = (a * b) / x_45_scale_m
	tmp = 0
	if y_45_scale <= 1.4e-228:
		tmp = ((((-4.0 / y_45_scale) / x_45_scale_m) * (a * b)) * (a * b)) / (y_45_scale * x_45_scale_m)
	else:
		tmp = t_0 * (((-4.0 * t_0) / y_45_scale) / y_45_scale)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(a * b) / x_45_scale_m)
	tmp = 0.0
	if (y_45_scale <= 1.4e-228)
		tmp = Float64(Float64(Float64(Float64(Float64(-4.0 / y_45_scale) / x_45_scale_m) * Float64(a * b)) * Float64(a * b)) / Float64(y_45_scale * x_45_scale_m));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(-4.0 * t_0) / y_45_scale) / y_45_scale));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = (a * b) / x_45_scale_m;
	tmp = 0.0;
	if (y_45_scale <= 1.4e-228)
		tmp = ((((-4.0 / y_45_scale) / x_45_scale_m) * (a * b)) * (a * b)) / (y_45_scale * x_45_scale_m);
	else
		tmp = t_0 * (((-4.0 * t_0) / y_45_scale) / y_45_scale);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(a * b), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]}, If[LessEqual[y$45$scale, 1.4e-228], N[(N[(N[(N[(N[(-4.0 / y$45$scale), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(-4.0 * t$95$0), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.4000000000000001e-228

   1. Initial program 18.5%

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 76.8%

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

   9. Applied rewrites 82.5%

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

   11. Applied rewrites 87.7%

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

   if 1.4000000000000001e-228 < y-scale

   1. Initial program 29.9%

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

   5. Applied rewrites 58.9%

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

   7. Applied rewrites 83.1%

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

   9. Applied rewrites 86.8%

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

   11. Applied rewrites 95.3%

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

Alternative 3: 89.7% accurate, 32.3× speedup

Math

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

FPCore

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

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return (a * b) * (((-4.0 * ((a * b) / x_45_scale_m)) / y_45_scale) / (y_45_scale * x_45_scale_m));
}

Fortran

x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = (a * b) * ((((-4.0d0) * ((a * b) / x_45scale_m)) / y_45scale) / (y_45scale * x_45scale_m))
end function

Java

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

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	return (a * b) * (((-4.0 * ((a * b) / x_45_scale_m)) / y_45_scale) / (y_45_scale * x_45_scale_m))

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	return Float64(Float64(a * b) * Float64(Float64(Float64(-4.0 * Float64(Float64(a * b) / x_45_scale_m)) / y_45_scale) / Float64(y_45_scale * x_45_scale_m)))
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = (a * b) * (((-4.0 * ((a * b) / x_45_scale_m)) / y_45_scale) / (y_45_scale * x_45_scale_m));
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := N[(N[(a * b), $MachinePrecision] * N[(N[(N[(-4.0 * N[(N[(a * b), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.7%

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

5. Applied rewrites 54.2%

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

7. Applied rewrites 79.7%

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

9. Applied rewrites 84.5%

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

11. Applied rewrites 88.9%

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

Alternative 4: 83.4% accurate, 35.9× speedup

Math

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

FPCore

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

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return (a * b) * ((-4.0 * ((a * b) / x_45_scale_m)) / ((y_45_scale * x_45_scale_m) * y_45_scale));
}

Fortran

x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = (a * b) * (((-4.0d0) * ((a * b) / x_45scale_m)) / ((y_45scale * x_45scale_m) * y_45scale))
end function

Java

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

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	return (a * b) * ((-4.0 * ((a * b) / x_45_scale_m)) / ((y_45_scale * x_45_scale_m) * y_45_scale))

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	return Float64(Float64(a * b) * Float64(Float64(-4.0 * Float64(Float64(a * b) / x_45_scale_m)) / Float64(Float64(y_45_scale * x_45_scale_m) * y_45_scale)))
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = (a * b) * ((-4.0 * ((a * b) / x_45_scale_m)) / ((y_45_scale * x_45_scale_m) * y_45_scale));
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := N[(N[(a * b), $MachinePrecision] * N[(N[(-4.0 * N[(N[(a * b), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale$95$m), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.7%

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

5. Applied rewrites 54.2%

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

7. Applied rewrites 79.7%

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

9. Applied rewrites 84.5%

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

11. Applied rewrites 83.5%

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

Alternative 5: 75.3% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = b / (y_45scale * x_45scale_m)
    code = ((-4.0d0) * (t_0 * t_0)) * (a * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.7%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 47.4%

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

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 47.3%

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

10. Applied rewrites 73.7%

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

Alternative 6: 61.0% accurate, 40.5× speedup

Math

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

FPCore

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

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return (-4.0 * ((b * b) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m)))) * (a * a);
}

Fortran

x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = ((-4.0d0) * ((b * b) / ((y_45scale * x_45scale_m) * (y_45scale * x_45scale_m)))) * (a * a)
end function

Java

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

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	return (-4.0 * ((b * b) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m)))) * (a * a)

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	return Float64(Float64(-4.0 * Float64(Float64(b * b) / Float64(Float64(y_45_scale * x_45_scale_m) * Float64(y_45_scale * x_45_scale_m)))) * Float64(a * a))
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = (-4.0 * ((b * b) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m)))) * (a * a);
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := N[(N[(-4.0 * N[(N[(b * b), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.7%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 47.4%

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

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 47.3%

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

10. Applied rewrites 60.7%

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

Alternative 7: 48.3% accurate, 40.5× speedup

Math

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

FPCore

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

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return (-4.0 * ((b * b) / ((x_45_scale_m * x_45_scale_m) * (y_45_scale * y_45_scale)))) * (a * a);
}

Fortran

x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = ((-4.0d0) * ((b * b) / ((x_45scale_m * x_45scale_m) * (y_45scale * y_45scale)))) * (a * a)
end function

Java

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

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	return (-4.0 * ((b * b) / ((x_45_scale_m * x_45_scale_m) * (y_45_scale * y_45_scale)))) * (a * a)

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	return Float64(Float64(-4.0 * Float64(Float64(b * b) / Float64(Float64(x_45_scale_m * x_45_scale_m) * Float64(y_45_scale * y_45_scale)))) * Float64(a * a))
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = (-4.0 * ((b * b) / ((x_45_scale_m * x_45_scale_m) * (y_45_scale * y_45_scale)))) * (a * a);
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := N[(N[(-4.0 * N[(N[(b * b), $MachinePrecision] / N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.7%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 47.4%

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

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 47.3%

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

Reproduce

herbie shell --seed 2025018 
(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))))