Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.9% → 91.8%

Time: 21.8s

Alternatives: 9

Speedup: 40.5×

• 33.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 24.9% 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: 91.8% accurate, 26.8× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{b \cdot a}{x-scale}\\ t_1 := \frac{-4}{y-scale\_m} \cdot t\_0\\ \mathbf{if}\;y-scale\_m \leq 5 \cdot 10^{-179}:\\ \;\;\;\;\frac{b \cdot a}{y-scale\_m} \cdot \frac{t\_1}{x-scale}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{t\_1}{y-scale\_m}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0 (/ (* b a) x-scale)) (t_1 (* (/ -4.0 y-scale_m) t_0)))
   (if (<= y-scale_m 5e-179)
     (* (/ (* b a) y-scale_m) (/ t_1 x-scale))
     (* t_0 (/ t_1 y-scale_m)))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (b * a) / x_45_scale;
	double t_1 = (-4.0 / y_45_scale_m) * t_0;
	double tmp;
	if (y_45_scale_m <= 5e-179) {
		tmp = ((b * a) / y_45_scale_m) * (t_1 / x_45_scale);
	} else {
		tmp = t_0 * (t_1 / y_45_scale_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * a) / x_45scale
    t_1 = ((-4.0d0) / y_45scale_m) * t_0
    if (y_45scale_m <= 5d-179) then
        tmp = ((b * a) / y_45scale_m) * (t_1 / x_45scale)
    else
        tmp = t_0 * (t_1 / y_45scale_m)
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (b * a) / x_45_scale;
	double t_1 = (-4.0 / y_45_scale_m) * t_0;
	double tmp;
	if (y_45_scale_m <= 5e-179) {
		tmp = ((b * a) / y_45_scale_m) * (t_1 / x_45_scale);
	} else {
		tmp = t_0 * (t_1 / y_45_scale_m);
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	t_0 = (b * a) / x_45_scale
	t_1 = (-4.0 / y_45_scale_m) * t_0
	tmp = 0
	if y_45_scale_m <= 5e-179:
		tmp = ((b * a) / y_45_scale_m) * (t_1 / x_45_scale)
	else:
		tmp = t_0 * (t_1 / y_45_scale_m)
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64(Float64(b * a) / x_45_scale)
	t_1 = Float64(Float64(-4.0 / y_45_scale_m) * t_0)
	tmp = 0.0
	if (y_45_scale_m <= 5e-179)
		tmp = Float64(Float64(Float64(b * a) / y_45_scale_m) * Float64(t_1 / x_45_scale));
	else
		tmp = Float64(t_0 * Float64(t_1 / y_45_scale_m));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = (b * a) / x_45_scale;
	t_1 = (-4.0 / y_45_scale_m) * t_0;
	tmp = 0.0;
	if (y_45_scale_m <= 5e-179)
		tmp = ((b * a) / y_45_scale_m) * (t_1 / x_45_scale);
	else
		tmp = t_0 * (t_1 / y_45_scale_m);
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(b * a), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 / y$45$scale$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 5e-179], N[(N[(N[(b * a), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision] * N[(t$95$1 / x$45$scale), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$1 / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.9999999999999998e-179

   1. Initial program 17.5%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 48.6

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

   5. Applied rewrites 48.6%

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

   7. Applied rewrites 77.7%

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

   9. Applied rewrites 80.6%

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

   11. Applied rewrites 88.7%

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

   if 4.9999999999999998e-179 < y-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 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 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 79.8%

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

   9. Applied rewrites 86.4%

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

   11. Applied rewrites 90.2%

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

Alternative 2: 75.0% accurate, 29.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 2.15 \cdot 10^{-142} \lor \neg \left(x-scale \leq 6 \cdot 10^{+119}\right):\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale\_m \cdot x-scale} \cdot \frac{a}{y-scale\_m \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y-scale\_m \cdot y-scale\_m} \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{x-scale \cdot x-scale}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (if (or (<= x-scale 2.15e-142) (not (<= x-scale 6e+119)))
   (*
    (* (/ (* -4.0 a) (* y-scale_m x-scale)) (/ a (* y-scale_m x-scale)))
    (* b b))
   (*
    (/ -4.0 (* y-scale_m y-scale_m))
    (/ (* (* a b) (* a b)) (* x-scale x-scale)))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if ((x_45_scale <= 2.15e-142) || !(x_45_scale <= 6e+119)) {
		tmp = (((-4.0 * a) / (y_45_scale_m * x_45_scale)) * (a / (y_45_scale_m * x_45_scale))) * (b * b);
	} else {
		tmp = (-4.0 / (y_45_scale_m * y_45_scale_m)) * (((a * b) * (a * b)) / (x_45_scale * x_45_scale));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
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_m
    real(8) :: tmp
    if ((x_45scale <= 2.15d-142) .or. (.not. (x_45scale <= 6d+119))) then
        tmp = ((((-4.0d0) * a) / (y_45scale_m * x_45scale)) * (a / (y_45scale_m * x_45scale))) * (b * b)
    else
        tmp = ((-4.0d0) / (y_45scale_m * y_45scale_m)) * (((a * b) * (a * b)) / (x_45scale * x_45scale))
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if ((x_45_scale <= 2.15e-142) || !(x_45_scale <= 6e+119)) {
		tmp = (((-4.0 * a) / (y_45_scale_m * x_45_scale)) * (a / (y_45_scale_m * x_45_scale))) * (b * b);
	} else {
		tmp = (-4.0 / (y_45_scale_m * y_45_scale_m)) * (((a * b) * (a * b)) / (x_45_scale * x_45_scale));
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	tmp = 0
	if (x_45_scale <= 2.15e-142) or not (x_45_scale <= 6e+119):
		tmp = (((-4.0 * a) / (y_45_scale_m * x_45_scale)) * (a / (y_45_scale_m * x_45_scale))) * (b * b)
	else:
		tmp = (-4.0 / (y_45_scale_m * y_45_scale_m)) * (((a * b) * (a * b)) / (x_45_scale * x_45_scale))
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0
	if ((x_45_scale <= 2.15e-142) || !(x_45_scale <= 6e+119))
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) / Float64(y_45_scale_m * x_45_scale)) * Float64(a / Float64(y_45_scale_m * x_45_scale))) * Float64(b * b));
	else
		tmp = Float64(Float64(-4.0 / Float64(y_45_scale_m * y_45_scale_m)) * Float64(Float64(Float64(a * b) * Float64(a * b)) / Float64(x_45_scale * x_45_scale)));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0;
	if ((x_45_scale <= 2.15e-142) || ~((x_45_scale <= 6e+119)))
		tmp = (((-4.0 * a) / (y_45_scale_m * x_45_scale)) * (a / (y_45_scale_m * x_45_scale))) * (b * b);
	else
		tmp = (-4.0 / (y_45_scale_m * y_45_scale_m)) * (((a * b) * (a * b)) / (x_45_scale * x_45_scale));
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := If[Or[LessEqual[x$45$scale, 2.15e-142], N[Not[LessEqual[x$45$scale, 6e+119]], $MachinePrecision]], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.1499999999999999e-142 or 6.00000000000000002e119 < x-scale

   1. Initial program 21.5%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 44.5%

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

   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 57.3%

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

   10. Applied rewrites 71.8%

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

   if 2.1499999999999999e-142 < x-scale < 6.00000000000000002e119

   1. Initial program 21.8%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 82.6%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 2.15 \cdot 10^{-142} \lor \neg \left(x-scale \leq 6 \cdot 10^{+119}\right):\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y-scale \cdot y-scale} \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{x-scale \cdot x-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 29.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 8.5 \cdot 10^{-26} \lor \neg \left(x-scale \leq 6 \cdot 10^{+119}\right):\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale\_m \cdot x-scale} \cdot \frac{a}{y-scale\_m \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(a \cdot b\right) \cdot a\right) \cdot b\right) \cdot 4}{\left(x-scale \cdot x-scale\right) \cdot \left(\left(-y-scale\_m\right) \cdot y-scale\_m\right)}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (if (or (<= x-scale 8.5e-26) (not (<= x-scale 6e+119)))
   (*
    (* (/ (* -4.0 a) (* y-scale_m x-scale)) (/ a (* y-scale_m x-scale)))
    (* b b))
   (/
    (* (* (* (* a b) a) b) 4.0)
    (* (* x-scale x-scale) (* (- y-scale_m) y-scale_m)))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if ((x_45_scale <= 8.5e-26) || !(x_45_scale <= 6e+119)) {
		tmp = (((-4.0 * a) / (y_45_scale_m * x_45_scale)) * (a / (y_45_scale_m * x_45_scale))) * (b * b);
	} else {
		tmp = ((((a * b) * a) * b) * 4.0) / ((x_45_scale * x_45_scale) * (-y_45_scale_m * y_45_scale_m));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
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_m
    real(8) :: tmp
    if ((x_45scale <= 8.5d-26) .or. (.not. (x_45scale <= 6d+119))) then
        tmp = ((((-4.0d0) * a) / (y_45scale_m * x_45scale)) * (a / (y_45scale_m * x_45scale))) * (b * b)
    else
        tmp = ((((a * b) * a) * b) * 4.0d0) / ((x_45scale * x_45scale) * (-y_45scale_m * y_45scale_m))
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if ((x_45_scale <= 8.5e-26) || !(x_45_scale <= 6e+119)) {
		tmp = (((-4.0 * a) / (y_45_scale_m * x_45_scale)) * (a / (y_45_scale_m * x_45_scale))) * (b * b);
	} else {
		tmp = ((((a * b) * a) * b) * 4.0) / ((x_45_scale * x_45_scale) * (-y_45_scale_m * y_45_scale_m));
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	tmp = 0
	if (x_45_scale <= 8.5e-26) or not (x_45_scale <= 6e+119):
		tmp = (((-4.0 * a) / (y_45_scale_m * x_45_scale)) * (a / (y_45_scale_m * x_45_scale))) * (b * b)
	else:
		tmp = ((((a * b) * a) * b) * 4.0) / ((x_45_scale * x_45_scale) * (-y_45_scale_m * y_45_scale_m))
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0
	if ((x_45_scale <= 8.5e-26) || !(x_45_scale <= 6e+119))
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) / Float64(y_45_scale_m * x_45_scale)) * Float64(a / Float64(y_45_scale_m * x_45_scale))) * Float64(b * b));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * b) * a) * b) * 4.0) / Float64(Float64(x_45_scale * x_45_scale) * Float64(Float64(-y_45_scale_m) * y_45_scale_m)));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0;
	if ((x_45_scale <= 8.5e-26) || ~((x_45_scale <= 6e+119)))
		tmp = (((-4.0 * a) / (y_45_scale_m * x_45_scale)) * (a / (y_45_scale_m * x_45_scale))) * (b * b);
	else
		tmp = ((((a * b) * a) * b) * 4.0) / ((x_45_scale * x_45_scale) * (-y_45_scale_m * y_45_scale_m));
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := If[Or[LessEqual[x$45$scale, 8.5e-26], N[Not[LessEqual[x$45$scale, 6e+119]], $MachinePrecision]], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * b), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] * 4.0), $MachinePrecision] / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[((-y$45$scale$95$m) * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 8.50000000000000004e-26 or 6.00000000000000002e119 < x-scale

   1. Initial program 20.8%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 45.0%

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

   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 57.9%

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

   10. Applied rewrites 72.3%

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

   if 8.50000000000000004e-26 < x-scale < 6.00000000000000002e119

   1. Initial program 27.9%

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

   3. Taylor expanded in angle around 0

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

      1. 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 55.7

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 82.7%

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

   9. Applied rewrites 79.4%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 8.5 \cdot 10^{-26} \lor \neg \left(x-scale \leq 6 \cdot 10^{+119}\right):\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(a \cdot b\right) \cdot a\right) \cdot b\right) \cdot 4}{\left(x-scale \cdot x-scale\right) \cdot \left(\left(-y-scale\right) \cdot y-scale\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
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_m
    real(8) :: t_0
    t_0 = (b * a) / x_45scale
    code = t_0 * ((((-4.0d0) / y_45scale_m) * t_0) / y_45scale_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 52.5

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

5. Applied rewrites 52.5%

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

7. Applied rewrites 78.5%

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

9. Applied rewrites 83.0%

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

11. Applied rewrites 90.8%

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

Alternative 5: 62.2% accurate, 31.2× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 10^{-165} \lor \neg \left(a \leq 1.25 \cdot 10^{+121}\right):\\ \;\;\;\;\frac{\left(\left(\left(a \cdot b\right) \cdot a\right) \cdot b\right) \cdot 4}{\left(x-scale \cdot x-scale\right) \cdot \left(\left(-y-scale\_m\right) \cdot y-scale\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-4 \cdot a\right) \cdot a}{\left(y-scale\_m \cdot x-scale\right) \cdot \left(y-scale\_m \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (if (or (<= a 1e-165) (not (<= a 1.25e+121)))
   (/
    (* (* (* (* a b) a) b) 4.0)
    (* (* x-scale x-scale) (* (- y-scale_m) y-scale_m)))
   (*
    (/ (* (* -4.0 a) a) (* (* y-scale_m x-scale) (* y-scale_m x-scale)))
    (* b b))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if ((a <= 1e-165) || !(a <= 1.25e+121)) {
		tmp = ((((a * b) * a) * b) * 4.0) / ((x_45_scale * x_45_scale) * (-y_45_scale_m * y_45_scale_m));
	} else {
		tmp = (((-4.0 * a) * a) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
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_m
    real(8) :: tmp
    if ((a <= 1d-165) .or. (.not. (a <= 1.25d+121))) then
        tmp = ((((a * b) * a) * b) * 4.0d0) / ((x_45scale * x_45scale) * (-y_45scale_m * y_45scale_m))
    else
        tmp = ((((-4.0d0) * a) * a) / ((y_45scale_m * x_45scale) * (y_45scale_m * x_45scale))) * (b * b)
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if ((a <= 1e-165) || !(a <= 1.25e+121)) {
		tmp = ((((a * b) * a) * b) * 4.0) / ((x_45_scale * x_45_scale) * (-y_45_scale_m * y_45_scale_m));
	} else {
		tmp = (((-4.0 * a) * a) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	tmp = 0
	if (a <= 1e-165) or not (a <= 1.25e+121):
		tmp = ((((a * b) * a) * b) * 4.0) / ((x_45_scale * x_45_scale) * (-y_45_scale_m * y_45_scale_m))
	else:
		tmp = (((-4.0 * a) * a) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b)
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0
	if ((a <= 1e-165) || !(a <= 1.25e+121))
		tmp = Float64(Float64(Float64(Float64(Float64(a * b) * a) * b) * 4.0) / Float64(Float64(x_45_scale * x_45_scale) * Float64(Float64(-y_45_scale_m) * y_45_scale_m)));
	else
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) * a) / Float64(Float64(y_45_scale_m * x_45_scale) * Float64(y_45_scale_m * x_45_scale))) * Float64(b * b));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0;
	if ((a <= 1e-165) || ~((a <= 1.25e+121)))
		tmp = ((((a * b) * a) * b) * 4.0) / ((x_45_scale * x_45_scale) * (-y_45_scale_m * y_45_scale_m));
	else
		tmp = (((-4.0 * a) * a) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := If[Or[LessEqual[a, 1e-165], N[Not[LessEqual[a, 1.25e+121]], $MachinePrecision]], N[(N[(N[(N[(N[(a * b), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] * 4.0), $MachinePrecision] / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[((-y$45$scale$95$m) * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] * a), $MachinePrecision] / N[(N[(y$45$scale$95$m * x$45$scale), $MachinePrecision] * N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1e-165 or 1.25000000000000002e121 < a

   1. Initial program 20.0%

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

   3. Taylor expanded in 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 50.7

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

   5. Applied rewrites 50.7%

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

   7. Applied rewrites 64.4%

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

   9. Applied rewrites 63.4%

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

   if 1e-165 < a < 1.25000000000000002e121

   1. Initial program 27.2%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 50.8%

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

   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 70.4%

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

   10. Applied rewrites 70.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 10^{-165} \lor \neg \left(a \leq 1.25 \cdot 10^{+121}\right):\\ \;\;\;\;\frac{\left(\left(\left(a \cdot b\right) \cdot a\right) \cdot b\right) \cdot 4}{\left(x-scale \cdot x-scale\right) \cdot \left(\left(-y-scale\right) \cdot y-scale\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-4 \cdot a\right) \cdot a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.5% accurate, 32.3× speedup

Math

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

FPCore

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

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return (((-4.0 / y_45_scale_m) * ((b * a) / x_45_scale)) * (b * a)) / (y_45_scale_m * x_45_scale);
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
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_m
    code = ((((-4.0d0) / y_45scale_m) * ((b * a) / x_45scale)) * (b * a)) / (y_45scale_m * x_45scale)
end function

Java

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

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return (((-4.0 / y_45_scale_m) * ((b * a) / x_45_scale)) * (b * a)) / (y_45_scale_m * x_45_scale)

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(Float64(Float64(-4.0 / y_45_scale_m) * Float64(Float64(b * a) / x_45_scale)) * Float64(b * a)) / Float64(y_45_scale_m * x_45_scale))
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = (((-4.0 / y_45_scale_m) * ((b * a) / x_45_scale)) * (b * a)) / (y_45_scale_m * x_45_scale);
end

Wolfram

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

Derivation

1. Initial program 21.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 52.5

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

5. Applied rewrites 52.5%

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

7. Applied rewrites 78.5%

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

9. Applied rewrites 83.0%

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

11. Applied rewrites 89.0%

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

Alternative 7: 90.0% accurate, 32.3× speedup

Math

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

FPCore

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

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return (b * a) * (((-4.0 / y_45_scale_m) * ((b * a) / x_45_scale)) / (y_45_scale_m * x_45_scale));
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
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_m
    code = (b * a) * ((((-4.0d0) / y_45scale_m) * ((b * a) / x_45scale)) / (y_45scale_m * x_45scale))
end function

Java

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

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return (b * a) * (((-4.0 / y_45_scale_m) * ((b * a) / x_45_scale)) / (y_45_scale_m * x_45_scale))

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(b * a) * Float64(Float64(Float64(-4.0 / y_45_scale_m) * Float64(Float64(b * a) / x_45_scale)) / Float64(y_45_scale_m * x_45_scale)))
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = (b * a) * (((-4.0 / y_45_scale_m) * ((b * a) / x_45_scale)) / (y_45_scale_m * x_45_scale));
end

Wolfram

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

Derivation

1. Initial program 21.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 52.5

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

5. Applied rewrites 52.5%

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

7. Applied rewrites 78.5%

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

9. Applied rewrites 83.0%

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

11. Applied rewrites 88.3%

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

Alternative 8: 61.6% accurate, 40.5× speedup

Math

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

FPCore

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

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return (((-4.0 * a) * a) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
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_m
    code = ((((-4.0d0) * a) * a) / ((y_45scale_m * x_45scale) * (y_45scale_m * x_45scale))) * (b * b)
end function

Java

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

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return (((-4.0 * a) * a) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b)

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(Float64(Float64(-4.0 * a) * a) / Float64(Float64(y_45_scale_m * x_45_scale) * Float64(y_45_scale_m * x_45_scale))) * Float64(b * b))
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = (((-4.0 * a) * a) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
end

Wolfram

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

Derivation

1. Initial program 21.6%

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

3. Taylor expanded in 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 45.8%

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

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 57.7%

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

10. Applied rewrites 57.7%

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

Alternative 9: 61.6% accurate, 40.5× speedup

Math

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

FPCore

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

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return ((-4.0 * (a * a)) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
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_m
    code = (((-4.0d0) * (a * a)) / ((y_45scale_m * x_45scale) * (y_45scale_m * x_45scale))) * (b * b)
end function

Java

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

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return ((-4.0 * (a * a)) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b)

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(Float64(-4.0 * Float64(a * a)) / Float64(Float64(y_45_scale_m * x_45_scale) * Float64(y_45_scale_m * x_45_scale))) * Float64(b * b))
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = ((-4.0 * (a * a)) / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
end

Wolfram

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

Derivation

1. Initial program 21.6%

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

3. Taylor expanded in 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 45.8%

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

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 57.7%

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

Reproduce

herbie shell --seed 2024354 
(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))))