Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.5% → 89.6%

Time: 20.4s

Alternatives: 9

Speedup: 40.5×

• 33.5% 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: 24.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) (PI)))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

Alternative 1: 89.6% accurate, 12.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 1.95 \cdot 10^{+201}:\\ \;\;\;\;\frac{b \cdot a}{y-scale} \cdot \frac{\frac{-4}{y-scale} \cdot \frac{b \cdot a}{x-scale\_m}}{x-scale\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(a \cdot b\right)}^{2} \cdot \frac{\frac{-4}{y-scale}}{x-scale\_m}}{y-scale \cdot x-scale\_m}\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (if (<= x-scale_m 1.95e+201)
   (*
    (/ (* b a) y-scale)
    (/ (* (/ -4.0 y-scale) (/ (* b a) x-scale_m)) x-scale_m))
   (/
    (* (pow (* a b) 2.0) (/ (/ -4.0 y-scale) x-scale_m))
    (* y-scale x-scale_m))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (x_45_scale_m <= 1.95e+201) {
		tmp = ((b * a) / y_45_scale) * (((-4.0 / y_45_scale) * ((b * a) / x_45_scale_m)) / x_45_scale_m);
	} else {
		tmp = (pow((a * b), 2.0) * ((-4.0 / y_45_scale) / x_45_scale_m)) / (y_45_scale * x_45_scale_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (x_45scale_m <= 1.95d+201) then
        tmp = ((b * a) / y_45scale) * ((((-4.0d0) / y_45scale) * ((b * a) / x_45scale_m)) / x_45scale_m)
    else
        tmp = (((a * b) ** 2.0d0) * (((-4.0d0) / y_45scale) / x_45scale_m)) / (y_45scale * x_45scale_m)
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (x_45_scale_m <= 1.95e+201) {
		tmp = ((b * a) / y_45_scale) * (((-4.0 / y_45_scale) * ((b * a) / x_45_scale_m)) / x_45_scale_m);
	} else {
		tmp = (Math.pow((a * b), 2.0) * ((-4.0 / y_45_scale) / x_45_scale_m)) / (y_45_scale * x_45_scale_m);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	tmp = 0
	if x_45_scale_m <= 1.95e+201:
		tmp = ((b * a) / y_45_scale) * (((-4.0 / y_45_scale) * ((b * a) / x_45_scale_m)) / x_45_scale_m)
	else:
		tmp = (math.pow((a * b), 2.0) * ((-4.0 / y_45_scale) / x_45_scale_m)) / (y_45_scale * x_45_scale_m)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (x_45_scale_m <= 1.95e+201)
		tmp = Float64(Float64(Float64(b * a) / y_45_scale) * Float64(Float64(Float64(-4.0 / y_45_scale) * Float64(Float64(b * a) / x_45_scale_m)) / x_45_scale_m));
	else
		tmp = Float64(Float64((Float64(a * b) ^ 2.0) * Float64(Float64(-4.0 / y_45_scale) / x_45_scale_m)) / Float64(y_45_scale * x_45_scale_m));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0;
	if (x_45_scale_m <= 1.95e+201)
		tmp = ((b * a) / y_45_scale) * (((-4.0 / y_45_scale) * ((b * a) / x_45_scale_m)) / x_45_scale_m);
	else
		tmp = (((a * b) ^ 2.0) * ((-4.0 / y_45_scale) / x_45_scale_m)) / (y_45_scale * x_45_scale_m);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[x$45$scale$95$m, 1.95e+201], N[(N[(N[(b * a), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(N[(N[(-4.0 / y$45$scale), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(-4.0 / y$45$scale), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.95e201

   1. Initial program 25.2%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 55.4

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

   5. Applied rewrites 55.4%

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

   7. Applied rewrites 86.4%

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

   9. Applied rewrites 89.3%

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

   11. Applied rewrites 94.0%

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

   if 1.95e201 < x-scale

   1. Initial program 14.1%

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

   3. Taylor expanded in angle around 0

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

      1. 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 37.2

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

   5. Applied rewrites 37.2%

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

   7. Applied rewrites 65.2%

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

   9. Applied rewrites 77.5%

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

Alternative 2: 89.7% accurate, 12.7× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 1.95 \cdot 10^{+201}:\\ \;\;\;\;\frac{b \cdot a}{y-scale} \cdot \frac{\frac{-4}{y-scale} \cdot \frac{b \cdot a}{x-scale\_m}}{x-scale\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(a \cdot b\right)}^{2} \cdot -4}{y-scale \cdot x-scale\_m}}{y-scale \cdot x-scale\_m}\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (if (<= x-scale_m 1.95e+201)
   (*
    (/ (* b a) y-scale)
    (/ (* (/ -4.0 y-scale) (/ (* b a) x-scale_m)) x-scale_m))
   (/
    (/ (* (pow (* a b) 2.0) -4.0) (* y-scale x-scale_m))
    (* y-scale x-scale_m))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (x_45_scale_m <= 1.95e+201) {
		tmp = ((b * a) / y_45_scale) * (((-4.0 / y_45_scale) * ((b * a) / x_45_scale_m)) / x_45_scale_m);
	} else {
		tmp = ((pow((a * b), 2.0) * -4.0) / (y_45_scale * x_45_scale_m)) / (y_45_scale * x_45_scale_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (x_45scale_m <= 1.95d+201) then
        tmp = ((b * a) / y_45scale) * ((((-4.0d0) / y_45scale) * ((b * a) / x_45scale_m)) / x_45scale_m)
    else
        tmp = ((((a * b) ** 2.0d0) * (-4.0d0)) / (y_45scale * x_45scale_m)) / (y_45scale * x_45scale_m)
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (x_45_scale_m <= 1.95e+201) {
		tmp = ((b * a) / y_45_scale) * (((-4.0 / y_45_scale) * ((b * a) / x_45_scale_m)) / x_45_scale_m);
	} else {
		tmp = ((Math.pow((a * b), 2.0) * -4.0) / (y_45_scale * x_45_scale_m)) / (y_45_scale * x_45_scale_m);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	tmp = 0
	if x_45_scale_m <= 1.95e+201:
		tmp = ((b * a) / y_45_scale) * (((-4.0 / y_45_scale) * ((b * a) / x_45_scale_m)) / x_45_scale_m)
	else:
		tmp = ((math.pow((a * b), 2.0) * -4.0) / (y_45_scale * x_45_scale_m)) / (y_45_scale * x_45_scale_m)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (x_45_scale_m <= 1.95e+201)
		tmp = Float64(Float64(Float64(b * a) / y_45_scale) * Float64(Float64(Float64(-4.0 / y_45_scale) * Float64(Float64(b * a) / x_45_scale_m)) / x_45_scale_m));
	else
		tmp = Float64(Float64(Float64((Float64(a * b) ^ 2.0) * -4.0) / Float64(y_45_scale * x_45_scale_m)) / Float64(y_45_scale * x_45_scale_m));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0;
	if (x_45_scale_m <= 1.95e+201)
		tmp = ((b * a) / y_45_scale) * (((-4.0 / y_45_scale) * ((b * a) / x_45_scale_m)) / x_45_scale_m);
	else
		tmp = ((((a * b) ^ 2.0) * -4.0) / (y_45_scale * x_45_scale_m)) / (y_45_scale * x_45_scale_m);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[x$45$scale$95$m, 1.95e+201], N[(N[(N[(b * a), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(N[(N[(-4.0 / y$45$scale), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision] * -4.0), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.95e201

   1. Initial program 25.2%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 55.4

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

   5. Applied rewrites 55.4%

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

   7. Applied rewrites 86.4%

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

   9. Applied rewrites 89.3%

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

   11. Applied rewrites 94.0%

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

   if 1.95e201 < x-scale

   1. Initial program 14.1%

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

   3. Taylor expanded in angle around 0

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

      1. 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 37.2

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

   5. Applied rewrites 37.2%

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

   7. Applied rewrites 65.2%

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

   9. Applied rewrites 77.4%

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

Alternative 3: 90.4% accurate, 26.8× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 7.5 \cdot 10^{+241}:\\ \;\;\;\;\frac{b \cdot a}{y-scale} \cdot \frac{\frac{-4}{y-scale} \cdot \frac{b \cdot a}{x-scale\_m}}{x-scale\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(a \cdot b\right) \cdot \frac{a \cdot b}{x-scale\_m}\right) \cdot -4}{\left(y-scale \cdot x-scale\_m\right) \cdot y-scale}\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (if (<= x-scale_m 7.5e+241)
   (*
    (/ (* b a) y-scale)
    (/ (* (/ -4.0 y-scale) (/ (* b a) x-scale_m)) x-scale_m))
   (/
    (* (* (* a b) (/ (* a b) x-scale_m)) -4.0)
    (* (* y-scale x-scale_m) y-scale))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (x_45_scale_m <= 7.5e+241) {
		tmp = ((b * a) / y_45_scale) * (((-4.0 / y_45_scale) * ((b * a) / x_45_scale_m)) / x_45_scale_m);
	} else {
		tmp = (((a * b) * ((a * b) / x_45_scale_m)) * -4.0) / ((y_45_scale * x_45_scale_m) * y_45_scale);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (x_45scale_m <= 7.5d+241) then
        tmp = ((b * a) / y_45scale) * ((((-4.0d0) / y_45scale) * ((b * a) / x_45scale_m)) / x_45scale_m)
    else
        tmp = (((a * b) * ((a * b) / x_45scale_m)) * (-4.0d0)) / ((y_45scale * x_45scale_m) * y_45scale)
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (x_45_scale_m <= 7.5e+241) {
		tmp = ((b * a) / y_45_scale) * (((-4.0 / y_45_scale) * ((b * a) / x_45_scale_m)) / x_45_scale_m);
	} else {
		tmp = (((a * b) * ((a * b) / x_45_scale_m)) * -4.0) / ((y_45_scale * x_45_scale_m) * y_45_scale);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	tmp = 0
	if x_45_scale_m <= 7.5e+241:
		tmp = ((b * a) / y_45_scale) * (((-4.0 / y_45_scale) * ((b * a) / x_45_scale_m)) / x_45_scale_m)
	else:
		tmp = (((a * b) * ((a * b) / x_45_scale_m)) * -4.0) / ((y_45_scale * x_45_scale_m) * y_45_scale)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (x_45_scale_m <= 7.5e+241)
		tmp = Float64(Float64(Float64(b * a) / y_45_scale) * Float64(Float64(Float64(-4.0 / y_45_scale) * Float64(Float64(b * a) / x_45_scale_m)) / x_45_scale_m));
	else
		tmp = Float64(Float64(Float64(Float64(a * b) * Float64(Float64(a * b) / x_45_scale_m)) * -4.0) / Float64(Float64(y_45_scale * x_45_scale_m) * y_45_scale));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0;
	if (x_45_scale_m <= 7.5e+241)
		tmp = ((b * a) / y_45_scale) * (((-4.0 / y_45_scale) * ((b * a) / x_45_scale_m)) / x_45_scale_m);
	else
		tmp = (((a * b) * ((a * b) / x_45_scale_m)) * -4.0) / ((y_45_scale * x_45_scale_m) * y_45_scale);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[x$45$scale$95$m, 7.5e+241], N[(N[(N[(b * a), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(N[(N[(-4.0 / y$45$scale), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * b), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale$95$m), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 7.5000000000000001e241

   1. Initial program 24.9%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 85.7%

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

   9. Applied rewrites 88.4%

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

   11. Applied rewrites 93.1%

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

   if 7.5000000000000001e241 < x-scale

   1. Initial program 14.1%

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

   3. Taylor expanded in angle around 0

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

      1. 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 41.2

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

   5. Applied rewrites 41.2%

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

   7. Applied rewrites 67.0%

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

   9. Applied rewrites 81.5%

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

Alternative 4: 67.3% accurate, 32.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-126} \lor \neg \left(b \leq 2.2 \cdot 10^{+126}\right):\\ \;\;\;\;\frac{\left(\left(b \cdot a\right) \cdot a\right) \cdot -4}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\_m\right) \cdot x-scale\_m} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot a\right)}{\left(y-scale \cdot x-scale\_m\right) \cdot \left(y-scale \cdot x-scale\_m\right)} \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (if (or (<= b 4.6e-126) (not (<= b 2.2e+126)))
   (*
    (/ (* (* (* b a) a) -4.0) (* (* (* y-scale y-scale) x-scale_m) x-scale_m))
    b)
   (*
    (/ (* -4.0 (* a a)) (* (* y-scale x-scale_m) (* y-scale x-scale_m)))
    (* b b))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if ((b <= 4.6e-126) || !(b <= 2.2e+126)) {
		tmp = ((((b * a) * a) * -4.0) / (((y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * b;
	} else {
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * (b * b);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if ((b <= 4.6d-126) .or. (.not. (b <= 2.2d+126))) then
        tmp = ((((b * a) * a) * (-4.0d0)) / (((y_45scale * y_45scale) * x_45scale_m) * x_45scale_m)) * b
    else
        tmp = (((-4.0d0) * (a * a)) / ((y_45scale * x_45scale_m) * (y_45scale * x_45scale_m))) * (b * b)
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if ((b <= 4.6e-126) || !(b <= 2.2e+126)) {
		tmp = ((((b * a) * a) * -4.0) / (((y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * b;
	} else {
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * (b * b);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	tmp = 0
	if (b <= 4.6e-126) or not (b <= 2.2e+126):
		tmp = ((((b * a) * a) * -4.0) / (((y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * b
	else:
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * (b * b)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if ((b <= 4.6e-126) || !(b <= 2.2e+126))
		tmp = Float64(Float64(Float64(Float64(Float64(b * a) * a) * -4.0) / Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * b);
	else
		tmp = Float64(Float64(Float64(-4.0 * Float64(a * a)) / Float64(Float64(y_45_scale * x_45_scale_m) * Float64(y_45_scale * x_45_scale_m))) * Float64(b * b));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0;
	if ((b <= 4.6e-126) || ~((b <= 2.2e+126)))
		tmp = ((((b * a) * a) * -4.0) / (((y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * b;
	else
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale_m) * (y_45_scale * x_45_scale_m))) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := If[Or[LessEqual[b, 4.6e-126], N[Not[LessEqual[b, 2.2e+126]], $MachinePrecision]], N[(N[(N[(N[(N[(b * a), $MachinePrecision] * a), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[(N[(-4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.60000000000000021e-126 or 2.19999999999999999e126 < b

   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 45.3%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 59.6%

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

   10. Applied rewrites 75.7%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 66.4%

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

   if 4.60000000000000021e-126 < b < 2.19999999999999999e126

   1. Initial program 35.4%

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

   3. Taylor expanded in 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 64.4%

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

   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 82.4%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-126} \lor \neg \left(b \leq 2.2 \cdot 10^{+126}\right):\\ \;\;\;\;\frac{\left(\left(b \cdot a\right) \cdot a\right) \cdot -4}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot a\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.2% accurate, 32.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-173}:\\ \;\;\;\;\frac{\left(\left(b \cdot a\right) \cdot a\right) \cdot -4}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\_m\right) \cdot x-scale\_m} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale\_m} \cdot \frac{a}{y-scale \cdot x-scale\_m}\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (if (<= b 4.6e-173)
   (*
    (/ (* (* (* b a) a) -4.0) (* (* (* y-scale y-scale) x-scale_m) x-scale_m))
    b)
   (*
    (* (/ (* -4.0 a) (* y-scale x-scale_m)) (/ a (* y-scale x-scale_m)))
    (* b b))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b <= 4.6e-173) {
		tmp = ((((b * a) * a) * -4.0) / (((y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * b;
	} else {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale_m)) * (a / (y_45_scale * x_45_scale_m))) * (b * b);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b <= 4.6d-173) then
        tmp = ((((b * a) * a) * (-4.0d0)) / (((y_45scale * y_45scale) * x_45scale_m) * x_45scale_m)) * b
    else
        tmp = ((((-4.0d0) * a) / (y_45scale * x_45scale_m)) * (a / (y_45scale * x_45scale_m))) * (b * b)
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b <= 4.6e-173) {
		tmp = ((((b * a) * a) * -4.0) / (((y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * b;
	} else {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale_m)) * (a / (y_45_scale * x_45_scale_m))) * (b * b);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	tmp = 0
	if b <= 4.6e-173:
		tmp = ((((b * a) * a) * -4.0) / (((y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * b
	else:
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale_m)) * (a / (y_45_scale * x_45_scale_m))) * (b * b)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (b <= 4.6e-173)
		tmp = Float64(Float64(Float64(Float64(Float64(b * a) * a) * -4.0) / Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * b);
	else
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) / Float64(y_45_scale * x_45_scale_m)) * Float64(a / Float64(y_45_scale * x_45_scale_m))) * Float64(b * b));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0;
	if (b <= 4.6e-173)
		tmp = ((((b * a) * a) * -4.0) / (((y_45_scale * y_45_scale) * x_45_scale_m) * x_45_scale_m)) * b;
	else
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale_m)) * (a / (y_45_scale * x_45_scale_m))) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[b, 4.6e-173], N[(N[(N[(N[(N[(b * a), $MachinePrecision] * a), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(a / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.59999999999999976e-173

   1. Initial program 24.0%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 45.1%

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

   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 61.1%

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

   10. Applied rewrites 75.9%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 66.3%

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

   if 4.59999999999999976e-173 < b

   1. Initial program 24.6%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 57.3%

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

   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 71.2%

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

   10. Applied rewrites 84.1%

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

Alternative 6: 90.3% accurate, 32.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.2%

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

3. Taylor expanded in 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 53.9

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

5. Applied rewrites 53.9%

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

7. Applied rewrites 84.6%

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

9. Applied rewrites 87.7%

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

11. Applied rewrites 91.2%

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

Alternative 7: 89.6% accurate, 32.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.2%

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

3. Taylor expanded in 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 53.9

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

5. Applied rewrites 53.9%

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

7. Applied rewrites 84.6%

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

9. Applied rewrites 87.7%

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

11. Applied rewrites 91.0%

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

Alternative 8: 82.6% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.2%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 49.8%

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

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 65.0%

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

10. Applied rewrites 79.2%

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

12. Applied rewrites 84.6%

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

Alternative 9: 65.6% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.2%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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 49.8%

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

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 65.0%

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

10. Applied rewrites 79.2%

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

11. Taylor expanded in angle around 0

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

13. Applied rewrites 66.7%

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

Reproduce

herbie shell --seed 2025010 
(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))))