Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 4.1s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{\frac{-x}{s}}} \end{array} \]

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

\[\begin{array}{l} \\ \frac{1}{1 + e^{\frac{-x}{s}}} \end{array} \]

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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot -1} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (exp (* (log1p (exp (/ (- x) s))) -1.0)))

C

float code(float x, float s) {
	return expf((log1pf(expf((-x / s))) * -1.0f));
}

Julia

function code(x, s)
	return exp(Float32(log1p(exp(Float32(Float32(-x) / s))) * Float32(-1.0)))
end

Derivation

1. Initial program 99.7%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. inv-pow N/A

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

   7. pow-to-exp N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-log1p.f32 N/A

       \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)} \cdot -1} \]

   11. lift-/.f32 N/A

       \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right) \cdot -1} \]

   12. lift-neg.f32 N/A

       \[\leadsto e^{\mathsf{log1p}\left(e^{\frac{\color{blue}{-x}}{s}}\right) \cdot -1} \]

   13. lift-exp.f32 99.8

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{e^{\frac{-x}{s}}}\right) \cdot -1} \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot -1}} \]
5. Add Preprocessing

Alternative 2: 64.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{\frac{-x}{s}}\\ \mathbf{if}\;t\_0 \leq 1.0049999952316284:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 100:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.25}{s}, x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x \cdot x\right) \cdot 0.5}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (exp (/ (- x) s)))))
   (if (<= t_0 1.0049999952316284)
     0.5
     (if (<= t_0 100.0)
       (fma (/ 0.25 s) x 0.5)
       (/ 1.0 (/ (* (* x x) 0.5) (* s s)))))))

C

float code(float x, float s) {
	float t_0 = 1.0f + expf((-x / s));
	float tmp;
	if (t_0 <= 1.0049999952316284f) {
		tmp = 0.5f;
	} else if (t_0 <= 100.0f) {
		tmp = fmaf((0.25f / s), x, 0.5f);
	} else {
		tmp = 1.0f / (((x * x) * 0.5f) / (s * s));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) + exp(Float32(Float32(-x) / s)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.0049999952316284))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(100.0))
		tmp = fma(Float32(Float32(0.25) / s), x, Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(x * x) * Float32(0.5)) / Float32(s * s)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.005

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

   if 1.005 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 100

   1. Initial program 99.3%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. lift-exp.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 99.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{expm1}\left(\frac{-x}{s} \cdot 2\right)}{\mathsf{expm1}\left(\frac{-x}{s}\right)}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + x \cdot \left(\frac{-1}{48} \cdot \frac{{x}^{2}}{{s}^{3}} + \frac{1}{4} \cdot \frac{1}{s}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(\frac{-1}{48} \cdot \frac{{x}^{2}}{{s}^{3}} + \frac{1}{4} \cdot \frac{1}{s}\right) + \color{blue}{\frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{48} \cdot \frac{{x}^{2}}{{s}^{3}} + \frac{1}{4} \cdot \frac{1}{s}\right) \cdot x + \frac{1}{2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{48} \cdot \frac{{x}^{2}}{{s}^{3}} + \frac{1}{4} \cdot \frac{1}{s}, \color{blue}{x}, \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2}}{{s}^{3}} \cdot \frac{-1}{48} + \frac{1}{4} \cdot \frac{1}{s}, x, \frac{1}{2}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{{x}^{2}}{{s}^{3}}, \frac{-1}{48}, \frac{1}{4} \cdot \frac{1}{s}\right), x, \frac{1}{2}\right) \]

      6. lower-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{{x}^{2}}{{s}^{3}}, \frac{-1}{48}, \frac{1}{4} \cdot \frac{1}{s}\right), x, \frac{1}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{{s}^{3}}, \frac{-1}{48}, \frac{1}{4} \cdot \frac{1}{s}\right), x, \frac{1}{2}\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{{s}^{3}}, \frac{-1}{48}, \frac{1}{4} \cdot \frac{1}{s}\right), x, \frac{1}{2}\right) \]

      9. lower-pow.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{{s}^{3}}, \frac{-1}{48}, \frac{1}{4} \cdot \frac{1}{s}\right), x, \frac{1}{2}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{{s}^{3}}, \frac{-1}{48}, \frac{\frac{1}{4} \cdot 1}{s}\right), x, \frac{1}{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{{s}^{3}}, \frac{-1}{48}, \frac{\frac{1}{4}}{s}\right), x, \frac{1}{2}\right) \]

      12. lower-/.f32 58.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{{s}^{3}}, -0.020833333333333332, \frac{0.25}{s}\right), x, 0.5\right) \]

   7. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{{s}^{3}}, -0.020833333333333332, \frac{0.25}{s}\right), x, 0.5\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{4}}{s}, x, \frac{1}{2}\right) \]
   9. Step-by-step derivation

      1. lift-/.f32 91.5

         \[\leadsto \mathsf{fma}\left(\frac{0.25}{s}, x, 0.5\right) \]

   10. Applied rewrites 91.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.25}{s}, x, 0.5\right) \]

   if 100 < (+.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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower--.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}, x, 2\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{{s}^{2}} \cdot \frac{1}{2} - \frac{1}{s}, x, 2\right)} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{{s}^{2}} \cdot \frac{1}{2} - \frac{1}{s}, x, 2\right)} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{{s}^{2}} \cdot \frac{1}{2} - \frac{1}{s}, x, 2\right)} \]

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-/.f32 83.1

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{s \cdot s} \cdot 0.5 - \frac{1}{s}, x, 2\right)} \]

   5. Applied rewrites 83.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{x}{s \cdot s} \cdot 0.5 - \frac{1}{s}, x, 2\right)}} \]

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 78.2

         \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot 0.5}{s \cdot s}} \]

   8. Applied rewrites 78.2%

      \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot 0.5}{\color{blue}{s \cdot s}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.0049999952316284:\\ \;\;\;\;0.5\\ \mathbf{elif}\;1 + e^{\frac{-x}{s}} \leq 100:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.25}{s}, x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x \cdot x\right) \cdot 0.5}{s \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ t_1 := 1 + e^{t\_0}\\ \mathbf{if}\;t\_1 \leq 1.0049999952316284:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.25}{s}, x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)) (t_1 (+ 1.0 (exp t_0))))
   (if (<= t_1 1.0049999952316284)
     0.5
     (if (<= t_1 5.0) (fma (/ 0.25 s) x 0.5) (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float t_1 = 1.0f + expf(t_0);
	float tmp;
	if (t_1 <= 1.0049999952316284f) {
		tmp = 0.5f;
	} else if (t_1 <= 5.0f) {
		tmp = fmaf((0.25f / s), x, 0.5f);
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	t_1 = Float32(Float32(1.0) + exp(t_0))
	tmp = Float32(0.0)
	if (t_1 <= Float32(1.0049999952316284))
		tmp = Float32(0.5);
	elseif (t_1 <= Float32(5.0))
		tmp = fma(Float32(Float32(0.25) / s), x, Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.005

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

   if 1.005 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 5

   1. Initial program 99.4%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. lift-exp.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 99.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{expm1}\left(\frac{-x}{s} \cdot 2\right)}{\mathsf{expm1}\left(\frac{-x}{s}\right)}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + x \cdot \left(\frac{-1}{48} \cdot \frac{{x}^{2}}{{s}^{3}} + \frac{1}{4} \cdot \frac{1}{s}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(\frac{-1}{48} \cdot \frac{{x}^{2}}{{s}^{3}} + \frac{1}{4} \cdot \frac{1}{s}\right) + \color{blue}{\frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{48} \cdot \frac{{x}^{2}}{{s}^{3}} + \frac{1}{4} \cdot \frac{1}{s}\right) \cdot x + \frac{1}{2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{48} \cdot \frac{{x}^{2}}{{s}^{3}} + \frac{1}{4} \cdot \frac{1}{s}, \color{blue}{x}, \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{{x}^{2}}{{s}^{3}} \cdot \frac{-1}{48} + \frac{1}{4} \cdot \frac{1}{s}, x, \frac{1}{2}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{{x}^{2}}{{s}^{3}}, \frac{-1}{48}, \frac{1}{4} \cdot \frac{1}{s}\right), x, \frac{1}{2}\right) \]

      6. lower-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{{x}^{2}}{{s}^{3}}, \frac{-1}{48}, \frac{1}{4} \cdot \frac{1}{s}\right), x, \frac{1}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{{s}^{3}}, \frac{-1}{48}, \frac{1}{4} \cdot \frac{1}{s}\right), x, \frac{1}{2}\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{{s}^{3}}, \frac{-1}{48}, \frac{1}{4} \cdot \frac{1}{s}\right), x, \frac{1}{2}\right) \]

      9. lower-pow.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{{s}^{3}}, \frac{-1}{48}, \frac{1}{4} \cdot \frac{1}{s}\right), x, \frac{1}{2}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{{s}^{3}}, \frac{-1}{48}, \frac{\frac{1}{4} \cdot 1}{s}\right), x, \frac{1}{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{{s}^{3}}, \frac{-1}{48}, \frac{\frac{1}{4}}{s}\right), x, \frac{1}{2}\right) \]

      12. lower-/.f32 58.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{{s}^{3}}, -0.020833333333333332, \frac{0.25}{s}\right), x, 0.5\right) \]

   7. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x \cdot x}{{s}^{3}}, -0.020833333333333332, \frac{0.25}{s}\right), x, 0.5\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{4}}{s}, x, \frac{1}{2}\right) \]
   9. Step-by-step derivation

      1. lift-/.f32 92.7

         \[\leadsto \mathsf{fma}\left(\frac{0.25}{s}, x, 0.5\right) \]

   10. Applied rewrites 92.7%

       \[\leadsto \mathsf{fma}\left(\frac{0.25}{s}, x, 0.5\right) \]

   if 5 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 43.6

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

   5. Applied rewrites 43.6%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. lift-/.f32 N/A

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

      4. lift-neg.f32 43.6

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

   8. Applied rewrites 43.6%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.0049999952316284:\\ \;\;\;\;0.5\\ \mathbf{elif}\;1 + e^{\frac{-x}{s}} \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.25}{s}, x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{{\left(e^{-2}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} + 1} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ (pow (exp -2.0) (/ (/ x s) 2.0)) 1.0)))

C

float code(float x, float s) {
	return 1.0f / (powf(expf(-2.0f), ((x / s) / 2.0f)) + 1.0f);
}

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 / ((exp((-2.0e0)) ** ((x / s) / 2.0e0)) + 1.0e0)
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32((exp(Float32(-2.0)) ^ Float32(Float32(x / s) / Float32(2.0))) + Float32(1.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((exp(single(-2.0)) ^ ((x / s) / single(2.0))) + single(1.0));
end

Derivation

1. Initial program 99.7%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. +-commutative N/A

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

   6. distribute-frac-neg N/A

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

   7. mul-1-neg N/A

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

   8. exp-prod N/A

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

   9. sqr-pow N/A

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

   10. lower-fma.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({\left(e^{-1}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}, {\left(e^{-1}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}, 1\right)}} \]

   11. lower-pow.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}}, {\left(e^{-1}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}, 1\right)} \]

   12. lower-exp.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left({\color{blue}{\left(e^{-1}\right)}}^{\left(\frac{\frac{x}{s}}{2}\right)}, {\left(e^{-1}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}, 1\right)} \]

   13. lower-/.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left({\left(e^{-1}\right)}^{\color{blue}{\left(\frac{\frac{x}{s}}{2}\right)}}, {\left(e^{-1}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}, 1\right)} \]

   14. lower-/.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left({\left(e^{-1}\right)}^{\left(\frac{\color{blue}{\frac{x}{s}}}{2}\right)}, {\left(e^{-1}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}, 1\right)} \]

   15. lower-pow.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left({\left(e^{-1}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}, \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}}, 1\right)} \]

   16. lower-exp.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left({\left(e^{-1}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}, {\color{blue}{\left(e^{-1}\right)}}^{\left(\frac{\frac{x}{s}}{2}\right)}, 1\right)} \]

   17. lower-/.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left({\left(e^{-1}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}, {\left(e^{-1}\right)}^{\color{blue}{\left(\frac{\frac{x}{s}}{2}\right)}}, 1\right)} \]

   18. lower-/.f32 99.7

       \[\leadsto \frac{1}{\mathsf{fma}\left({\left(e^{-1}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}, {\left(e^{-1}\right)}^{\left(\frac{\color{blue}{\frac{x}{s}}}{2}\right)}, 1\right)} \]

4. Applied rewrites 99.7%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({\left(e^{-1}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}, {\left(e^{-1}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}, 1\right)}} \]
5. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-pow.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lower-+.f32 N/A

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

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1}{{\left(e^{-2}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} + 1}} \]
7. Add Preprocessing

Alternative 5: 49.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;1 + e^{t\_0} \leq 1.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= (+ 1.0 (exp t_0)) 1.5) 0.5 (/ 1.0 (+ t_0 2.0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if ((1.0f + expf(t_0)) <= 1.5f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (t_0 + 2.0f);
	}
	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 ((1.0e0 + exp(t_0)) <= 1.5e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (t_0 + 2.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(t_0)) <= Float32(1.5))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(t_0 + Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if ((single(1.0) + exp(t_0)) <= single(1.5))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (t_0 + single(2.0));
	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))) < 1.5

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

   if 1.5 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 62.2

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

   5. Applied rewrites 62.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-1, \frac{x}{s}, 2\right)}} \]
   6. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-/.f32 N/A

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

      7. lift-neg.f32 62.2

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

   7. Applied rewrites 62.2%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s} + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{\frac{-x}{s}}} \end{array} \]

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

Derivation

1. Initial program 99.7%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 7: 63.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, x, -s\right)}{s \cdot s}, x, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.9999999593223797e-31)
   0.5
   (/ 1.0 (fma (/ (fma 0.5 x (- s)) (* s s)) x 2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.9999999593223797e-31f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf((fmaf(0.5f, x, -s) / (s * s)), x, 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.9999999593223797e-31))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(Float32(fma(Float32(0.5), x, Float32(-s)) / Float32(s * s)), x, Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1.99999996e-31

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 46.4%

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

   if 1.99999996e-31 < (neg.f32 x)

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower--.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}, x, 2\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{{s}^{2}} \cdot \frac{1}{2} - \frac{1}{s}, x, 2\right)} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{{s}^{2}} \cdot \frac{1}{2} - \frac{1}{s}, x, 2\right)} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{{s}^{2}} \cdot \frac{1}{2} - \frac{1}{s}, x, 2\right)} \]

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-/.f32 85.0

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{s \cdot s} \cdot 0.5 - \frac{1}{s}, x, 2\right)} \]

   5. Applied rewrites 85.0%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{x}{s \cdot s} \cdot 0.5 - \frac{1}{s}, x, 2\right)}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{-1 \cdot s + \frac{1}{2} \cdot x}{{s}^{2}}, x, 2\right)} \]
   7. Step-by-step derivation

      1. lower-/.f32 N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-neg.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{2}, x, -s\right)}{{s}^{2}}, x, 2\right)} \]

      6. pow2 N/A

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

      7. lift-*.f32 85.0

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, x, -s\right)}{s \cdot s}, x, 2\right)} \]

   8. Applied rewrites 85.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, x, -s\right)}{s \cdot s}, x, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, x, -s\right)}{s \cdot s}, x, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.3% accurate, 128.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

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 99.7%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 34.1%

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

6. Final simplification 34.1%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))