Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 6.7s

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: 99.8% 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 2: 67.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   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. 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 5.3

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 28.1%

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

   if 0.100000001 < (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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. 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 83.1%

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

   7. Applied rewrites 92.9%

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

   9. Applied rewrites 92.9%

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

Alternative 3: 63.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   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. 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 5.3

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 28.1%

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

   if 0.100000001 < (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 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. metadata-eval N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. distribute-frac-neg N/A

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

      12. distribute-neg-in N/A

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

      13. div-add N/A

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

      14. mul-1-neg N/A

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

      15. +-commutative N/A

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 84.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}} \]
   9. 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

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

   10. Applied rewrites 87.5%

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

Alternative 4: 62.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right)} \]

      2. div0 N/A

         \[\leadsto \color{blue}{\frac{0}{\mathsf{neg}\left({s}^{2}\right)}} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      3. +-inverses N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot {x}^{2} - \frac{-1}{8} \cdot {x}^{2}}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot {x}^{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)} \cdot {x}^{2}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

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

         \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right)\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} + \frac{1}{2}\right) + \frac{1}{4} \cdot \frac{x}{s}} \]

      9. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right)} + \frac{1}{4} \cdot \frac{x}{s} \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot \frac{x}{s} \]

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

          \[\leadsto \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right) - \frac{-1}{4} \cdot \frac{x}{s}} \]

      12. associate--l+ N/A

          \[\leadsto \color{blue}{\frac{1}{2} + \left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} - \frac{-1}{4} \cdot \frac{x}{s}\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} - \frac{-1}{4} \cdot \frac{x}{s}\right) + \frac{1}{2}} \]

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 8.7%

      \[\leadsto \frac{0.25}{s} \cdot \color{blue}{x} \]

   9. Taylor expanded in s around 0

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

   11. Applied rewrites 45.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 52.9%

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

   if 2 < (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

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}} \]
   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 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 84.6%

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

Alternative 5: 49.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.20000000298023224:\\ \;\;\;\;\frac{1}{\frac{2}{x} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 0.20000000298023224)
   (/ 1.0 (* (/ 2.0 x) x))
   (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 0.20000000298023224f) {
		tmp = 1.0f / ((2.0f / x) * x);
	} 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 (exp((-x / s)) <= 0.20000000298023224e0) then
        tmp = 1.0e0 / ((2.0e0 / x) * x)
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(2.0) / x) * x));
	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 (exp((-x / s)) <= single(0.20000000298023224))
		tmp = single(1.0) / ((single(2.0) / x) * x);
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 0.200000003

   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. 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 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 28.1%

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

   if 0.200000003 < (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. 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 63.2

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

   5. Applied rewrites 63.2%

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

Alternative 6: 47.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;1 + e^{t\_0} \leq 5:\\ \;\;\;\;\frac{0.5 \cdot s}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if ((1.0f + expf(t_0)) <= 5.0f) {
		tmp = (0.5f * s) / s;
	} else {
		tmp = 1.0f / t_0;
	}
	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)) <= 5.0e0) then
        tmp = (0.5e0 * s) / s
    else
        tmp = 1.0e0 / t_0
    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(5.0))
		tmp = Float32(Float32(Float32(0.5) * s) / s);
	else
		tmp = Float32(Float32(1.0) / t_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(5.0))
		tmp = (single(0.5) * s) / s;
	else
		tmp = single(1.0) / t_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))) < 5

   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. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right)} \]

      2. div0 N/A

         \[\leadsto \color{blue}{\frac{0}{\mathsf{neg}\left({s}^{2}\right)}} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      3. +-inverses N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot {x}^{2} - \frac{-1}{8} \cdot {x}^{2}}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot {x}^{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)} \cdot {x}^{2}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

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

         \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right)\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} + \frac{1}{2}\right) + \frac{1}{4} \cdot \frac{x}{s}} \]

      9. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right)} + \frac{1}{4} \cdot \frac{x}{s} \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot \frac{x}{s} \]

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

          \[\leadsto \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right) - \frac{-1}{4} \cdot \frac{x}{s}} \]

      12. associate--l+ N/A

          \[\leadsto \color{blue}{\frac{1}{2} + \left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} - \frac{-1}{4} \cdot \frac{x}{s}\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} - \frac{-1}{4} \cdot \frac{x}{s}\right) + \frac{1}{2}} \]

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 8.7%

      \[\leadsto \frac{0.25}{s} \cdot \color{blue}{x} \]

   9. Taylor expanded in s around 0

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

   11. Applied rewrites 45.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 52.6%

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

   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. 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 39.4

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 39.4%

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

Alternative 7: 61.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 2000:\\ \;\;\;\;\frac{0.5 \cdot s}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(0.5 \cdot x - s\right)}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 2000.0)
   (/ (* 0.5 s) s)
   (/ 1.0 (/ (* x (- (* 0.5 x) s)) (* s s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 2000.0f) {
		tmp = (0.5f * s) / s;
	} else {
		tmp = 1.0f / ((x * ((0.5f * x) - s)) / (s * s));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 2000.0e0) then
        tmp = (0.5e0 * s) / s
    else
        tmp = 1.0e0 / ((x * ((0.5e0 * x) - s)) / (s * s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(2000.0))
		tmp = Float32(Float32(Float32(0.5) * s) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(Float32(0.5) * x) - s)) / Float32(s * s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(2000.0))
		tmp = (single(0.5) * s) / s;
	else
		tmp = single(1.0) / ((x * ((single(0.5) * x) - s)) / (s * s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 2e3

   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. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right)} \]

      2. div0 N/A

         \[\leadsto \color{blue}{\frac{0}{\mathsf{neg}\left({s}^{2}\right)}} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      3. +-inverses N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot {x}^{2} - \frac{-1}{8} \cdot {x}^{2}}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot {x}^{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)} \cdot {x}^{2}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

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

         \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right)\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} + \frac{1}{2}\right) + \frac{1}{4} \cdot \frac{x}{s}} \]

      9. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right)} + \frac{1}{4} \cdot \frac{x}{s} \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot \frac{x}{s} \]

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

          \[\leadsto \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right) - \frac{-1}{4} \cdot \frac{x}{s}} \]

      12. associate--l+ N/A

          \[\leadsto \color{blue}{\frac{1}{2} + \left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} - \frac{-1}{4} \cdot \frac{x}{s}\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} - \frac{-1}{4} \cdot \frac{x}{s}\right) + \frac{1}{2}} \]

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 8.6%

      \[\leadsto \frac{0.25}{s} \cdot \color{blue}{x} \]

   9. Taylor expanded in s around 0

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

   11. Applied rewrites 44.3%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 51.8%

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

   if 2e3 < (/.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. metadata-eval N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. distribute-frac-neg N/A

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

      12. distribute-neg-in N/A

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

      13. div-add N/A

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

      14. mul-1-neg N/A

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

      15. +-commutative N/A

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

   5. Applied rewrites 77.7%

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

   7. Applied rewrites 77.7%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 84.7%

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

Alternative 8: 51.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;\frac{0.5 \cdot s}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot s - x}{s \cdot x} \cdot x}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.10000000149011612f) {
		tmp = (0.5f * s) / s;
	} else {
		tmp = 1.0f / ((((2.0f * s) - x) / (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 ((-x / s) <= 0.10000000149011612e0) then
        tmp = (0.5e0 * s) / s
    else
        tmp = 1.0e0 / ((((2.0e0 * s) - x) / (s * x)) * x)
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(0.10000000149011612))
		tmp = (single(0.5) * s) / s;
	else
		tmp = single(1.0) / ((((single(2.0) * s) - x) / (s * x)) * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. 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. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right)} \]

      2. div0 N/A

         \[\leadsto \color{blue}{\frac{0}{\mathsf{neg}\left({s}^{2}\right)}} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      3. +-inverses N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot {x}^{2} - \frac{-1}{8} \cdot {x}^{2}}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot {x}^{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)} \cdot {x}^{2}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

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

         \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right)\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} + \frac{1}{2}\right) + \frac{1}{4} \cdot \frac{x}{s}} \]

      9. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right)} + \frac{1}{4} \cdot \frac{x}{s} \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot \frac{x}{s} \]

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

          \[\leadsto \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right) - \frac{-1}{4} \cdot \frac{x}{s}} \]

      12. associate--l+ N/A

          \[\leadsto \color{blue}{\frac{1}{2} + \left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} - \frac{-1}{4} \cdot \frac{x}{s}\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} - \frac{-1}{4} \cdot \frac{x}{s}\right) + \frac{1}{2}} \]

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 8.7%

      \[\leadsto \frac{0.25}{s} \cdot \color{blue}{x} \]

   9. Taylor expanded in s around 0

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

   11. Applied rewrites 45.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 52.9%

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

   if 0.100000001 < (/.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. 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 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 39.3%

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

   10. Applied rewrites 47.3%

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

Alternative 9: 49.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -1

   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. 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 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 28.1%

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

   if -1 < (/.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 63.2

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

   5. Applied rewrites 63.2%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 63.2%

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

Alternative 10: 49.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -1.0f) {
		tmp = 1.0f / ((2.0f / x) * x);
	} 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(-x) / s) <= Float32(-1.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(2.0) / x) * x));
	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 (neg.f32 x) s) < -1

   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. 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 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 28.1%

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

   if -1 < (/.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

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

      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 80.3

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

   5. Applied rewrites 80.3%

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

   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 63.2%

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

Alternative 11: 49.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -1

   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. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right)} \]

      2. div0 N/A

         \[\leadsto \color{blue}{\frac{0}{\mathsf{neg}\left({s}^{2}\right)}} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      3. +-inverses N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot {x}^{2} - \frac{-1}{8} \cdot {x}^{2}}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot {x}^{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)} \cdot {x}^{2}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

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

         \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right)\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} + \frac{1}{2}\right) + \frac{1}{4} \cdot \frac{x}{s}} \]

      9. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right)} + \frac{1}{4} \cdot \frac{x}{s} \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot \frac{x}{s} \]

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

          \[\leadsto \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right) - \frac{-1}{4} \cdot \frac{x}{s}} \]

      12. associate--l+ N/A

          \[\leadsto \color{blue}{\frac{1}{2} + \left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} - \frac{-1}{4} \cdot \frac{x}{s}\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} - \frac{-1}{4} \cdot \frac{x}{s}\right) + \frac{1}{2}} \]

   5. Applied rewrites 9.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 9.4%

      \[\leadsto \frac{0.25}{s} \cdot \color{blue}{x} \]

   9. Taylor expanded in s around 0

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

   11. Applied rewrites 9.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 28.1%

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

   if -1 < (/.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 63.2

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

   5. Applied rewrites 63.2%

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

Alternative 12: 35.1% 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. +-lft-identity N/A

      \[\leadsto \color{blue}{0 + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right)} \]

   2. div0 N/A

      \[\leadsto \color{blue}{\frac{0}{\mathsf{neg}\left({s}^{2}\right)}} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

   3. +-inverses N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot {x}^{2} - \frac{-1}{8} \cdot {x}^{2}}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot {x}^{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)} \cdot {x}^{2}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

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

      \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}}{\mathsf{neg}\left({s}^{2}\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right)\right)} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

   7. mul-1-neg N/A

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}} + \left(\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}\right) \]

   8. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} + \frac{1}{2}\right) + \frac{1}{4} \cdot \frac{x}{s}} \]

   9. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right)} + \frac{1}{4} \cdot \frac{x}{s} \]

   10. metadata-eval N/A

       \[\leadsto \left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot \frac{x}{s} \]

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

       \[\leadsto \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}\right) - \frac{-1}{4} \cdot \frac{x}{s}} \]

   12. associate--l+ N/A

       \[\leadsto \color{blue}{\frac{1}{2} + \left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} - \frac{-1}{4} \cdot \frac{x}{s}\right)} \]

   13. +-commutative N/A

       \[\leadsto \color{blue}{\left(-1 \cdot \frac{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}} - \frac{-1}{4} \cdot \frac{x}{s}\right) + \frac{1}{2}} \]

5. Applied rewrites 30.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 7.0%

   \[\leadsto \frac{0.25}{s} \cdot \color{blue}{x} \]

9. Taylor expanded in s around 0

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

11. Applied rewrites 30.7%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 36.5%

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

Alternative 13: 35.1% 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 36.5%

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

Reproduce

herbie shell --seed 2024360 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))