Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 4.5s

Alternatives: 15

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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. 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 99.9

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

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot -1}} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-log1p.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. log-pow-rev N/A

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

   8. inv-pow N/A

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

   9. neg-log N/A

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

   10. lower-neg.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-neg.f32 N/A

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

   13. lift-exp.f32 N/A

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

   14. lift-log1p.f32 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 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 99.8%

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

Alternative 3: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.4000000059604645:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s \cdot s}}{s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot x}{s}, 0.16666666666666666, \left(x \cdot x\right) \cdot 0.5\right)}{s}\right) + \left(-x\right)}{s}\right) + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 0.4000000059604645)
   (/
    1.0
    (fma
     (-
      (* (fma (/ (/ x (* s s)) s) -0.16666666666666666 (/ 0.5 (* s s))) x)
      (/ 1.0 s))
     x
     2.0))
   (/
    1.0
    (+
     1.0
     (/
      1.0
      (+
       (-
        (/
         (+
          (-
           (/ (fma (/ (* (* x x) x) s) 0.16666666666666666 (* (* x x) 0.5)) s))
          (- x))
         s))
       1.0))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 0.4000000059604645f) {
		tmp = 1.0f / fmaf(((fmaf(((x / (s * s)) / s), -0.16666666666666666f, (0.5f / (s * s))) * x) - (1.0f / s)), x, 2.0f);
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (-((-(fmaf((((x * x) * x) / s), 0.16666666666666666f, ((x * x) * 0.5f)) / s) + -x) / s) + 1.0f)));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s)))) <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(fma(Float32(Float32(x / Float32(s * s)) / s), Float32(-0.16666666666666666), Float32(Float32(0.5) / Float32(s * s))) * x) - Float32(Float32(1.0) / s)), x, Float32(2.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(-Float32(Float32(Float32(-Float32(fma(Float32(Float32(Float32(x * x) * x) / s), Float32(0.16666666666666666), Float32(Float32(x * x) * Float32(0.5))) / s)) + Float32(-x)) / s)) + Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.400000006

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) \cdot x + 2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 89.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, \frac{-1}{6}, \frac{\frac{1}{2}}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, \frac{-1}{6}, \frac{\frac{1}{2}}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)} \]

      3. lift-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, \frac{-1}{6}, \frac{\frac{1}{2}}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)} \]

      4. pow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{{s}^{2} \cdot s}, \frac{-1}{6}, \frac{\frac{1}{2}}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{{s}^{2}}}{s}, \frac{-1}{6}, \frac{\frac{1}{2}}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)} \]

      6. lower-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{{s}^{2}}}{s}, \frac{-1}{6}, \frac{\frac{1}{2}}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{{s}^{2}}}{s}, \frac{-1}{6}, \frac{\frac{1}{2}}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s \cdot s}}{s}, \frac{-1}{6}, \frac{\frac{1}{2}}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)} \]

      9. lift-*.f32 89.4

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s \cdot s}}{s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)} \]

   6. Applied rewrites 89.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s \cdot s}}{s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)} \]

   if 0.400000006 < (/.f32 #s(literal 1 binary32) (+.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. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + -1 \cdot \frac{-1 \cdot x + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}}} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{1 + \frac{1}{-1 \cdot \frac{-1 \cdot x + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s} + \color{blue}{1}}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{1}{1 + \frac{1}{-1 \cdot \frac{-1 \cdot x + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s} + \color{blue}{1}}} \]

   6. Applied rewrites 96.6%

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot x}{s}, 0.16666666666666666, \left(x \cdot x\right) \cdot 0.5\right)}{s}\right) + \left(-x\right)}{s}\right) + 1}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.6000000238418579:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{x \cdot x}{s}, -0.16666666666666666, 0.5 \cdot x\right)}{s}\right) + 1}{s}, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, \frac{x}{\left(s \cdot s\right) \cdot s}, \frac{0.5}{s \cdot s}\right), x, \frac{1}{s}\right), x, 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 0.6000000238418579)
   (/
    1.0
    (fma
     (-
      (/
       (+ (- (/ (fma (/ (* x x) s) -0.16666666666666666 (* 0.5 x)) s)) 1.0)
       s))
     x
     2.0))
   (/
    1.0
    (+
     1.0
     (/
      1.0
      (fma
       (fma
        (fma 0.16666666666666666 (/ x (* (* s s) s)) (/ 0.5 (* s s)))
        x
        (/ 1.0 s))
       x
       1.0))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 0.6000000238418579f) {
		tmp = 1.0f / fmaf(-((-(fmaf(((x * x) / s), -0.16666666666666666f, (0.5f * x)) / s) + 1.0f) / s), x, 2.0f);
	} else {
		tmp = 1.0f / (1.0f + (1.0f / fmaf(fmaf(fmaf(0.16666666666666666f, (x / ((s * s) * s)), (0.5f / (s * s))), x, (1.0f / s)), x, 1.0f)));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s)))) <= Float32(0.6000000238418579))
		tmp = Float32(Float32(1.0) / fma(Float32(-Float32(Float32(Float32(-Float32(fma(Float32(Float32(x * x) / s), Float32(-0.16666666666666666), Float32(Float32(0.5) * x)) / s)) + Float32(1.0)) / s)), x, Float32(2.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / fma(fma(fma(Float32(0.16666666666666666), Float32(x / Float32(Float32(s * s) * s)), Float32(Float32(0.5) / Float32(s * s))), x, Float32(Float32(1.0) / s)), x, Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.600000024

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) \cdot x + 2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 79.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)}} \]

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{2}}{s} + \frac{1}{2} \cdot x}{s}}{s}\right), x, 2\right)} \]

      2. lower-neg.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(-\frac{1 + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{2}}{s} + \frac{1}{2} \cdot x}{s}}{s}, x, 2\right)} \]

      3. lower-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(-\frac{1 + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{2}}{s} + \frac{1}{2} \cdot x}{s}}{s}, x, 2\right)} \]

   7. Applied rewrites 90.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{x \cdot x}{s}, -0.16666666666666666, 0.5 \cdot x\right)}{s}\right) + 1}{s}, x, 2\right)} \]

   if 0.600000024 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

      1. lift-exp.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{1 + \frac{1}{x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) + \frac{1}{s}\right) + \color{blue}{1}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{1 + \frac{1}{\left(x \cdot \left(\frac{1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) + \frac{1}{s}\right) \cdot x + 1}} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{1}{1 + \frac{1}{\mathsf{fma}\left(x \cdot \left(\frac{1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) + \frac{1}{s}, \color{blue}{x}, 1\right)}} \]

   6. Applied rewrites 97.9%

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, \frac{x}{\left(s \cdot s\right) \cdot s}, \frac{0.5}{s \cdot s}\right), x, \frac{1}{s}\right), x, 1\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.6000000238418579:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{x \cdot x}{s}, -0.16666666666666666, 0.5 \cdot x\right)}{s}\right) + 1}{s}, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{s \cdot s}, 0.5, \frac{1}{s}\right), x, 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 0.6000000238418579)
   (/
    1.0
    (fma
     (-
      (/
       (+ (- (/ (fma (/ (* x x) s) -0.16666666666666666 (* 0.5 x)) s)) 1.0)
       s))
     x
     2.0))
   (/ 1.0 (+ 1.0 (/ 1.0 (fma (fma (/ x (* s s)) 0.5 (/ 1.0 s)) x 1.0))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 0.6000000238418579f) {
		tmp = 1.0f / fmaf(-((-(fmaf(((x * x) / s), -0.16666666666666666f, (0.5f * x)) / s) + 1.0f) / s), x, 2.0f);
	} else {
		tmp = 1.0f / (1.0f + (1.0f / fmaf(fmaf((x / (s * s)), 0.5f, (1.0f / s)), x, 1.0f)));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s)))) <= Float32(0.6000000238418579))
		tmp = Float32(Float32(1.0) / fma(Float32(-Float32(Float32(Float32(-Float32(fma(Float32(Float32(x * x) / s), Float32(-0.16666666666666666), Float32(Float32(0.5) * x)) / s)) + Float32(1.0)) / s)), x, Float32(2.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / fma(fma(Float32(x / Float32(s * s)), Float32(0.5), Float32(Float32(1.0) / s)), x, Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.600000024

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) \cdot x + 2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 79.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)}} \]

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{2}}{s} + \frac{1}{2} \cdot x}{s}}{s}\right), x, 2\right)} \]

      2. lower-neg.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(-\frac{1 + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{2}}{s} + \frac{1}{2} \cdot x}{s}}{s}, x, 2\right)} \]

      3. lower-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(-\frac{1 + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{2}}{s} + \frac{1}{2} \cdot x}{s}}{s}, x, 2\right)} \]

   7. Applied rewrites 90.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{x \cdot x}{s}, -0.16666666666666666, 0.5 \cdot x\right)}{s}\right) + 1}{s}, x, 2\right)} \]

   if 0.600000024 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

      1. lift-exp.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. lower-/.f32 97.3

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

   6. Applied rewrites 97.3%

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{s \cdot s}, 0.5, \frac{1}{s}\right), x, 1\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\frac{x}{s} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{x}{s}, -0.16666666666666666, 0.5\right)}{s \cdot s} \cdot x - \frac{1}{s}, x, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 0.10000000149011612)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ (/ x s) 1.0))))
   (/
    1.0
    (fma
     (- (* (/ (fma (/ x s) -0.16666666666666666 0.5) (* s s)) x) (/ 1.0 s))
     x
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.10000000149011612f) {
		tmp = 1.0f / (1.0f + (1.0f / ((x / s) + 1.0f)));
	} else {
		tmp = 1.0f / fmaf((((fmaf((x / s), -0.16666666666666666f, 0.5f) / (s * s)) * x) - (1.0f / s)), x, 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0)))));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(Float32(fma(Float32(x / s), Float32(-0.16666666666666666), Float32(0.5)) / Float32(s * s)) * x) - Float32(Float32(1.0) / s)), x, Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.100000001

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lift-/.f32 95.2

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

   6. Applied rewrites 95.2%

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

   if 0.100000001 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) \cdot x + 2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 89.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)}} \]

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{1}{2} + \frac{-1}{6} \cdot \frac{x}{s}}{{s}^{2}} \cdot x - \frac{1}{s}, x, 2\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{-1}{6} \cdot \frac{x}{s} + \frac{1}{2}}{{s}^{2}} \cdot x - \frac{1}{s}, x, 2\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{x}{s} \cdot \frac{-1}{6} + \frac{1}{2}}{{s}^{2}} \cdot x - \frac{1}{s}, x, 2\right)} \]

      4. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{-1}{6}, \frac{1}{2}\right)}{{s}^{2}} \cdot x - \frac{1}{s}, x, 2\right)} \]

      5. lift-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{-1}{6}, \frac{1}{2}\right)}{{s}^{2}} \cdot x - \frac{1}{s}, x, 2\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{-1}{6}, \frac{1}{2}\right)}{s \cdot s} \cdot x - \frac{1}{s}, x, 2\right)} \]

      7. lift-*.f32 88.7

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{x}{s}, -0.16666666666666666, 0.5\right)}{s \cdot s} \cdot x - \frac{1}{s}, x, 2\right)} \]

   7. Applied rewrites 88.7%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{x}{s}, -0.16666666666666666, 0.5\right)}{s \cdot s} \cdot x - \frac{1}{s}, x, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 92.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 5:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\frac{x}{s} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot -0.16666666666666666}{s}, x, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 5.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ (/ x s) 1.0))))
   (/ 1.0 (fma (/ (* (* x (/ x (* s s))) -0.16666666666666666) s) x 2.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 5.0f) {
		tmp = 1.0f / (1.0f + (1.0f / ((x / s) + 1.0f)));
	} else {
		tmp = 1.0f / fmaf((((x * (x / (s * s))) * -0.16666666666666666f) / s), x, 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(5.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0)))));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(Float32(x * Float32(x / Float32(s * s))) * Float32(-0.16666666666666666)) / s), x, Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lift-/.f32 94.0

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

   6. Applied rewrites 94.0%

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

   if 5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) \cdot x + 2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 91.0%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{-1}{6} \cdot \frac{{x}^{2}}{{s}^{3}}, x, 2\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{-1}{6} \cdot {x}^{2}}{{s}^{3}}, x, 2\right)} \]

      2. lower-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{-1}{6} \cdot {x}^{2}}{{s}^{3}}, x, 2\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{{x}^{2} \cdot \frac{-1}{6}}{{s}^{3}}, x, 2\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{{x}^{2} \cdot \frac{-1}{6}}{{s}^{3}}, x, 2\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{{s}^{3}}, x, 2\right)} \]

      6. lift-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{{s}^{3}}, x, 2\right)} \]

      7. pow3 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{\left(s \cdot s\right) \cdot s}, x, 2\right)} \]

      8. lift-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{\left(s \cdot s\right) \cdot s}, x, 2\right)} \]

      9. lift-*.f32 86.3

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot -0.16666666666666666}{\left(s \cdot s\right) \cdot s}, x, 2\right)} \]

   7. Applied rewrites 86.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot -0.16666666666666666}{\left(s \cdot s\right) \cdot s}, x, 2\right)} \]
   8. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{\left(s \cdot s\right) \cdot s}, x, 2\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{\left(s \cdot s\right) \cdot s}, x, 2\right)} \]

      3. lift-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{\left(s \cdot s\right) \cdot s}, x, 2\right)} \]

      4. pow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{{s}^{2} \cdot s}, x, 2\right)} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{{s}^{2}}}{s}, x, 2\right)} \]

      6. lift-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{{s}^{2}}}{s}, x, 2\right)} \]

      7. lift-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{{s}^{2}}}{s}, x, 2\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{{x}^{2} \cdot \frac{-1}{6}}{{s}^{2}}}{s}, x, 2\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{\frac{-1}{6} \cdot {x}^{2}}{{s}^{2}}}{s}, x, 2\right)} \]

      10. associate-*r/ N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{-1}{6} \cdot \frac{{x}^{2}}{{s}^{2}}}{s}, x, 2\right)} \]

      11. lower-/.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{-1}{6} \cdot \frac{{x}^{2}}{{s}^{2}}}{s}, x, 2\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{{x}^{2}}{{s}^{2}} \cdot \frac{-1}{6}}{s}, x, 2\right)} \]

      13. lower-*.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{{x}^{2}}{{s}^{2}} \cdot \frac{-1}{6}}{s}, x, 2\right)} \]

      14. pow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{x \cdot x}{{s}^{2}} \cdot \frac{-1}{6}}{s}, x, 2\right)} \]

      15. associate-/l* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot \frac{x}{{s}^{2}}\right) \cdot \frac{-1}{6}}{s}, x, 2\right)} \]

      16. lower-*.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot \frac{x}{{s}^{2}}\right) \cdot \frac{-1}{6}}{s}, x, 2\right)} \]

      17. lower-/.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot \frac{x}{{s}^{2}}\right) \cdot \frac{-1}{6}}{s}, x, 2\right)} \]

      18. pow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot \frac{-1}{6}}{s}, x, 2\right)} \]

      19. lift-*.f32 88.8

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot -0.16666666666666666}{s}, x, 2\right)} \]

   9. Applied rewrites 88.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot -0.16666666666666666}{s}, x, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 91.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\frac{x}{s} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{x}{s} \cdot \left(x \cdot \frac{x}{s \cdot s}\right)\right) \cdot -0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 50.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ (/ x s) 1.0))))
   (/ 1.0 (* (* (/ x s) (* x (/ x (* s s)))) -0.16666666666666666))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 50.0f) {
		tmp = 1.0f / (1.0f + (1.0f / ((x / s) + 1.0f)));
	} else {
		tmp = 1.0f / (((x / s) * (x * (x / (s * s)))) * -0.16666666666666666f);
	}
	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))) <= 50.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / ((x / s) + 1.0e0)))
    else
        tmp = 1.0e0 / (((x / s) * (x * (x / (s * s)))) * (-0.16666666666666666e0))
    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(50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(x / s) * Float32(x * Float32(x / Float32(s * s)))) * Float32(-0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(50.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / ((x / s) + single(1.0))));
	else
		tmp = single(1.0) / (((x / s) * (x * (x / (s * s)))) * single(-0.16666666666666666));
	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))) < 50

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lift-/.f32 94.0

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

   6. Applied rewrites 94.0%

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

   if 50 < (+.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

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) \cdot x + 2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 90.9%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\frac{-1}{6} \cdot \color{blue}{\frac{{x}^{3}}{{s}^{3}}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{\frac{{x}^{3}}{{s}^{3}} \cdot \frac{-1}{6}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{1}{\frac{{x}^{3}}{{s}^{3}} \cdot \frac{-1}{6}} \]

      3. lower-/.f32 N/A

         \[\leadsto \frac{1}{\frac{{x}^{3}}{{s}^{3}} \cdot \frac{-1}{6}} \]

      4. pow3 N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot x}{{s}^{3}} \cdot \frac{-1}{6}} \]

      5. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot x}{{s}^{3}} \cdot \frac{-1}{6}} \]

      6. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot x}{{s}^{3}} \cdot \frac{-1}{6}} \]

      7. pow3 N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot x}{\left(s \cdot s\right) \cdot s} \cdot \frac{-1}{6}} \]

      8. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot x}{\left(s \cdot s\right) \cdot s} \cdot \frac{-1}{6}} \]

      9. lift-*.f32 80.4

         \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot x}{\left(s \cdot s\right) \cdot s} \cdot -0.16666666666666666} \]

   7. Applied rewrites 80.4%

      \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot x}{\left(s \cdot s\right) \cdot s} \cdot \color{blue}{-0.16666666666666666}} \]
   8. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot x}{\left(s \cdot s\right) \cdot s} \cdot \frac{-1}{6}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot x}{\left(s \cdot s\right) \cdot s} \cdot \frac{-1}{6}} \]

      3. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot x}{\left(s \cdot s\right) \cdot s} \cdot \frac{-1}{6}} \]

      4. pow3 N/A

         \[\leadsto \frac{1}{\frac{{x}^{3}}{\left(s \cdot s\right) \cdot s} \cdot \frac{-1}{6}} \]

      5. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{{x}^{3}}{\left(s \cdot s\right) \cdot s} \cdot \frac{-1}{6}} \]

      6. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{{x}^{3}}{\left(s \cdot s\right) \cdot s} \cdot \frac{-1}{6}} \]

      7. pow3 N/A

         \[\leadsto \frac{1}{\frac{{x}^{3}}{{s}^{3}} \cdot \frac{-1}{6}} \]

      8. cube-div-rev N/A

         \[\leadsto \frac{1}{{\left(\frac{x}{s}\right)}^{3} \cdot \frac{-1}{6}} \]

      9. cube-mult N/A

         \[\leadsto \frac{1}{\left(\frac{x}{s} \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)\right) \cdot \frac{-1}{6}} \]

      10. times-frac N/A

          \[\leadsto \frac{1}{\left(\frac{x}{s} \cdot \frac{x \cdot x}{s \cdot s}\right) \cdot \frac{-1}{6}} \]

      11. pow2 N/A

          \[\leadsto \frac{1}{\left(\frac{x}{s} \cdot \frac{{x}^{2}}{s \cdot s}\right) \cdot \frac{-1}{6}} \]

      12. pow2 N/A

          \[\leadsto \frac{1}{\left(\frac{x}{s} \cdot \frac{{x}^{2}}{{s}^{2}}\right) \cdot \frac{-1}{6}} \]

      13. lower-*.f32 N/A

          \[\leadsto \frac{1}{\left(\frac{x}{s} \cdot \frac{{x}^{2}}{{s}^{2}}\right) \cdot \frac{-1}{6}} \]

      14. lift-/.f32 N/A

          \[\leadsto \frac{1}{\left(\frac{x}{s} \cdot \frac{{x}^{2}}{{s}^{2}}\right) \cdot \frac{-1}{6}} \]

      15. pow2 N/A

          \[\leadsto \frac{1}{\left(\frac{x}{s} \cdot \frac{x \cdot x}{{s}^{2}}\right) \cdot \frac{-1}{6}} \]

      16. associate-/l* N/A

          \[\leadsto \frac{1}{\left(\frac{x}{s} \cdot \left(x \cdot \frac{x}{{s}^{2}}\right)\right) \cdot \frac{-1}{6}} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{\left(\frac{x}{s} \cdot \left(x \cdot \frac{x}{{s}^{2}}\right)\right) \cdot \frac{-1}{6}} \]

      18. lower-/.f32 N/A

          \[\leadsto \frac{1}{\left(\frac{x}{s} \cdot \left(x \cdot \frac{x}{{s}^{2}}\right)\right) \cdot \frac{-1}{6}} \]

      19. pow2 N/A

          \[\leadsto \frac{1}{\left(\frac{x}{s} \cdot \left(x \cdot \frac{x}{s \cdot s}\right)\right) \cdot \frac{-1}{6}} \]

      20. lift-*.f32 88.0

          \[\leadsto \frac{1}{\left(\frac{x}{s} \cdot \left(x \cdot \frac{x}{s \cdot s}\right)\right) \cdot -0.16666666666666666} \]

   9. Applied rewrites 88.0%

      \[\leadsto \frac{1}{\left(\frac{x}{s} \cdot \left(x \cdot \frac{x}{s \cdot s}\right)\right) \cdot -0.16666666666666666} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 91.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1000000000:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\frac{x}{s} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(\left(x \cdot x\right) \cdot \frac{x}{\left(s \cdot s\right) \cdot s}\right) \cdot -0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 1000000000.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ (/ x s) 1.0))))
   (/ 1.0 (+ 2.0 (* (* (* x x) (/ x (* (* s s) s))) -0.16666666666666666)))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1000000000.0f) {
		tmp = 1.0f / (1.0f + (1.0f / ((x / s) + 1.0f)));
	} else {
		tmp = 1.0f / (2.0f + (((x * x) * (x / ((s * s) * s))) * -0.16666666666666666f));
	}
	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))) <= 1000000000.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / ((x / s) + 1.0e0)))
    else
        tmp = 1.0e0 / (2.0e0 + (((x * x) * (x / ((s * s) * s))) * (-0.16666666666666666e0)))
    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(1000000000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(x * x) * Float32(x / Float32(Float32(s * s) * s))) * Float32(-0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(1000000000.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / ((x / s) + single(1.0))));
	else
		tmp = single(1.0) / (single(2.0) + (((x * x) * (x / ((s * s) * s))) * single(-0.16666666666666666)));
	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))) < 1e9

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lift-/.f32 93.5

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

   6. Applied rewrites 93.5%

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

   if 1e9 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) \cdot x + 2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 91.9%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)}} \]

   5. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

         \[\leadsto \frac{1}{\left(2 + -1 \cdot \frac{x}{s}\right) + \color{blue}{\left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{1}{\left(2 + -1 \cdot \frac{x}{s}\right) + \color{blue}{\left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\left(-1 \cdot \frac{x}{s} + 2\right) + \left(\color{blue}{\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{1}{\left(-1 \cdot \frac{x}{s} + 2\right) + \left(\color{blue}{\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{\left(\left(\mathsf{neg}\left(\frac{x}{s}\right)\right) + 2\right) + \left(\color{blue}{\frac{-1}{6}} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

      6. distribute-frac-neg N/A

         \[\leadsto \frac{1}{\left(\frac{\mathsf{neg}\left(x\right)}{s} + 2\right) + \left(\color{blue}{\frac{-1}{6}} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{1}{\left(\frac{\mathsf{neg}\left(x\right)}{s} + 2\right) + \left(\color{blue}{\frac{-1}{6}} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

      8. lift-neg.f32 N/A

         \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \left(\frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}} + \color{blue}{\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \left(\frac{{x}^{2}}{{s}^{2}} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{6}} \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]

      11. lower-fma.f32 N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{{x}^{2}}{{s}^{2}}, \color{blue}{\frac{1}{2}}, \frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]

      12. lower-/.f32 N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{{x}^{2}}{{s}^{2}}, \frac{1}{2}, \frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]

      13. pow2 N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{x \cdot x}{{s}^{2}}, \frac{1}{2}, \frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]

      14. lift-*.f32 N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{x \cdot x}{{s}^{2}}, \frac{1}{2}, \frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]

      15. pow2 N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{x \cdot x}{s \cdot s}, \frac{1}{2}, \frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]

      16. lift-*.f32 N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{x \cdot x}{s \cdot s}, \frac{1}{2}, \frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{x \cdot x}{s \cdot s}, \frac{1}{2}, \frac{{x}^{3}}{{s}^{3}} \cdot \frac{-1}{6}\right)} \]

      18. lower-*.f32 N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{x \cdot x}{s \cdot s}, \frac{1}{2}, \frac{{x}^{3}}{{s}^{3}} \cdot \frac{-1}{6}\right)} \]

   7. Applied rewrites 81.3%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{x \cdot x}{s \cdot s}, 0.5, \frac{\left(x \cdot x\right) \cdot x}{\left(s \cdot s\right) \cdot s} \cdot -0.16666666666666666\right)}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 81.3%

       \[\leadsto \frac{1}{2 + \mathsf{fma}\left(\color{blue}{\frac{x \cdot x}{s \cdot s}}, 0.5, \frac{\left(x \cdot x\right) \cdot x}{\left(s \cdot s\right) \cdot s} \cdot -0.16666666666666666\right)} \]

   11. Taylor expanded in x around inf

       \[\leadsto \frac{1}{2 + \frac{-1}{6} \cdot \color{blue}{\frac{{x}^{3}}{{s}^{3}}}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{2 + \frac{{x}^{3}}{{s}^{3}} \cdot \frac{-1}{6}} \]

       2. lower-*.f32 N/A

          \[\leadsto \frac{1}{2 + \frac{{x}^{3}}{{s}^{3}} \cdot \frac{-1}{6}} \]

       3. pow3 N/A

          \[\leadsto \frac{1}{2 + \frac{\left(x \cdot x\right) \cdot x}{{s}^{3}} \cdot \frac{-1}{6}} \]

       4. pow2 N/A

          \[\leadsto \frac{1}{2 + \frac{{x}^{2} \cdot x}{{s}^{3}} \cdot \frac{-1}{6}} \]

       5. pow3 N/A

          \[\leadsto \frac{1}{2 + \frac{{x}^{2} \cdot x}{\left(s \cdot s\right) \cdot s} \cdot \frac{-1}{6}} \]

       6. associate-/l* N/A

          \[\leadsto \frac{1}{2 + \left({x}^{2} \cdot \frac{x}{\left(s \cdot s\right) \cdot s}\right) \cdot \frac{-1}{6}} \]

       7. pow3 N/A

          \[\leadsto \frac{1}{2 + \left({x}^{2} \cdot \frac{x}{{s}^{3}}\right) \cdot \frac{-1}{6}} \]

       8. lower-*.f32 N/A

          \[\leadsto \frac{1}{2 + \left({x}^{2} \cdot \frac{x}{{s}^{3}}\right) \cdot \frac{-1}{6}} \]

       9. pow2 N/A

          \[\leadsto \frac{1}{2 + \left(\left(x \cdot x\right) \cdot \frac{x}{{s}^{3}}\right) \cdot \frac{-1}{6}} \]

       10. lift-*.f32 N/A

           \[\leadsto \frac{1}{2 + \left(\left(x \cdot x\right) \cdot \frac{x}{{s}^{3}}\right) \cdot \frac{-1}{6}} \]

       11. lower-/.f32 N/A

           \[\leadsto \frac{1}{2 + \left(\left(x \cdot x\right) \cdot \frac{x}{{s}^{3}}\right) \cdot \frac{-1}{6}} \]

       12. pow3 N/A

           \[\leadsto \frac{1}{2 + \left(\left(x \cdot x\right) \cdot \frac{x}{\left(s \cdot s\right) \cdot s}\right) \cdot \frac{-1}{6}} \]

       13. lift-*.f32 N/A

           \[\leadsto \frac{1}{2 + \left(\left(x \cdot x\right) \cdot \frac{x}{\left(s \cdot s\right) \cdot s}\right) \cdot \frac{-1}{6}} \]

       14. lift-*.f32 87.7

           \[\leadsto \frac{1}{2 + \left(\left(x \cdot x\right) \cdot \frac{x}{\left(s \cdot s\right) \cdot s}\right) \cdot -0.16666666666666666} \]

   13. Applied rewrites 87.7%

       \[\leadsto \frac{1}{2 + \left(\left(x \cdot x\right) \cdot \frac{x}{\left(s \cdot s\right) \cdot s}\right) \cdot \color{blue}{-0.16666666666666666}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 89.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-exp.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lift-/.f32 95.2

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

   6. Applied rewrites 95.2%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) \cdot x + 2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 79.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)}} \]

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lift-/.f32 84.6

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

   7. Applied rewrites 84.6%

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

Alternative 11: 89.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 5:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\frac{x}{s} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(x \cdot \frac{x}{s \cdot s}\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 5.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ (/ x s) 1.0))))
   (/ 1.0 (+ 2.0 (* (* x (/ x (* s s))) 0.5)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 5.0f) {
		tmp = 1.0f / (1.0f + (1.0f / ((x / s) + 1.0f)));
	} else {
		tmp = 1.0f / (2.0f + ((x * (x / (s * s))) * 0.5f));
	}
	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 / s) <= 5.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / ((x / s) + 1.0e0)))
    else
        tmp = 1.0e0 / (2.0e0 + ((x * (x / (s * s))) * 0.5e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(5.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(x * Float32(x / Float32(s * s))) * Float32(0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(5.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / ((x / s) + single(1.0))));
	else
		tmp = single(1.0) / (single(2.0) + ((x * (x / (s * s))) * single(0.5)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lift-/.f32 94.0

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

   6. Applied rewrites 94.0%

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

   if 5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) \cdot x + 2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 91.0%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)}} \]

   5. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

         \[\leadsto \frac{1}{\left(2 + -1 \cdot \frac{x}{s}\right) + \color{blue}{\left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{1}{\left(2 + -1 \cdot \frac{x}{s}\right) + \color{blue}{\left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\left(-1 \cdot \frac{x}{s} + 2\right) + \left(\color{blue}{\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{1}{\left(-1 \cdot \frac{x}{s} + 2\right) + \left(\color{blue}{\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{\left(\left(\mathsf{neg}\left(\frac{x}{s}\right)\right) + 2\right) + \left(\color{blue}{\frac{-1}{6}} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

      6. distribute-frac-neg N/A

         \[\leadsto \frac{1}{\left(\frac{\mathsf{neg}\left(x\right)}{s} + 2\right) + \left(\color{blue}{\frac{-1}{6}} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{1}{\left(\frac{\mathsf{neg}\left(x\right)}{s} + 2\right) + \left(\color{blue}{\frac{-1}{6}} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

      8. lift-neg.f32 N/A

         \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \left(\frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}} + \color{blue}{\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \left(\frac{{x}^{2}}{{s}^{2}} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{6}} \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]

      11. lower-fma.f32 N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{{x}^{2}}{{s}^{2}}, \color{blue}{\frac{1}{2}}, \frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]

      12. lower-/.f32 N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{{x}^{2}}{{s}^{2}}, \frac{1}{2}, \frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]

      13. pow2 N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{x \cdot x}{{s}^{2}}, \frac{1}{2}, \frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]

      14. lift-*.f32 N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{x \cdot x}{{s}^{2}}, \frac{1}{2}, \frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]

      15. pow2 N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{x \cdot x}{s \cdot s}, \frac{1}{2}, \frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]

      16. lift-*.f32 N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{x \cdot x}{s \cdot s}, \frac{1}{2}, \frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{x \cdot x}{s \cdot s}, \frac{1}{2}, \frac{{x}^{3}}{{s}^{3}} \cdot \frac{-1}{6}\right)} \]

      18. lower-*.f32 N/A

          \[\leadsto \frac{1}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{x \cdot x}{s \cdot s}, \frac{1}{2}, \frac{{x}^{3}}{{s}^{3}} \cdot \frac{-1}{6}\right)} \]

   7. Applied rewrites 80.5%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{-x}{s} + 2\right) + \mathsf{fma}\left(\frac{x \cdot x}{s \cdot s}, 0.5, \frac{\left(x \cdot x\right) \cdot x}{\left(s \cdot s\right) \cdot s} \cdot -0.16666666666666666\right)}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 80.5%

       \[\leadsto \frac{1}{2 + \mathsf{fma}\left(\color{blue}{\frac{x \cdot x}{s \cdot s}}, 0.5, \frac{\left(x \cdot x\right) \cdot x}{\left(s \cdot s\right) \cdot s} \cdot -0.16666666666666666\right)} \]

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. pow2 N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f32 N/A

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

       6. lower-/.f32 N/A

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

       7. pow2 N/A

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

       8. lift-*.f32 82.5

          \[\leadsto \frac{1}{2 + \left(x \cdot \frac{x}{s \cdot s}\right) \cdot 0.5} \]

   13. Applied rewrites 82.5%

       \[\leadsto \frac{1}{2 + \left(x \cdot \frac{x}{s \cdot s}\right) \cdot \color{blue}{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 88.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\frac{x}{s} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 50.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ (/ x s) 1.0))))
   (/ 1.0 (* (* x (/ x (* s s))) 0.5))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 50.0f) {
		tmp = 1.0f / (1.0f + (1.0f / ((x / s) + 1.0f)));
	} else {
		tmp = 1.0f / ((x * (x / (s * s))) * 0.5f);
	}
	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))) <= 50.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / ((x / s) + 1.0e0)))
    else
        tmp = 1.0e0 / ((x * (x / (s * s))) * 0.5e0)
    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(50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(x / Float32(s * s))) * Float32(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(50.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / ((x / s) + single(1.0))));
	else
		tmp = single(1.0) / ((x * (x / (s * s))) * single(0.5));
	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))) < 50

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lift-/.f32 94.0

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

   6. Applied rewrites 94.0%

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

   if 50 < (+.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

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) \cdot x + 2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 90.9%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)}} \]

   5. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot \left(s \cdot {x}^{2}\right)}{{s}^{\color{blue}{3}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{1}{2} \cdot \left(s \cdot {x}^{2}\right) + \frac{-1}{6} \cdot {x}^{3}}{{s}^{3}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\left(s \cdot {x}^{2}\right) \cdot \frac{1}{2} + \frac{-1}{6} \cdot {x}^{3}}{{s}^{3}}} \]

      4. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(s \cdot {x}^{2}, \frac{1}{2}, \frac{-1}{6} \cdot {x}^{3}\right)}{{s}^{3}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left({x}^{2} \cdot s, \frac{1}{2}, \frac{-1}{6} \cdot {x}^{3}\right)}{{s}^{3}}} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left({x}^{2} \cdot s, \frac{1}{2}, \frac{-1}{6} \cdot {x}^{3}\right)}{{s}^{3}}} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, \frac{-1}{6} \cdot {x}^{3}\right)}{{s}^{3}}} \]

      8. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, \frac{-1}{6} \cdot {x}^{3}\right)}{{s}^{3}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, {x}^{3} \cdot \frac{-1}{6}\right)}{{s}^{3}}} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, {x}^{3} \cdot \frac{-1}{6}\right)}{{s}^{3}}} \]

      11. pow3 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right)}{{s}^{3}}} \]

      12. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right)}{{s}^{3}}} \]

      13. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right)}{{s}^{3}}} \]

      14. pow3 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right)}{\left(s \cdot s\right) \cdot s}} \]

      15. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right)}{\left(s \cdot s\right) \cdot s}} \]

      16. lift-*.f32 80.2

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, 0.5, \left(\left(x \cdot x\right) \cdot x\right) \cdot -0.16666666666666666\right)}{\left(s \cdot s\right) \cdot s}} \]

   7. Applied rewrites 80.2%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, 0.5, \left(\left(x \cdot x\right) \cdot x\right) \cdot -0.16666666666666666\right)}{\color{blue}{\left(s \cdot s\right) \cdot s}}} \]

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. pow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 82.4

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

   10. Applied rewrites 82.4%

       \[\leadsto \frac{1}{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 62.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 50:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 50.0)
   0.5
   (/ 1.0 (* (* x (/ x (* s s))) 0.5))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 50.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((x * (x / (s * s))) * 0.5f);
	}
	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))) <= 50.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((x * (x / (s * s))) * 0.5e0)
    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(50.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(x / Float32(s * s))) * Float32(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(50.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((x * (x / (s * s))) * single(0.5));
	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))) < 50

   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 51.1%

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

   if 50 < (+.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

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) \cdot x + 2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 90.9%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right) \cdot x - \frac{1}{s}, x, 2\right)}} \]

   5. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot \left(s \cdot {x}^{2}\right)}{{s}^{\color{blue}{3}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{1}{2} \cdot \left(s \cdot {x}^{2}\right) + \frac{-1}{6} \cdot {x}^{3}}{{s}^{3}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\left(s \cdot {x}^{2}\right) \cdot \frac{1}{2} + \frac{-1}{6} \cdot {x}^{3}}{{s}^{3}}} \]

      4. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(s \cdot {x}^{2}, \frac{1}{2}, \frac{-1}{6} \cdot {x}^{3}\right)}{{s}^{3}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left({x}^{2} \cdot s, \frac{1}{2}, \frac{-1}{6} \cdot {x}^{3}\right)}{{s}^{3}}} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left({x}^{2} \cdot s, \frac{1}{2}, \frac{-1}{6} \cdot {x}^{3}\right)}{{s}^{3}}} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, \frac{-1}{6} \cdot {x}^{3}\right)}{{s}^{3}}} \]

      8. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, \frac{-1}{6} \cdot {x}^{3}\right)}{{s}^{3}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, {x}^{3} \cdot \frac{-1}{6}\right)}{{s}^{3}}} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, {x}^{3} \cdot \frac{-1}{6}\right)}{{s}^{3}}} \]

      11. pow3 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right)}{{s}^{3}}} \]

      12. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right)}{{s}^{3}}} \]

      13. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right)}{{s}^{3}}} \]

      14. pow3 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right)}{\left(s \cdot s\right) \cdot s}} \]

      15. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, \frac{1}{2}, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right)}{\left(s \cdot s\right) \cdot s}} \]

      16. lift-*.f32 80.2

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, 0.5, \left(\left(x \cdot x\right) \cdot x\right) \cdot -0.16666666666666666\right)}{\left(s \cdot s\right) \cdot s}} \]

   7. Applied rewrites 80.2%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot s, 0.5, \left(\left(x \cdot x\right) \cdot x\right) \cdot -0.16666666666666666\right)}{\color{blue}{\left(s \cdot s\right) \cdot s}}} \]

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. pow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 82.4

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

   10. Applied rewrites 82.4%

       \[\leadsto \frac{1}{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 48.6% 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 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 28.2%

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

   if 1.5 < (+.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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. mul-1-neg N/A

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

      4. distribute-frac-neg N/A

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

      5. lift-/.f32 N/A

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

      6. lift-neg.f32 61.5

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

   4. Applied rewrites 61.5%

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

Alternative 15: 34.6% 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 99.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 34.6%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))