Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 3.9s

Alternatives: 9

Speedup: 1.0×

Specification

\[0 \leq s \land s \leq 1.0651631\]

Math

\[\frac{1}{1 + e^{\frac{-x}{s}}} \]

FPCore

(FPCore (x s)
  :precision binary32
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / s)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + exp((-x / s)))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + exp((-x / s)));
end

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\frac{1}{1 + e^{\frac{-x}{s}}} \]

FPCore

(FPCore (x s)
  :precision binary32
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / s)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + exp((-x / s)))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + exp((-x / s)));
end

Alternative 1: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ t_1 := x \cdot \frac{x}{s \cdot s}\\ \mathbf{if}\;t\_0 \leq 0.00019999999494757503:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot 2 - \sqrt{t\_1 \cdot t\_1}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  (let* ((t_0 (/ (- x) s)) (t_1 (* x (/ x (* s s)))))
  (if (<= t_0 0.00019999999494757503)
    0.5
    (if (<= t_0 1.9999999360571385e+38)
      (/ 1.0 (/ (- (* 2.0 2.0) (sqrt (* t_1 t_1))) (+ 2.0 (/ x s))))
      (/ 1.0 (- 2.0 (/ x s)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float t_1 = x * (x / (s * s));
	float tmp;
	if (t_0 <= 0.00019999999494757503f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.9999999360571385e+38f) {
		tmp = 1.0f / (((2.0f * 2.0f) - sqrtf((t_1 * t_1))) / (2.0f + (x / s)));
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = -x / s
    t_1 = x * (x / (s * s))
    if (t_0 <= 0.00019999999494757503e0) then
        tmp = 0.5e0
    else if (t_0 <= 1.9999999360571385e+38) then
        tmp = 1.0e0 / (((2.0e0 * 2.0e0) - sqrt((t_1 * t_1))) / (2.0e0 + (x / s)))
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	t_1 = Float32(x * Float32(x / Float32(s * s)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00019999999494757503))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.9999999360571385e+38))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(2.0) * Float32(2.0)) - sqrt(Float32(t_1 * t_1))) / Float32(Float32(2.0) + Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	t_1 = x * (x / (s * s));
	tmp = single(0.0);
	if (t_0 <= single(0.00019999999494757503))
		tmp = single(0.5);
	elseif (t_0 <= single(1.9999999360571385e+38))
		tmp = single(1.0) / (((single(2.0) * single(2.0)) - sqrt((t_1 * t_1))) / (single(2.0) + (x / s)));
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < 1.99999995e-4

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   3. Step-by-step derivation

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{0.5} \]

   if 1.99999995e-4 < (/.f32 (neg.f32 x) s) < 1.99999994e38

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   3. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. lower-/.f32 41.3%

         \[\leadsto \frac{1}{2 + -1 \cdot \frac{x}{\color{blue}{s}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{1}{2 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}}} \]

      4. flip-- N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}\right)}{\color{blue}{2 + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}}}} \]

      5. lower-unsound-/.f32 N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}\right)}{\color{blue}{2 + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}}}} \]

   6. Applied rewrites 38.9%

      \[\leadsto \frac{1}{\frac{2 \cdot 2 - \frac{x}{s} \cdot \frac{x}{s}}{\color{blue}{2 + \frac{x}{s}}}} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\frac{x}{s} \cdot \frac{x}{s}} \cdot \sqrt{\frac{x}{s} \cdot \frac{x}{s}}}{2 + \frac{x}{s}}} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      4. lower-unsound-*.f32 N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      5. lower-sqrt.f32 N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      6. lower-unsound-*.f32 44.9%

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      7. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      8. lift-/.f32 N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      9. lift-/.f32 N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      10. frac-times N/A

          \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\frac{x \cdot x}{s \cdot s} \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      11. associate-/l* N/A

          \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      12. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      13. lower-/.f32 N/A

          \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      14. lower-*.f32 43.4%

          \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      15. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      16. lift-/.f32 N/A

          \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      17. lift-/.f32 N/A

          \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{2 + \frac{x}{s}}} \]

      18. frac-times N/A

          \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot \frac{x \cdot x}{s \cdot s}}}{2 + \frac{x}{s}}} \]

      19. associate-/l* N/A

          \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}}{2 + \frac{x}{s}}} \]

      20. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}}{2 + \frac{x}{s}}} \]

      21. lower-/.f32 N/A

          \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}}{2 + \frac{x}{s}}} \]

      22. lower-*.f32 43.4%

          \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}}{2 + \frac{x}{s}}} \]

   8. Applied rewrites 43.4%

      \[\leadsto \frac{1}{\frac{2 \cdot 2 - \sqrt{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}}{2 + \frac{x}{s}}} \]

   if 1.99999994e38 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   3. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. lower-/.f32 41.3%

         \[\leadsto \frac{1}{2 + -1 \cdot \frac{x}{\color{blue}{s}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

      5. lower--.f32 41.3%

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

   6. Applied rewrites 41.3%

      \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 63.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq 0.00019999999494757503:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{x}{s \cdot s}}{\frac{x}{s} - -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  (let* ((t_0 (/ (- x) s)))
  (if (<= t_0 0.00019999999494757503)
    0.5
    (if (<= t_0 1.9999999360571385e+38)
      (/ 1.0 (/ (- 4.0 (* x (/ x (* s s)))) (- (/ x s) -2.0)))
      (/ 1.0 (- 2.0 (/ x s)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 0.00019999999494757503f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.9999999360571385e+38f) {
		tmp = 1.0f / ((4.0f - (x * (x / (s * s)))) / ((x / s) - -2.0f));
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= 0.00019999999494757503e0) then
        tmp = 0.5e0
    else if (t_0 <= 1.9999999360571385e+38) then
        tmp = 1.0e0 / ((4.0e0 - (x * (x / (s * s)))) / ((x / s) - (-2.0e0)))
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00019999999494757503))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.9999999360571385e+38))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x * Float32(x / Float32(s * s)))) / Float32(Float32(x / s) - Float32(-2.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(0.00019999999494757503))
		tmp = single(0.5);
	elseif (t_0 <= single(1.9999999360571385e+38))
		tmp = single(1.0) / ((single(4.0) - (x * (x / (s * s)))) / ((x / s) - single(-2.0)));
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < 1.99999995e-4

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   3. Step-by-step derivation

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{0.5} \]

   if 1.99999995e-4 < (/.f32 (neg.f32 x) s) < 1.99999994e38

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   3. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. lower-/.f32 41.3%

         \[\leadsto \frac{1}{2 + -1 \cdot \frac{x}{\color{blue}{s}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{1}{2 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}}} \]

      4. flip-- N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}\right)}{\color{blue}{2 + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}}}} \]

      5. lower-unsound-/.f32 N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}\right)}{\color{blue}{2 + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}}}} \]

   6. Applied rewrites 38.9%

      \[\leadsto \frac{1}{\frac{2 \cdot 2 - \frac{x}{s} \cdot \frac{x}{s}}{\color{blue}{2 + \frac{x}{s}}}} \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}} \]

      2. metadata-eval 38.9%

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}} \]

      3. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}} \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}} \]

      5. lift-/.f32 N/A

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}} \]

      6. frac-times N/A

         \[\leadsto \frac{1}{\frac{4 - \frac{x \cdot x}{s \cdot s}}{2 + \frac{x}{s}}} \]

      7. associate-/l* N/A

         \[\leadsto \frac{1}{\frac{4 - x \cdot \frac{x}{s \cdot s}}{2 + \frac{x}{s}}} \]

      8. lower-*.f32 N/A

         \[\leadsto \frac{1}{\frac{4 - x \cdot \frac{x}{s \cdot s}}{2 + \frac{x}{s}}} \]

      9. lower-/.f32 N/A

         \[\leadsto \frac{1}{\frac{4 - x \cdot \frac{x}{s \cdot s}}{2 + \frac{x}{s}}} \]

      10. lower-*.f32 39.8%

          \[\leadsto \frac{1}{\frac{4 - x \cdot \frac{x}{s \cdot s}}{2 + \frac{x}{s}}} \]

      11. lift-+.f32 N/A

          \[\leadsto \frac{1}{\frac{4 - x \cdot \frac{x}{s \cdot s}}{2 + \color{blue}{\frac{x}{s}}}} \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\frac{4 - x \cdot \frac{x}{s \cdot s}}{\frac{x}{s} + \color{blue}{2}}} \]

      13. add-flip N/A

          \[\leadsto \frac{1}{\frac{4 - x \cdot \frac{x}{s \cdot s}}{\frac{x}{s} - \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{4 - x \cdot \frac{x}{s \cdot s}}{\frac{x}{s} - -2}} \]

      15. lower--.f32 39.8%

          \[\leadsto \frac{1}{\frac{4 - x \cdot \frac{x}{s \cdot s}}{\frac{x}{s} - \color{blue}{-2}}} \]

   8. Applied rewrites 39.8%

      \[\leadsto \frac{1}{\frac{4 - x \cdot \frac{x}{s \cdot s}}{\color{blue}{\frac{x}{s} - -2}}} \]

   if 1.99999994e38 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   3. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. lower-/.f32 41.3%

         \[\leadsto \frac{1}{2 + -1 \cdot \frac{x}{\color{blue}{s}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

      5. lower--.f32 41.3%

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

   6. Applied rewrites 41.3%

      \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 63.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq 0.00019999999494757503:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \left(4 - x \cdot \frac{x}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  (let* ((t_0 (/ (- x) s)))
  (if (<= t_0 0.00019999999494757503)
    0.5
    (if (<= t_0 1.9999999360571385e+38)
      (/ 1.0 (* (/ s (+ (+ s s) x)) (- 4.0 (* x (/ x (* s s))))))
      (/ 1.0 (- 2.0 (/ x s)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 0.00019999999494757503f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.9999999360571385e+38f) {
		tmp = 1.0f / ((s / ((s + s) + x)) * (4.0f - (x * (x / (s * s)))));
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= 0.00019999999494757503e0) then
        tmp = 0.5e0
    else if (t_0 <= 1.9999999360571385e+38) then
        tmp = 1.0e0 / ((s / ((s + s) + x)) * (4.0e0 - (x * (x / (s * s)))))
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00019999999494757503))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.9999999360571385e+38))
		tmp = Float32(Float32(1.0) / Float32(Float32(s / Float32(Float32(s + s) + x)) * Float32(Float32(4.0) - Float32(x * Float32(x / Float32(s * s))))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(0.00019999999494757503))
		tmp = single(0.5);
	elseif (t_0 <= single(1.9999999360571385e+38))
		tmp = single(1.0) / ((s / ((s + s) + x)) * (single(4.0) - (x * (x / (s * s)))));
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < 1.99999995e-4

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   3. Step-by-step derivation

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{0.5} \]

   if 1.99999995e-4 < (/.f32 (neg.f32 x) s) < 1.99999994e38

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   3. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. lower-/.f32 41.3%

         \[\leadsto \frac{1}{2 + -1 \cdot \frac{x}{\color{blue}{s}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{1}{2 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}}} \]

      4. flip-- N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}\right)}{\color{blue}{2 + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}}}} \]

      5. lower-unsound-/.f32 N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}\right)}{\color{blue}{2 + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x}{s}}}} \]

   6. Applied rewrites 38.9%

      \[\leadsto \frac{1}{\frac{2 \cdot 2 - \frac{x}{s} \cdot \frac{x}{s}}{\color{blue}{2 + \frac{x}{s}}}} \]
   7. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \frac{x}{s} \cdot \frac{x}{s}}{\color{blue}{2 + \frac{x}{s}}}} \]

      2. mult-flip N/A

         \[\leadsto \frac{1}{\left(2 \cdot 2 - \frac{x}{s} \cdot \frac{x}{s}\right) \cdot \color{blue}{\frac{1}{2 + \frac{x}{s}}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\frac{1}{2 + \frac{x}{s}} \cdot \color{blue}{\left(2 \cdot 2 - \frac{x}{s} \cdot \frac{x}{s}\right)}} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{1}{\frac{1}{2 + \frac{x}{s}} \cdot \color{blue}{\left(2 \cdot 2 - \frac{x}{s} \cdot \frac{x}{s}\right)}} \]

      5. lift-+.f32 N/A

         \[\leadsto \frac{1}{\frac{1}{2 + \frac{x}{s}} \cdot \left(2 \cdot \color{blue}{2} - \frac{x}{s} \cdot \frac{x}{s}\right)} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{1}{\frac{1}{2 + \frac{x}{s}} \cdot \left(2 \cdot 2 - \frac{x}{s} \cdot \frac{x}{s}\right)} \]

      7. add-to-fraction N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{2 \cdot s + x}{s}} \cdot \left(2 \cdot \color{blue}{2} - \frac{x}{s} \cdot \frac{x}{s}\right)} \]

      8. div-flip-rev N/A

         \[\leadsto \frac{1}{\frac{s}{2 \cdot s + x} \cdot \left(\color{blue}{2 \cdot 2} - \frac{x}{s} \cdot \frac{x}{s}\right)} \]

      9. lower-/.f32 N/A

         \[\leadsto \frac{1}{\frac{s}{2 \cdot s + x} \cdot \left(\color{blue}{2 \cdot 2} - \frac{x}{s} \cdot \frac{x}{s}\right)} \]

      10. count-2 N/A

          \[\leadsto \frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \left(2 \cdot 2 - \frac{x}{s} \cdot \frac{x}{s}\right)} \]

      11. lift-+.f32 N/A

          \[\leadsto \frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \left(2 \cdot 2 - \frac{x}{s} \cdot \frac{x}{s}\right)} \]

      12. lower-+.f32 42.8%

          \[\leadsto \frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \left(2 \cdot \color{blue}{2} - \frac{x}{s} \cdot \frac{x}{s}\right)} \]

      13. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \left(2 \cdot 2 - \color{blue}{\frac{x}{s}} \cdot \frac{x}{s}\right)} \]

      14. metadata-eval 42.8%

          \[\leadsto \frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \left(4 - \color{blue}{\frac{x}{s}} \cdot \frac{x}{s}\right)} \]

      15. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \left(4 - \frac{x}{s} \cdot \color{blue}{\frac{x}{s}}\right)} \]

      16. lift-/.f32 N/A

          \[\leadsto \frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \left(4 - \frac{x}{s} \cdot \frac{\color{blue}{x}}{s}\right)} \]

      17. lift-/.f32 N/A

          \[\leadsto \frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \left(4 - \frac{x}{s} \cdot \frac{x}{\color{blue}{s}}\right)} \]

      18. frac-times N/A

          \[\leadsto \frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \left(4 - \frac{x \cdot x}{\color{blue}{s \cdot s}}\right)} \]

      19. associate-/l* N/A

          \[\leadsto \frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \left(4 - x \cdot \color{blue}{\frac{x}{s \cdot s}}\right)} \]

      20. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \left(4 - x \cdot \color{blue}{\frac{x}{s \cdot s}}\right)} \]

      21. lower-/.f32 N/A

          \[\leadsto \frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \left(4 - x \cdot \frac{x}{\color{blue}{s \cdot s}}\right)} \]

      22. lower-*.f32 43.7%

          \[\leadsto \frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \left(4 - x \cdot \frac{x}{s \cdot \color{blue}{s}}\right)} \]

   8. Applied rewrites 43.7%

      \[\leadsto \frac{1}{\frac{s}{\left(s + s\right) + x} \cdot \color{blue}{\left(4 - x \cdot \frac{x}{s \cdot s}\right)}} \]

   if 1.99999994e38 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   3. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. lower-/.f32 41.3%

         \[\leadsto \frac{1}{2 + -1 \cdot \frac{x}{\color{blue}{s}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

      5. lower--.f32 41.3%

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

   6. Applied rewrites 41.3%

      \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 53.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;-x \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(s + s\right) \cdot s - s \cdot x}{s \cdot s}}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  (if (<= (- x) 1.999999936531045e-21)
  0.5
  (/ 1.0 (/ (- (* (+ s s) s) (* s x)) (* s s)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.999999936531045e-21f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((((s + s) * s) - (s * x)) / (s * s));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 1.999999936531045e-21) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((((s + s) * s) - (s * x)) / (s * s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.999999936531045e-21))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(s + s) * s) - Float32(s * x)) / Float32(s * s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(1.999999936531045e-21))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((((s + s) * s) - (s * x)) / (s * s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1.99999994e-21

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   3. Step-by-step derivation

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{0.5} \]

   if 1.99999994e-21 < (neg.f32 x)

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   3. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. lower-/.f32 41.3%

         \[\leadsto \frac{1}{2 + -1 \cdot \frac{x}{\color{blue}{s}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)} \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{1}{2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)} \]

      5. distribute-frac-neg N/A

         \[\leadsto \frac{1}{2 + \frac{\mathsf{neg}\left(x\right)}{\color{blue}{s}}} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{1}{2 + \frac{-x}{s}} \]

      7. add-to-fraction N/A

         \[\leadsto \frac{1}{\frac{2 \cdot s + \left(-x\right)}{\color{blue}{s}}} \]

      8. mult-flip N/A

         \[\leadsto \frac{1}{\left(2 \cdot s + \left(-x\right)\right) \cdot \color{blue}{\frac{1}{s}}} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{1}{\left(2 \cdot s + \left(-x\right)\right) \cdot \color{blue}{\frac{1}{s}}} \]

      10. lift-neg.f32 N/A

          \[\leadsto \frac{1}{\left(2 \cdot s + \left(\mathsf{neg}\left(x\right)\right)\right) \cdot \frac{1}{s}} \]

      11. sub-flip-reverse N/A

          \[\leadsto \frac{1}{\left(2 \cdot s - x\right) \cdot \frac{\color{blue}{1}}{s}} \]

      12. lower--.f32 N/A

          \[\leadsto \frac{1}{\left(2 \cdot s - x\right) \cdot \frac{\color{blue}{1}}{s}} \]

      13. count-2-rev N/A

          \[\leadsto \frac{1}{\left(\left(s + s\right) - x\right) \cdot \frac{1}{s}} \]

      14. lower-+.f32 N/A

          \[\leadsto \frac{1}{\left(\left(s + s\right) - x\right) \cdot \frac{1}{s}} \]

      15. lower-/.f32 41.3%

          \[\leadsto \frac{1}{\left(\left(s + s\right) - x\right) \cdot \frac{1}{\color{blue}{s}}} \]

   6. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\left(\left(s + s\right) - x\right) \cdot \color{blue}{\frac{1}{s}}} \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{1}{\left(\left(s + s\right) - x\right) \cdot \color{blue}{\frac{1}{s}}} \]

      2. lift-/.f32 N/A

         \[\leadsto \frac{1}{\left(\left(s + s\right) - x\right) \cdot \frac{1}{\color{blue}{s}}} \]

      3. mult-flip-rev N/A

         \[\leadsto \frac{1}{\frac{\left(s + s\right) - x}{\color{blue}{s}}} \]

      4. lift--.f32 N/A

         \[\leadsto \frac{1}{\frac{\left(s + s\right) - x}{s}} \]

      5. div-sub N/A

         \[\leadsto \frac{1}{\frac{s + s}{s} - \color{blue}{\frac{x}{s}}} \]

      6. frac-sub N/A

         \[\leadsto \frac{1}{\frac{\left(s + s\right) \cdot s - s \cdot x}{\color{blue}{s \cdot s}}} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{1}{\frac{\left(s + s\right) \cdot s - s \cdot x}{\color{blue}{s \cdot s}}} \]

      8. lower--.f32 N/A

         \[\leadsto \frac{1}{\frac{\left(s + s\right) \cdot s - s \cdot x}{\color{blue}{s} \cdot s}} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\left(s + s\right) \cdot s - s \cdot x}{s \cdot s}} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(s + s\right) \cdot s - s \cdot x}{s \cdot s}} \]

      11. lower-*.f32 41.9%

          \[\leadsto \frac{1}{\frac{\left(s + s\right) \cdot s - s \cdot x}{s \cdot \color{blue}{s}}} \]

   8. Applied rewrites 41.9%

      \[\leadsto \frac{1}{\frac{\left(s + s\right) \cdot s - s \cdot x}{\color{blue}{s \cdot s}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 49.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.399999976158142:\\ \;\;\;\;\frac{1}{\left(2 \cdot s\right) \cdot \frac{1}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{s}{\left(s + s\right) - x}}}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  (if (<= (+ 1.0 (exp (/ (- x) s))) 1.399999976158142)
  (/ 1.0 (* (* 2.0 s) (/ 1.0 s)))
  (/ 1.0 (/ 1.0 (/ s (- (+ s s) x))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.399999976158142f) {
		tmp = 1.0f / ((2.0f * s) * (1.0f / s));
	} else {
		tmp = 1.0f / (1.0f / (s / ((s + s) - x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((1.0e0 + exp((-x / s))) <= 1.399999976158142e0) then
        tmp = 1.0e0 / ((2.0e0 * s) * (1.0e0 / s))
    else
        tmp = 1.0e0 / (1.0e0 / (s / ((s + s) - x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.399999976158142))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(2.0) * s) * Float32(Float32(1.0) / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) / Float32(s / Float32(Float32(s + s) - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(1.399999976158142))
		tmp = single(1.0) / ((single(2.0) * s) * (single(1.0) / s));
	else
		tmp = single(1.0) / (single(1.0) / (s / ((s + s) - x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.39999998

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   3. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. lower-/.f32 41.3%

         \[\leadsto \frac{1}{2 + -1 \cdot \frac{x}{\color{blue}{s}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)} \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{1}{2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)} \]

      5. distribute-frac-neg N/A

         \[\leadsto \frac{1}{2 + \frac{\mathsf{neg}\left(x\right)}{\color{blue}{s}}} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{1}{2 + \frac{-x}{s}} \]

      7. add-to-fraction N/A

         \[\leadsto \frac{1}{\frac{2 \cdot s + \left(-x\right)}{\color{blue}{s}}} \]

      8. mult-flip N/A

         \[\leadsto \frac{1}{\left(2 \cdot s + \left(-x\right)\right) \cdot \color{blue}{\frac{1}{s}}} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{1}{\left(2 \cdot s + \left(-x\right)\right) \cdot \color{blue}{\frac{1}{s}}} \]

      10. lift-neg.f32 N/A

          \[\leadsto \frac{1}{\left(2 \cdot s + \left(\mathsf{neg}\left(x\right)\right)\right) \cdot \frac{1}{s}} \]

      11. sub-flip-reverse N/A

          \[\leadsto \frac{1}{\left(2 \cdot s - x\right) \cdot \frac{\color{blue}{1}}{s}} \]

      12. lower--.f32 N/A

          \[\leadsto \frac{1}{\left(2 \cdot s - x\right) \cdot \frac{\color{blue}{1}}{s}} \]

      13. count-2-rev N/A

          \[\leadsto \frac{1}{\left(\left(s + s\right) - x\right) \cdot \frac{1}{s}} \]

      14. lower-+.f32 N/A

          \[\leadsto \frac{1}{\left(\left(s + s\right) - x\right) \cdot \frac{1}{s}} \]

      15. lower-/.f32 41.3%

          \[\leadsto \frac{1}{\left(\left(s + s\right) - x\right) \cdot \frac{1}{\color{blue}{s}}} \]

   6. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\left(\left(s + s\right) - x\right) \cdot \color{blue}{\frac{1}{s}}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\left(2 \cdot s\right) \cdot \frac{\color{blue}{1}}{s}} \]
   8. Step-by-step derivation

      1. lower-*.f32 35.5%

         \[\leadsto \frac{1}{\left(2 \cdot s\right) \cdot \frac{1}{s}} \]

   9. Applied rewrites 35.5%

      \[\leadsto \frac{1}{\left(2 \cdot s\right) \cdot \frac{\color{blue}{1}}{s}} \]

   if 1.39999998 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   3. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. lower-/.f32 41.3%

         \[\leadsto \frac{1}{2 + -1 \cdot \frac{x}{\color{blue}{s}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)} \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{1}{2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)} \]

      5. distribute-frac-neg N/A

         \[\leadsto \frac{1}{2 + \frac{\mathsf{neg}\left(x\right)}{\color{blue}{s}}} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{1}{2 + \frac{-x}{s}} \]

      7. add-to-fraction N/A

         \[\leadsto \frac{1}{\frac{2 \cdot s + \left(-x\right)}{\color{blue}{s}}} \]

      8. div-flip N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{s}{2 \cdot s + \left(-x\right)}}}} \]

      9. lower-unsound-/.f32 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{s}{2 \cdot s + \left(-x\right)}}}} \]

      10. lower-unsound-/.f32 N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{s}{\color{blue}{2 \cdot s + \left(-x\right)}}}} \]

      11. lift-neg.f32 N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{s}{2 \cdot s + \left(\mathsf{neg}\left(x\right)\right)}}} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{s}{2 \cdot s - \color{blue}{x}}}} \]

      13. lower--.f32 N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{s}{2 \cdot s - \color{blue}{x}}}} \]

      14. count-2-rev N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{s}{\left(s + s\right) - x}}} \]

      15. lower-+.f32 41.3%

          \[\leadsto \frac{1}{\frac{1}{\frac{s}{\left(s + s\right) - x}}} \]

   6. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{s}{\left(s + s\right) - x}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 49.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.399999976158142:\\ \;\;\;\;\frac{1}{\left(2 \cdot s\right) \cdot \frac{1}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  (if (<= (+ 1.0 (exp (/ (- x) s))) 1.399999976158142)
  (/ 1.0 (* (* 2.0 s) (/ 1.0 s)))
  (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.399999976158142f) {
		tmp = 1.0f / ((2.0f * s) * (1.0f / s));
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((1.0e0 + exp((-x / s))) <= 1.399999976158142e0) then
        tmp = 1.0e0 / ((2.0e0 * s) * (1.0e0 / s))
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.399999976158142))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(2.0) * s) * Float32(Float32(1.0) / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(1.399999976158142))
		tmp = single(1.0) / ((single(2.0) * s) * (single(1.0) / s));
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.39999998

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   3. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. lower-/.f32 41.3%

         \[\leadsto \frac{1}{2 + -1 \cdot \frac{x}{\color{blue}{s}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)} \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{1}{2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)} \]

      5. distribute-frac-neg N/A

         \[\leadsto \frac{1}{2 + \frac{\mathsf{neg}\left(x\right)}{\color{blue}{s}}} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{1}{2 + \frac{-x}{s}} \]

      7. add-to-fraction N/A

         \[\leadsto \frac{1}{\frac{2 \cdot s + \left(-x\right)}{\color{blue}{s}}} \]

      8. mult-flip N/A

         \[\leadsto \frac{1}{\left(2 \cdot s + \left(-x\right)\right) \cdot \color{blue}{\frac{1}{s}}} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{1}{\left(2 \cdot s + \left(-x\right)\right) \cdot \color{blue}{\frac{1}{s}}} \]

      10. lift-neg.f32 N/A

          \[\leadsto \frac{1}{\left(2 \cdot s + \left(\mathsf{neg}\left(x\right)\right)\right) \cdot \frac{1}{s}} \]

      11. sub-flip-reverse N/A

          \[\leadsto \frac{1}{\left(2 \cdot s - x\right) \cdot \frac{\color{blue}{1}}{s}} \]

      12. lower--.f32 N/A

          \[\leadsto \frac{1}{\left(2 \cdot s - x\right) \cdot \frac{\color{blue}{1}}{s}} \]

      13. count-2-rev N/A

          \[\leadsto \frac{1}{\left(\left(s + s\right) - x\right) \cdot \frac{1}{s}} \]

      14. lower-+.f32 N/A

          \[\leadsto \frac{1}{\left(\left(s + s\right) - x\right) \cdot \frac{1}{s}} \]

      15. lower-/.f32 41.3%

          \[\leadsto \frac{1}{\left(\left(s + s\right) - x\right) \cdot \frac{1}{\color{blue}{s}}} \]

   6. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\left(\left(s + s\right) - x\right) \cdot \color{blue}{\frac{1}{s}}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\left(2 \cdot s\right) \cdot \frac{\color{blue}{1}}{s}} \]
   8. Step-by-step derivation

      1. lower-*.f32 35.5%

         \[\leadsto \frac{1}{\left(2 \cdot s\right) \cdot \frac{1}{s}} \]

   9. Applied rewrites 35.5%

      \[\leadsto \frac{1}{\left(2 \cdot s\right) \cdot \frac{\color{blue}{1}}{s}} \]

   if 1.39999998 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   3. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. lower-/.f32 41.3%

         \[\leadsto \frac{1}{2 + -1 \cdot \frac{x}{\color{blue}{s}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

      5. lower--.f32 41.3%

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

   6. Applied rewrites 41.3%

      \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 49.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.399999976158142:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  (if (<= (+ 1.0 (exp (/ (- x) s))) 1.399999976158142)
  0.5
  (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.399999976158142f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((1.0e0 + exp((-x / s))) <= 1.399999976158142e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.399999976158142))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(1.399999976158142))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.39999998

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   3. Step-by-step derivation

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{0.5} \]

   if 1.39999998 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   3. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. lower-/.f32 41.3%

         \[\leadsto \frac{1}{2 + -1 \cdot \frac{x}{\color{blue}{s}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   5. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot \frac{x}{s}}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{2 + -1 \cdot \color{blue}{\frac{x}{s}}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

      5. lower--.f32 41.3%

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

   6. Applied rewrites 41.3%

      \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 35.5% accurate, 128.0× speedup

Math

\[0.5 \]

FPCore

(FPCore (x s)
  :precision binary32
  0.5)

C

float code(float x, float s) {
	return 0.5f;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0
end function

Julia

function code(x, s)
	return Float32(0.5)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5);
end

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
3. Step-by-step derivation

4. Applied rewrites 35.5%

   \[\leadsto \color{blue}{0.5} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))