Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 7.1s

Alternatives: 15

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 3504.652099609375
  2. s = 5.223013795330189e-5

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 99.9%

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

4. Applied rewrites 99.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f32 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 94.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.30000001192092896:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right)}{s \cdot s} \cdot x - \frac{1}{s}, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, \frac{x}{s}, 1\right)}{s}, x, 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 0.30000001192092896)
   (/
    1.0
    (fma
     (- (* (/ (fma -0.16666666666666666 (/ x s) 0.5) (* s s)) x) (/ 1.0 s))
     x
     2.0))
   (/ 1.0 (- 1.0 (/ -1.0 (fma (/ (fma 0.5 (/ x s) 1.0) s) x 1.0))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 0.30000001192092896f) {
		tmp = 1.0f / fmaf((((fmaf(-0.16666666666666666f, (x / s), 0.5f) / (s * s)) * x) - (1.0f / s)), x, 2.0f);
	} else {
		tmp = 1.0f / (1.0f - (-1.0f / fmaf((fmaf(0.5f, (x / s), 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(0.30000001192092896))
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(Float32(fma(Float32(-0.16666666666666666), Float32(x / s), 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) / fma(Float32(fma(Float32(0.5), Float32(x / s), 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)))) < 0.300000012

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

   5. Applied rewrites 92.0%

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

   if 0.300000012 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. *-lft-identity N/A

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

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

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

      4. distribute-lft-neg-in N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f32 N/A

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

      7. lift-exp.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. distribute-frac-neg N/A

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

      11. exp-neg N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. cosh-neg N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-neg.f32 N/A

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

      18. lift-/.f32 N/A

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

      19. cosh-neg-rev N/A

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

      20. remove-double-neg N/A

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

      21. lift-neg.f32 N/A

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

      22. distribute-frac-neg N/A

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

      23. lift-/.f32 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-/l* N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f32 N/A

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

      9. lower-fma.f32 N/A

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

      10. lower-/.f32 97.6

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

   7. Applied rewrites 97.6%

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

Alternative 3: 92.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.30000001192092896:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{\frac{\left(x \cdot \frac{x}{s}\right) \cdot -0.16666666666666666}{s} - 1}{s}, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, \frac{x}{s}, 1\right)}{s}, x, 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 0.30000001192092896)
   (/
    1.0
    (fma (/ (- (/ (* (* x (/ x s)) -0.16666666666666666) s) 1.0) s) x 2.0))
   (/ 1.0 (- 1.0 (/ -1.0 (fma (/ (fma 0.5 (/ x s) 1.0) s) x 1.0))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 0.30000001192092896f) {
		tmp = 1.0f / fmaf((((((x * (x / s)) * -0.16666666666666666f) / s) - 1.0f) / s), x, 2.0f);
	} else {
		tmp = 1.0f / (1.0f - (-1.0f / fmaf((fmaf(0.5f, (x / s), 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(0.30000001192092896))
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(Float32(Float32(Float32(x * Float32(x / s)) * Float32(-0.16666666666666666)) / s) - Float32(1.0)) / s), x, Float32(2.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(-1.0) / fma(Float32(fma(Float32(0.5), Float32(x / s), 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)))) < 0.300000012

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

   5. Applied rewrites 92.0%

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

   7. Applied rewrites 88.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 88.0%

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

   if 0.300000012 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. *-lft-identity N/A

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

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

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

      4. distribute-lft-neg-in N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f32 N/A

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

      7. lift-exp.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. distribute-frac-neg N/A

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

      11. exp-neg N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. cosh-neg N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-neg.f32 N/A

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

      18. lift-/.f32 N/A

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

      19. cosh-neg-rev N/A

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

      20. remove-double-neg N/A

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

      21. lift-neg.f32 N/A

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

      22. distribute-frac-neg N/A

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

      23. lift-/.f32 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-/l* N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f32 N/A

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

      9. lower-fma.f32 N/A

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

      10. lower-/.f32 97.6

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

   7. Applied rewrites 97.6%

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

Alternative 4: 91.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.30000001192092896:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \frac{1}{s}, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, \frac{x}{s}, 1\right)}{s}, x, 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 0.30000001192092896)
   (/ 1.0 (+ 1.0 (fma (- (* (/ 0.5 (* s s)) x) (/ 1.0 s)) x 1.0)))
   (/ 1.0 (- 1.0 (/ -1.0 (fma (/ (fma 0.5 (/ x s) 1.0) s) x 1.0))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 0.30000001192092896f) {
		tmp = 1.0f / (1.0f + fmaf((((0.5f / (s * s)) * x) - (1.0f / s)), x, 1.0f));
	} else {
		tmp = 1.0f / (1.0f - (-1.0f / fmaf((fmaf(0.5f, (x / s), 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(0.30000001192092896))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + fma(Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) - Float32(Float32(1.0) / s)), x, Float32(1.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(-1.0) / fma(Float32(fma(Float32(0.5), Float32(x / s), 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)))) < 0.300000012

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 82.2

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

   5. Applied rewrites 82.2%

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

   if 0.300000012 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. *-lft-identity N/A

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

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

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

      4. distribute-lft-neg-in N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f32 N/A

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

      7. lift-exp.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. distribute-frac-neg N/A

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

      11. exp-neg N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. cosh-neg N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-neg.f32 N/A

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

      18. lift-/.f32 N/A

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

      19. cosh-neg-rev N/A

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

      20. remove-double-neg N/A

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

      21. lift-neg.f32 N/A

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

      22. distribute-frac-neg N/A

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

      23. lift-/.f32 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-/l* N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f32 N/A

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

      9. lower-fma.f32 N/A

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

      10. lower-/.f32 97.6

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

   7. Applied rewrites 97.6%

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

Alternative 5: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.30000001192092896:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \frac{1}{s}, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\frac{x}{s} + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 0.30000001192092896)
   (/ 1.0 (+ 1.0 (fma (- (* (/ 0.5 (* s s)) x) (/ 1.0 s)) x 1.0)))
   (/ 1.0 (- 1.0 (/ -1.0 (+ (/ x s) 1.0))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 0.30000001192092896f) {
		tmp = 1.0f / (1.0f + fmaf((((0.5f / (s * s)) * x) - (1.0f / s)), x, 1.0f));
	} else {
		tmp = 1.0f / (1.0f - (-1.0f / ((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(0.30000001192092896))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + fma(Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) - Float32(Float32(1.0) / s)), x, Float32(1.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(-1.0) / Float32(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)))) < 0.300000012

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 82.2

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

   5. Applied rewrites 82.2%

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

   if 0.300000012 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. *-lft-identity N/A

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

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

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

      4. distribute-lft-neg-in N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f32 N/A

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

      7. lift-exp.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. distribute-frac-neg N/A

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

      11. exp-neg N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. cosh-neg N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-neg.f32 N/A

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

      18. lift-/.f32 N/A

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

      19. cosh-neg-rev N/A

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

      20. remove-double-neg N/A

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

      21. lift-neg.f32 N/A

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

      22. distribute-frac-neg N/A

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

      23. lift-/.f32 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 96.3

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

   7. Applied rewrites 96.3%

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

Alternative 6: 93.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.5:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, \frac{x}{s}, 1\right)}{s}, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right) \cdot x}{s} - 1}{s}, x, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 1.5)
   (/ 1.0 (- 1.0 (/ -1.0 (fma (/ (fma 0.5 (/ x s) 1.0) s) x 1.0))))
   (/
    1.0
    (fma
     (/ (- (/ (* (fma -0.16666666666666666 (/ x s) 0.5) x) s) 1.0) s)
     x
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.5f) {
		tmp = 1.0f / (1.0f - (-1.0f / fmaf((fmaf(0.5f, (x / s), 1.0f) / s), x, 1.0f)));
	} else {
		tmp = 1.0f / fmaf(((((fmaf(-0.16666666666666666f, (x / s), 0.5f) * x) / s) - 1.0f) / s), x, 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.5))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(-1.0) / fma(Float32(fma(Float32(0.5), Float32(x / s), Float32(1.0)) / s), x, Float32(1.0)))));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(Float32(Float32(fma(Float32(-0.16666666666666666), Float32(x / s), Float32(0.5)) * x) / s) - Float32(1.0)) / s), x, Float32(2.0)));
	end
	return 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. *-lft-identity N/A

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

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

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

      4. distribute-lft-neg-in N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f32 N/A

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

      7. lift-exp.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. distribute-frac-neg N/A

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

      11. exp-neg N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. cosh-neg N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-neg.f32 N/A

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

      18. lift-/.f32 N/A

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

      19. cosh-neg-rev N/A

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

      20. remove-double-neg N/A

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

      21. lift-neg.f32 N/A

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

      22. distribute-frac-neg N/A

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

      23. lift-/.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-/l* N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f32 N/A

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

      9. lower-fma.f32 N/A

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

      10. lower-/.f32 98.1

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

   7. Applied rewrites 98.1%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

   5. Applied rewrites 87.5%

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

   7. Applied rewrites 92.3%

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

Alternative 7: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.30000001192092896:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \frac{1}{s}, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\frac{x}{s} + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 0.30000001192092896)
   (/ 1.0 (fma (- (* (/ 0.5 (* s s)) x) (/ 1.0 s)) x 2.0))
   (/ 1.0 (- 1.0 (/ -1.0 (+ (/ x s) 1.0))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 0.30000001192092896f) {
		tmp = 1.0f / fmaf((((0.5f / (s * s)) * x) - (1.0f / s)), x, 2.0f);
	} else {
		tmp = 1.0f / (1.0f - (-1.0f / ((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(0.30000001192092896))
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(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(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)))) < 0.300000012

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 82.2

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

   5. Applied rewrites 82.2%

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

   if 0.300000012 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. *-lft-identity N/A

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

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

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

      4. distribute-lft-neg-in N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f32 N/A

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

      7. lift-exp.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. distribute-frac-neg N/A

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

      11. exp-neg N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. cosh-neg N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-neg.f32 N/A

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

      18. lift-/.f32 N/A

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

      19. cosh-neg-rev N/A

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

      20. remove-double-neg N/A

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

      21. lift-neg.f32 N/A

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

      22. distribute-frac-neg N/A

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

      23. lift-/.f32 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 96.3

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

   7. Applied rewrites 96.3%

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

Alternative 8: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{0.5}{s}}{s} \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\frac{x}{s} + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 0.0)
   (/ 1.0 (* (* (/ (/ 0.5 s) s) x) x))
   (/ 1.0 (- 1.0 (/ -1.0 (+ (/ x s) 1.0))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 0.0f) {
		tmp = 1.0f / ((((0.5f / s) / s) * x) * x);
	} else {
		tmp = 1.0f / (1.0f - (-1.0f / ((x / s) + 1.0f)));
	}
	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) :: tmp
    if ((1.0e0 / (1.0e0 + exp((-x / s)))) <= 0.0e0) then
        tmp = 1.0e0 / ((((0.5e0 / s) / s) * x) * x)
    else
        tmp = 1.0e0 / (1.0e0 - ((-1.0e0) / ((x / s) + 1.0e0)))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

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

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 84.1%

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

   if 0.0 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. *-lft-identity N/A

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

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

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

      4. distribute-lft-neg-in N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f32 N/A

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

      7. lift-exp.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. distribute-frac-neg N/A

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

      11. exp-neg N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. cosh-neg N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-neg.f32 N/A

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

      18. lift-/.f32 N/A

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

      19. cosh-neg-rev N/A

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

      20. remove-double-neg N/A

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

      21. lift-neg.f32 N/A

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

      22. distribute-frac-neg N/A

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

      23. lift-/.f32 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 94.9

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

   7. Applied rewrites 94.9%

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

Alternative 9: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\frac{x}{s} + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 0.0)
   (/ 1.0 (* (* (/ 0.5 (* s s)) x) x))
   (/ 1.0 (- 1.0 (/ -1.0 (+ (/ x s) 1.0))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 0.0f) {
		tmp = 1.0f / (((0.5f / (s * s)) * x) * x);
	} else {
		tmp = 1.0f / (1.0f - (-1.0f / ((x / s) + 1.0f)));
	}
	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) :: tmp
    if ((1.0e0 / (1.0e0 + exp((-x / s)))) <= 0.0e0) then
        tmp = 1.0e0 / (((0.5e0 / (s * s)) * x) * x)
    else
        tmp = 1.0e0 / (1.0e0 - ((-1.0e0) / ((x / s) + 1.0e0)))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

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

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 84.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 84.1%

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

   if 0.0 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. *-lft-identity N/A

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

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

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

      4. distribute-lft-neg-in N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f32 N/A

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

      7. lift-exp.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. distribute-frac-neg N/A

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

      11. exp-neg N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. cosh-neg N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-neg.f32 N/A

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

      18. lift-/.f32 N/A

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

      19. cosh-neg-rev N/A

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

      20. remove-double-neg N/A

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

      21. lift-neg.f32 N/A

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

      22. distribute-frac-neg N/A

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

      23. lift-/.f32 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 94.9

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

   7. Applied rewrites 94.9%

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

Alternative 10: 63.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 2000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 2000000.0)
   0.5
   (/ 1.0 (* (* (/ 0.5 (* s s)) x) x))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 2000000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (((0.5f / (s * s)) * x) * x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((1.0e0 + exp((-x / s))) <= 2000000.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (((0.5e0 / (s * s)) * x) * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(2000000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(2000000.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (((single(0.5) / (s * s)) * x) * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 2e6

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 54.3%

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

   if 2e6 < (+.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. Add Preprocessing

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

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

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 84.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 84.1%

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

Alternative 11: 49.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{-1}{s}, x, 2\right)}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.5f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf((-1.0f / s), x, 2.0f);
	}
	return tmp;
}

Julia

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.5%

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

Alternative 12: 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 99.9%

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

Alternative 13: 93.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.00000023350551 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right)}{s} \cdot x - 1}{s}, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, \frac{x}{s}, 1\right)}{s}, x, 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.00000023350551e-35)
   (/
    1.0
    (fma
     (/ (- (* (/ (fma -0.16666666666666666 (/ x s) 0.5) s) x) 1.0) s)
     x
     2.0))
   (/ 1.0 (- 1.0 (/ -1.0 (fma (/ (fma 0.5 (/ x s) 1.0) s) x 1.0))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.00000023350551e-35f) {
		tmp = 1.0f / fmaf(((((fmaf(-0.16666666666666666f, (x / s), 0.5f) / s) * x) - 1.0f) / s), x, 2.0f);
	} else {
		tmp = 1.0f / (1.0f - (-1.0f / fmaf((fmaf(0.5f, (x / s), 1.0f) / s), x, 1.0f)));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.00000023350551e-35))
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(Float32(Float32(fma(Float32(-0.16666666666666666), Float32(x / s), Float32(0.5)) / s) * x) - Float32(1.0)) / s), x, Float32(2.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(-1.0) / fma(Float32(fma(Float32(0.5), Float32(x / s), Float32(1.0)) / s), x, Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000023e-35

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

   5. Applied rewrites 90.1%

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

   7. Applied rewrites 90.4%

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

   if -5.00000023e-35 < x

   1. Initial program 100.0%

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

      1. lift-+.f32 N/A

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

      2. *-lft-identity N/A

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

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

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

      4. distribute-lft-neg-in N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f32 N/A

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

      7. lift-exp.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. distribute-frac-neg N/A

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

      11. exp-neg N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. cosh-neg N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-neg.f32 N/A

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

      18. lift-/.f32 N/A

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

      19. cosh-neg-rev N/A

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

      20. remove-double-neg N/A

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

      21. lift-neg.f32 N/A

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

      22. distribute-frac-neg N/A

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

      23. lift-/.f32 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-/l* N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f32 N/A

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

      9. lower-fma.f32 N/A

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

      10. lower-/.f32 98.6

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

   7. Applied rewrites 98.6%

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

Alternative 14: 49.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -5.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-5.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 66.5

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

   5. Applied rewrites 66.5%

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

Alternative 15: 35.4% 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 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 37.6%

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

Reproduce

herbie shell --seed 2024352 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))