Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.1% → 89.7%

Time: 21.7s

Alternatives: 9

Speedup: 40.5×

• 34.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 25.1% 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: 89.7% accurate, 32.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 58.5

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

5. Applied rewrites 58.5%

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

7. Applied rewrites 82.1%

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

9. Applied rewrites 87.5%

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

11. Applied rewrites 91.5%

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

Alternative 2: 77.7% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 1.02 \cdot 10^{-170} \lor \neg \left(y-scale \leq 4.8 \cdot 10^{+210}\right):\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a \cdot b}{x-scale} \cdot a\right) \cdot \frac{-4 \cdot b}{\left(y-scale \cdot y-scale\right) \cdot x-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (or (<= y-scale 1.02e-170) (not (<= y-scale 4.8e+210)))
   (* (* (/ (* -4.0 a) (* y-scale x-scale)) (/ a (* y-scale x-scale))) (* b b))
   (*
    (* (/ (* a b) x-scale) a)
    (/ (* -4.0 b) (* (* y-scale y-scale) x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if ((y_45_scale <= 1.02e-170) || !(y_45_scale <= 4.8e+210)) {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	} else {
		tmp = (((a * b) / x_45_scale) * a) * ((-4.0 * b) / ((y_45_scale * y_45_scale) * x_45_scale));
	}
	return tmp;
}

Fortran

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, 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
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if ((y_45scale <= 1.02d-170) .or. (.not. (y_45scale <= 4.8d+210))) then
        tmp = ((((-4.0d0) * a) / (y_45scale * x_45scale)) * (a / (y_45scale * x_45scale))) * (b * b)
    else
        tmp = (((a * b) / x_45scale) * a) * (((-4.0d0) * b) / ((y_45scale * y_45scale) * x_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 ((y_45_scale <= 1.02e-170) || !(y_45_scale <= 4.8e+210)) {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	} else {
		tmp = (((a * b) / x_45_scale) * a) * ((-4.0 * b) / ((y_45_scale * y_45_scale) * x_45_scale));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if (y_45_scale <= 1.02e-170) or not (y_45_scale <= 4.8e+210):
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b)
	else:
		tmp = (((a * b) / x_45_scale) * a) * ((-4.0 * b) / ((y_45_scale * y_45_scale) * x_45_scale))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if ((y_45_scale <= 1.02e-170) || !(y_45_scale <= 4.8e+210))
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) / Float64(y_45_scale * x_45_scale)) * Float64(a / Float64(y_45_scale * x_45_scale))) * Float64(b * b));
	else
		tmp = Float64(Float64(Float64(Float64(a * b) / x_45_scale) * a) * Float64(Float64(-4.0 * b) / Float64(Float64(y_45_scale * y_45_scale) * x_45_scale)));
	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 <= 1.02e-170) || ~((y_45_scale <= 4.8e+210)))
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	else
		tmp = (((a * b) / x_45_scale) * a) * ((-4.0 * b) / ((y_45_scale * y_45_scale) * x_45_scale));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[Or[LessEqual[y$45$scale, 1.02e-170], N[Not[LessEqual[y$45$scale, 4.8e+210]], $MachinePrecision]], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * b), $MachinePrecision] / x$45$scale), $MachinePrecision] * a), $MachinePrecision] * N[(N[(-4.0 * b), $MachinePrecision] / N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.02e-170 or 4.79999999999999977e210 < y-scale

   1. Initial program 29.5%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 45.2%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 45.6%

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

   10. Applied rewrites 75.5%

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

   if 1.02e-170 < y-scale < 4.79999999999999977e210

   1. Initial program 23.5%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 78.2%

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

   9. Applied rewrites 86.3%

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

   11. Applied rewrites 83.9%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.02 \cdot 10^{-170} \lor \neg \left(y-scale \leq 4.8 \cdot 10^{+210}\right):\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a \cdot b}{x-scale} \cdot a\right) \cdot \frac{-4 \cdot b}{\left(y-scale \cdot y-scale\right) \cdot x-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.7% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.12 \cdot 10^{-213} \lor \neg \left(b \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(-4 \cdot b\right) \cdot \frac{\frac{a \cdot b}{x-scale} \cdot a}{\left(y-scale \cdot y-scale\right) \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (or (<= b 1.12e-213) (not (<= b 1.35e+154)))
   (* (* -4.0 b) (/ (* (/ (* a b) x-scale) a) (* (* y-scale y-scale) x-scale)))
   (*
    (* (/ (* -4.0 a) (* y-scale x-scale)) (/ a (* y-scale x-scale)))
    (* b b))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if ((b <= 1.12e-213) || !(b <= 1.35e+154)) {
		tmp = (-4.0 * b) * ((((a * b) / x_45_scale) * a) / ((y_45_scale * y_45_scale) * x_45_scale));
	} else {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	}
	return tmp;
}

Fortran

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, 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
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if ((b <= 1.12d-213) .or. (.not. (b <= 1.35d+154))) then
        tmp = ((-4.0d0) * b) * ((((a * b) / x_45scale) * a) / ((y_45scale * y_45scale) * x_45scale))
    else
        tmp = ((((-4.0d0) * a) / (y_45scale * x_45scale)) * (a / (y_45scale * x_45scale))) * (b * b)
    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 ((b <= 1.12e-213) || !(b <= 1.35e+154)) {
		tmp = (-4.0 * b) * ((((a * b) / x_45_scale) * a) / ((y_45_scale * y_45_scale) * x_45_scale));
	} else {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if (b <= 1.12e-213) or not (b <= 1.35e+154):
		tmp = (-4.0 * b) * ((((a * b) / x_45_scale) * a) / ((y_45_scale * y_45_scale) * x_45_scale))
	else:
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if ((b <= 1.12e-213) || !(b <= 1.35e+154))
		tmp = Float64(Float64(-4.0 * b) * Float64(Float64(Float64(Float64(a * b) / x_45_scale) * a) / Float64(Float64(y_45_scale * y_45_scale) * x_45_scale)));
	else
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) / Float64(y_45_scale * x_45_scale)) * Float64(a / Float64(y_45_scale * x_45_scale))) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if ((b <= 1.12e-213) || ~((b <= 1.35e+154)))
		tmp = (-4.0 * b) * ((((a * b) / x_45_scale) * a) / ((y_45_scale * y_45_scale) * x_45_scale));
	else
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[Or[LessEqual[b, 1.12e-213], N[Not[LessEqual[b, 1.35e+154]], $MachinePrecision]], N[(N[(-4.0 * b), $MachinePrecision] * N[(N[(N[(N[(a * b), $MachinePrecision] / x$45$scale), $MachinePrecision] * a), $MachinePrecision] / N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1200000000000001e-213 or 1.35000000000000003e154 < b

   1. Initial program 25.7%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 55.4

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

   5. Applied rewrites 55.4%

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

   7. Applied rewrites 82.0%

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

   9. Applied rewrites 87.0%

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

   11. Applied rewrites 73.2%

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

   if 1.1200000000000001e-213 < b < 1.35000000000000003e154

   1. Initial program 31.3%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 53.1%

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

   10. Applied rewrites 86.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.12 \cdot 10^{-213} \lor \neg \left(b \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(-4 \cdot b\right) \cdot \frac{\frac{a \cdot b}{x-scale} \cdot a}{\left(y-scale \cdot y-scale\right) \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.2% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 1.5 \cdot 10^{-141} \lor \neg \left(x-scale \leq 1.62 \cdot 10^{+181}\right):\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot a\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale} \cdot \frac{a}{\left(x-scale \cdot x-scale\right) \cdot y-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (or (<= x-scale 1.5e-141) (not (<= x-scale 1.62e+181)))
   (* (/ (* -4.0 (* a a)) (* (* y-scale x-scale) (* y-scale x-scale))) (* b b))
   (*
    (* (/ (* -4.0 a) y-scale) (/ a (* (* x-scale x-scale) y-scale)))
    (* b b))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if ((x_45_scale <= 1.5e-141) || !(x_45_scale <= 1.62e+181)) {
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * b);
	} else {
		tmp = (((-4.0 * a) / y_45_scale) * (a / ((x_45_scale * x_45_scale) * y_45_scale))) * (b * b);
	}
	return tmp;
}

Fortran

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, 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
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if ((x_45scale <= 1.5d-141) .or. (.not. (x_45scale <= 1.62d+181))) then
        tmp = (((-4.0d0) * (a * a)) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * (b * b)
    else
        tmp = ((((-4.0d0) * a) / y_45scale) * (a / ((x_45scale * x_45scale) * y_45scale))) * (b * b)
    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 <= 1.5e-141) || !(x_45_scale <= 1.62e+181)) {
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * b);
	} else {
		tmp = (((-4.0 * a) / y_45_scale) * (a / ((x_45_scale * x_45_scale) * y_45_scale))) * (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if (x_45_scale <= 1.5e-141) or not (x_45_scale <= 1.62e+181):
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * b)
	else:
		tmp = (((-4.0 * a) / y_45_scale) * (a / ((x_45_scale * x_45_scale) * y_45_scale))) * (b * b)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if ((x_45_scale <= 1.5e-141) || !(x_45_scale <= 1.62e+181))
		tmp = Float64(Float64(Float64(-4.0 * Float64(a * a)) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * Float64(b * b));
	else
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) / y_45_scale) * Float64(a / Float64(Float64(x_45_scale * x_45_scale) * y_45_scale))) * Float64(b * b));
	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 <= 1.5e-141) || ~((x_45_scale <= 1.62e+181)))
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * (b * b);
	else
		tmp = (((-4.0 * a) / y_45_scale) * (a / ((x_45_scale * x_45_scale) * y_45_scale))) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[Or[LessEqual[x$45$scale, 1.5e-141], N[Not[LessEqual[x$45$scale, 1.62e+181]], $MachinePrecision]], N[(N[(N[(-4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(a / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.49999999999999992e-141 or 1.62000000000000001e181 < x-scale

   1. Initial program 27.7%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 44.8%

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

   10. Applied rewrites 60.7%

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

   if 1.49999999999999992e-141 < x-scale < 1.62000000000000001e181

   1. Initial program 26.8%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 59.1%

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

   10. Applied rewrites 79.5%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.5 \cdot 10^{-141} \lor \neg \left(x-scale \leq 1.62 \cdot 10^{+181}\right):\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot a\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale} \cdot \frac{a}{\left(x-scale \cdot x-scale\right) \cdot y-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 2 \cdot 10^{-215}:\\ \;\;\;\;\frac{\left(\left(a \cdot b\right) \cdot \frac{a \cdot b}{x-scale}\right) \cdot \frac{-4}{y-scale}}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \frac{\frac{b}{x-scale} \cdot \left(a \cdot \frac{-4}{y-scale}\right)}{y-scale \cdot x-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= angle 2e-215)
   (/ (* (* (* a b) (/ (* a b) x-scale)) (/ -4.0 y-scale)) (* y-scale x-scale))
   (*
    (* a b)
    (/ (* (/ b x-scale) (* a (/ -4.0 y-scale))) (* y-scale x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (angle <= 2e-215) {
		tmp = (((a * b) * ((a * b) / x_45_scale)) * (-4.0 / y_45_scale)) / (y_45_scale * x_45_scale);
	} else {
		tmp = (a * b) * (((b / x_45_scale) * (a * (-4.0 / y_45_scale))) / (y_45_scale * x_45_scale));
	}
	return tmp;
}

Fortran

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, 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
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (angle <= 2d-215) then
        tmp = (((a * b) * ((a * b) / x_45scale)) * ((-4.0d0) / y_45scale)) / (y_45scale * x_45scale)
    else
        tmp = (a * b) * (((b / x_45scale) * (a * ((-4.0d0) / y_45scale))) / (y_45scale * x_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 (angle <= 2e-215) {
		tmp = (((a * b) * ((a * b) / x_45_scale)) * (-4.0 / y_45_scale)) / (y_45_scale * x_45_scale);
	} else {
		tmp = (a * b) * (((b / x_45_scale) * (a * (-4.0 / y_45_scale))) / (y_45_scale * x_45_scale));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if angle <= 2e-215:
		tmp = (((a * b) * ((a * b) / x_45_scale)) * (-4.0 / y_45_scale)) / (y_45_scale * x_45_scale)
	else:
		tmp = (a * b) * (((b / x_45_scale) * (a * (-4.0 / y_45_scale))) / (y_45_scale * x_45_scale))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (angle <= 2e-215)
		tmp = Float64(Float64(Float64(Float64(a * b) * Float64(Float64(a * b) / x_45_scale)) * Float64(-4.0 / y_45_scale)) / Float64(y_45_scale * x_45_scale));
	else
		tmp = Float64(Float64(a * b) * Float64(Float64(Float64(b / x_45_scale) * Float64(a * Float64(-4.0 / y_45_scale))) / Float64(y_45_scale * x_45_scale)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (angle <= 2e-215)
		tmp = (((a * b) * ((a * b) / x_45_scale)) * (-4.0 / y_45_scale)) / (y_45_scale * x_45_scale);
	else
		tmp = (a * b) * (((b / x_45_scale) * (a * (-4.0 / y_45_scale))) / (y_45_scale * x_45_scale));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[angle, 2e-215], N[(N[(N[(N[(a * b), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(a * b), $MachinePrecision] * N[(N[(N[(b / x$45$scale), $MachinePrecision] * N[(a * N[(-4.0 / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.00000000000000008e-215

   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 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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 59.9

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 84.7%

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

   9. Applied rewrites 89.7%

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

   if 2.00000000000000008e-215 < angle

   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 78.2%

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

   9. Applied rewrites 84.3%

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

   11. Applied rewrites 86.6%

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

Alternative 6: 86.5% accurate, 32.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 58.5

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

5. Applied rewrites 58.5%

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

7. Applied rewrites 82.1%

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

9. Applied rewrites 87.5%

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

11. Applied rewrites 88.2%

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

Alternative 7: 76.3% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 46.8%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 48.3%

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

10. Applied rewrites 76.2%

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

Alternative 8: 62.2% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 46.8%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 48.3%

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

10. Applied rewrites 61.4%

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

Alternative 9: 60.5% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 46.8%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 48.3%

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

9. Taylor expanded in x-scale around 0

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

11. Applied rewrites 59.9%

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

Reproduce

herbie shell --seed 2024358 
(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))))