Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 3.5s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.9%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. inv-pow N/A

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

   7. pow-to-exp N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-log1p.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-neg.f32 N/A

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

   13. lift-exp.f32 99.9

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

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. lift-log1p.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. log-pow-rev N/A

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

   8. inv-pow N/A

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

   9. neg-log N/A

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

   10. lower-neg.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-neg.f32 N/A

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

   13. lift-exp.f32 N/A

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

   14. lift-log1p.f32 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 65.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{\frac{-x}{s}}\\ \mathbf{if}\;t\_0 \leq 1.2000000476837158:\\ \;\;\;\;\frac{0.5 \cdot s}{s}\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{x}{s \cdot s} \cdot 0.5 - \frac{1}{s}\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (exp (/ (- x) s)))))
   (if (<= t_0 1.2000000476837158)
     (/ (* 0.5 s) s)
     (if (<= t_0 5.0)
       (fma (/ x s) 0.25 0.5)
       (/ 1.0 (* (- (* (/ x (* s s)) 0.5) (/ 1.0 s)) x))))))

C

float code(float x, float s) {
	float t_0 = 1.0f + expf((-x / s));
	float tmp;
	if (t_0 <= 1.2000000476837158f) {
		tmp = (0.5f * s) / s;
	} else if (t_0 <= 5.0f) {
		tmp = fmaf((x / s), 0.25f, 0.5f);
	} else {
		tmp = 1.0f / ((((x / (s * s)) * 0.5f) - (1.0f / s)) * x);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) + exp(Float32(Float32(-x) / s)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.2000000476837158))
		tmp = Float32(Float32(Float32(0.5) * s) / s);
	elseif (t_0 <= Float32(5.0))
		tmp = fma(Float32(x / s), Float32(0.25), Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x / Float32(s * s)) * Float32(0.5)) - Float32(Float32(1.0) / s)) * x));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.20000005

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 9.0

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

   5. Applied rewrites 9.0%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 9.0

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

   8. Applied rewrites 9.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{2} \cdot s}{s} \]
   10. Step-by-step derivation

       1. lift-*.f32 28.1

          \[\leadsto \frac{0.5 \cdot s}{s} \]

   11. Applied rewrites 28.1%

       \[\leadsto \frac{0.5 \cdot s}{s} \]

   if 1.20000005 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 5

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 95.2

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

   5. Applied rewrites 95.2%

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

   if 5 < (+.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 75.3

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

   8. Applied rewrites 75.3%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower--.f32 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f32 N/A

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

       6. lower-/.f32 N/A

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

       7. pow2 N/A

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

       8. lift-*.f32 N/A

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

       9. lower-/.f32 82.6

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

   11. Applied rewrites 82.6%

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

Alternative 3: 49.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ t_1 := 1 + e^{t\_0}\\ \mathbf{if}\;t\_1 \leq 1.2000000476837158:\\ \;\;\;\;\frac{0.5 \cdot s}{s}\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)) (t_1 (+ 1.0 (exp t_0))))
   (if (<= t_1 1.2000000476837158)
     (/ (* 0.5 s) s)
     (if (<= t_1 5.0) (fma (/ x s) 0.25 0.5) (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float t_1 = 1.0f + expf(t_0);
	float tmp;
	if (t_1 <= 1.2000000476837158f) {
		tmp = (0.5f * s) / s;
	} else if (t_1 <= 5.0f) {
		tmp = fmaf((x / s), 0.25f, 0.5f);
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	t_1 = Float32(Float32(1.0) + exp(t_0))
	tmp = Float32(0.0)
	if (t_1 <= Float32(1.2000000476837158))
		tmp = Float32(Float32(Float32(0.5) * s) / s);
	elseif (t_1 <= Float32(5.0))
		tmp = fma(Float32(x / s), Float32(0.25), Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.20000005

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 9.0

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

   5. Applied rewrites 9.0%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 9.0

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

   8. Applied rewrites 9.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{2} \cdot s}{s} \]
   10. Step-by-step derivation

       1. lift-*.f32 28.1

          \[\leadsto \frac{0.5 \cdot s}{s} \]

   11. Applied rewrites 28.1%

       \[\leadsto \frac{0.5 \cdot s}{s} \]

   if 1.20000005 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 5

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 95.2

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

   5. Applied rewrites 95.2%

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

   if 5 < (+.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 x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 41.1

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

   5. Applied rewrites 41.1%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. lift-/.f32 N/A

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

      4. lift-neg.f32 41.1

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

   8. Applied rewrites 41.1%

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

Alternative 4: 63.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 2.0000500679016113:\\ \;\;\;\;\frac{0.5 \cdot s}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{x}{s \cdot s} \cdot 0.5 - \frac{1}{s}, x, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 2.0000500679016113)
   (/ (* 0.5 s) s)
   (/ 1.0 (fma (- (* (/ x (* s s)) 0.5) (/ 1.0 s)) x 2.0))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 2.0000500679016113f) {
		tmp = (0.5f * s) / s;
	} else {
		tmp = 1.0f / fmaf((((x / (s * s)) * 0.5f) - (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(2.0000500679016113))
		tmp = Float32(Float32(Float32(0.5) * s) / s);
	else
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(Float32(x / Float32(s * s)) * Float32(0.5)) - 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))) < 2.00005007

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 39.7

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

   5. Applied rewrites 39.7%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 39.6

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

   8. Applied rewrites 39.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{2} \cdot s}{s} \]
   10. Step-by-step derivation

       1. lift-*.f32 49.6

          \[\leadsto \frac{0.5 \cdot s}{s} \]

   11. Applied rewrites 49.6%

       \[\leadsto \frac{0.5 \cdot s}{s} \]

   if 2.00005007 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-/.f32 82.5

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

   5. Applied rewrites 82.5%

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

Alternative 5: 49.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if ((1.0f + expf(t_0)) <= 1.5f) {
		tmp = (0.5f * s) / s;
	} else {
		tmp = 1.0f / (t_0 + 2.0f);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if ((1.0e0 + exp(t_0)) <= 1.5e0) then
        tmp = (0.5e0 * s) / s
    else
        tmp = 1.0e0 / (t_0 + 2.0e0)
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 9.3

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

   5. Applied rewrites 9.3%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 9.3

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

   8. Applied rewrites 9.3%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{2} \cdot s}{s} \]
   10. Step-by-step derivation

       1. lift-*.f32 28.1

          \[\leadsto \frac{0.5 \cdot s}{s} \]

   11. Applied rewrites 28.1%

       \[\leadsto \frac{0.5 \cdot s}{s} \]

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 61.8

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

   5. Applied rewrites 61.8%

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

      1. lift-/.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-/.f32 N/A

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

      7. lift-neg.f32 61.8

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

   7. Applied rewrites 61.8%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

Alternative 7: 63.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\frac{0.5 \cdot s}{s}\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(x \cdot x, 0.5, s \cdot \left(-x\right)\right)}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -5.0)
     (/ (* 0.5 s) s)
     (if (<= t_0 500.0)
       (fma (/ x s) 0.25 0.5)
       (/ 1.0 (/ (fma (* x x) 0.5 (* s (- x))) (* s s)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = (0.5f * s) / s;
	} else if (t_0 <= 500.0f) {
		tmp = fmaf((x / s), 0.25f, 0.5f);
	} else {
		tmp = 1.0f / (fmaf((x * x), 0.5f, (s * -x)) / (s * s));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(Float32(Float32(0.5) * s) / s);
	elseif (t_0 <= Float32(500.0))
		tmp = fma(Float32(x / s), Float32(0.25), Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(fma(Float32(x * x), Float32(0.5), Float32(s * Float32(-x))) / Float32(s * s)));
	end
	return tmp
end

Derivation

1. Split input into 3 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 9.0

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

   5. Applied rewrites 9.0%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 9.0

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

   8. Applied rewrites 9.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{2} \cdot s}{s} \]
   10. Step-by-step derivation

       1. lift-*.f32 28.1

          \[\leadsto \frac{0.5 \cdot s}{s} \]

   11. Applied rewrites 28.1%

       \[\leadsto \frac{0.5 \cdot s}{s} \]

   if -5 < (/.f32 (neg.f32 x) s) < 500

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 91.2

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

   5. Applied rewrites 91.2%

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

   if 500 < (/.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 77.8

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

   8. Applied rewrites 77.8%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;\frac{0.5 \cdot s}{s}\\ \mathbf{elif}\;\frac{-x}{s} \leq 500:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(x \cdot x, 0.5, s \cdot \left(-x\right)\right)}{s \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\frac{0.5 \cdot s}{s}\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(0.5 \cdot x - s\right) \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -5.0)
     (/ (* 0.5 s) s)
     (if (<= t_0 500.0)
       (fma (/ x s) 0.25 0.5)
       (/ 1.0 (/ (* (- (* 0.5 x) s) x) (* s s)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = (0.5f * s) / s;
	} else if (t_0 <= 500.0f) {
		tmp = fmaf((x / s), 0.25f, 0.5f);
	} else {
		tmp = 1.0f / ((((0.5f * x) - s) * x) / (s * s));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(Float32(Float32(0.5) * s) / s);
	elseif (t_0 <= Float32(500.0))
		tmp = fma(Float32(x / s), Float32(0.25), Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(0.5) * x) - s) * x) / Float32(s * s)));
	end
	return tmp
end

Derivation

1. Split input into 3 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 9.0

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

   5. Applied rewrites 9.0%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 9.0

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

   8. Applied rewrites 9.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{2} \cdot s}{s} \]
   10. Step-by-step derivation

       1. lift-*.f32 28.1

          \[\leadsto \frac{0.5 \cdot s}{s} \]

   11. Applied rewrites 28.1%

       \[\leadsto \frac{0.5 \cdot s}{s} \]

   if -5 < (/.f32 (neg.f32 x) s) < 500

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 91.2

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

   5. Applied rewrites 91.2%

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

   if 500 < (/.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 77.8

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

   8. Applied rewrites 77.8%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower--.f32 N/A

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

       4. lower-*.f32 77.8

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

   11. Applied rewrites 77.8%

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

Alternative 9: 63.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\frac{0.5 \cdot s}{s}\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x \cdot x\right) \cdot 0.5}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -5.0)
     (/ (* 0.5 s) s)
     (if (<= t_0 500.0)
       (fma (/ x s) 0.25 0.5)
       (/ 1.0 (/ (* (* x x) 0.5) (* s s)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = (0.5f * s) / s;
	} else if (t_0 <= 500.0f) {
		tmp = fmaf((x / s), 0.25f, 0.5f);
	} else {
		tmp = 1.0f / (((x * x) * 0.5f) / (s * s));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(Float32(Float32(0.5) * s) / s);
	elseif (t_0 <= Float32(500.0))
		tmp = fma(Float32(x / s), Float32(0.25), Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(x * x) * Float32(0.5)) / Float32(s * s)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 9.0

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

   5. Applied rewrites 9.0%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 9.0

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

   8. Applied rewrites 9.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{2} \cdot s}{s} \]
   10. Step-by-step derivation

       1. lift-*.f32 28.1

          \[\leadsto \frac{0.5 \cdot s}{s} \]

   11. Applied rewrites 28.1%

       \[\leadsto \frac{0.5 \cdot s}{s} \]

   if -5 < (/.f32 (neg.f32 x) s) < 500

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 91.2

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

   5. Applied rewrites 91.2%

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

   if 500 < (/.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 77.8

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

   8. Applied rewrites 77.8%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. pow2 N/A

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

       4. lift-*.f32 77.8

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

   11. Applied rewrites 77.8%

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

Alternative 10: 54.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\frac{0.5 \cdot s}{s}\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{s \cdot \left(-x\right)}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -5.0)
     (/ (* 0.5 s) s)
     (if (<= t_0 500.0)
       (fma (/ x s) 0.25 0.5)
       (/ 1.0 (/ (* s (- x)) (* s s)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = (0.5f * s) / s;
	} else if (t_0 <= 500.0f) {
		tmp = fmaf((x / s), 0.25f, 0.5f);
	} else {
		tmp = 1.0f / ((s * -x) / (s * s));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(Float32(Float32(0.5) * s) / s);
	elseif (t_0 <= Float32(500.0))
		tmp = fma(Float32(x / s), Float32(0.25), Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(s * Float32(-x)) / Float32(s * s)));
	end
	return tmp
end

Derivation

1. Split input into 3 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 9.0

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

   5. Applied rewrites 9.0%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 9.0

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

   8. Applied rewrites 9.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{2} \cdot s}{s} \]
   10. Step-by-step derivation

       1. lift-*.f32 28.1

          \[\leadsto \frac{0.5 \cdot s}{s} \]

   11. Applied rewrites 28.1%

       \[\leadsto \frac{0.5 \cdot s}{s} \]

   if -5 < (/.f32 (neg.f32 x) s) < 500

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 91.2

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

   5. Applied rewrites 91.2%

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

   if 500 < (/.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 77.8

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

   8. Applied rewrites 77.8%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-rgt-neg-in N/A

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

       3. lower-*.f32 N/A

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

       4. lift-neg.f32 53.3

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

   11. Applied rewrites 53.3%

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

Alternative 11: 62.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.0000000195414814 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.5 \cdot s}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(-\frac{\mathsf{fma}\left(x \cdot x, -0.5, s \cdot x\right)}{s \cdot s}\right) + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.0000000195414814e-24)
   (/ (* 0.5 s) s)
   (/ 1.0 (+ (- (/ (fma (* x x) -0.5 (* s x)) (* s s))) 2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.0000000195414814e-24f) {
		tmp = (0.5f * s) / s;
	} else {
		tmp = 1.0f / (-(fmaf((x * x), -0.5f, (s * x)) / (s * s)) + 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.0000000195414814e-24))
		tmp = Float32(Float32(Float32(0.5) * s) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(-Float32(fma(Float32(x * x), Float32(-0.5), Float32(s * x)) / Float32(s * s))) + Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1.00000002e-24

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-/.f32 36.6

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

   5. Applied rewrites 36.6%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 36.5

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

   8. Applied rewrites 36.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{2} \cdot s}{s} \]
   10. Step-by-step derivation

       1. lift-*.f32 46.2

          \[\leadsto \frac{0.5 \cdot s}{s} \]

   11. Applied rewrites 46.2%

       \[\leadsto \frac{0.5 \cdot s}{s} \]

   if 1.00000002e-24 < (neg.f32 x)

   1. Initial program 99.8%

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

   3. Taylor expanded in s around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 75.6

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

   5. Applied rewrites 75.6%

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

      1. lift-fma.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. pow2 N/A

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

      6. lower-fma.f32 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-+.f32 N/A

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

   7. Applied rewrites 75.7%

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

   8. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 81.6

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

   10. Applied rewrites 81.6%

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

Alternative 12: 34.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot s}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (* 0.5 s) s))

C

float code(float x, float s) {
	return (0.5f * s) / 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 = (0.5e0 * s) / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{1}{4}}, \frac{1}{2}\right) \]

   4. lower-/.f32 29.1

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

5. Applied rewrites 29.1%

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

6. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 29.0

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

8. Applied rewrites 29.0%

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

9. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{1}{2} \cdot s}{s} \]
10. Step-by-step derivation

    1. lift-*.f32 34.2

       \[\leadsto \frac{0.5 \cdot s}{s} \]

11. Applied rewrites 34.2%

    \[\leadsto \frac{0.5 \cdot s}{s} \]
12. Add Preprocessing

Alternative 13: 34.8% 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 34.2%

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

Reproduce

herbie shell --seed 2025051 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))