Logistic distribution

Percentage Accurate: 99.5% → 99.4%

Time: 7.9s

Alternatives: 13

Speedup: 1.1×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

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) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

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) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-x\_m}{s}}\\ \frac{t\_0}{t\_0 + 1} \cdot \frac{1}{\mathsf{fma}\left(t\_0, s, s\right)} \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x_m) s))))
   (* (/ t_0 (+ t_0 1.0)) (/ 1.0 (fma t_0 s s)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((-x_m / s));
	return (t_0 / (t_0 + 1.0f)) * (1.0f / fmaf(t_0, s, s));
}

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(Float32(-x_m) / s))
	return Float32(Float32(t_0 / Float32(t_0 + Float32(1.0))) * Float32(Float32(1.0) / fma(t_0, s, s)))
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 88.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}{s}} \]

6. Applied rewrites 65.3%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{-x\_m}{s}\\ \frac{\frac{1}{e^{-\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{t\_0}\right), t\_0\right)}}}{s} \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (/ (- x_m) s)))
   (/ (/ 1.0 (exp (- (fma -2.0 (log1p (exp t_0)) t_0)))) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = -x_m / s;
	return (1.0f / expf(-fmaf(-2.0f, log1pf(expf(t_0)), t_0))) / s;
}

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = Float32(Float32(-x_m) / s)
	return Float32(Float32(Float32(1.0) / exp(Float32(-fma(Float32(-2.0), log1p(exp(t_0)), t_0)))) / s)
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 88.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}{s}} \]
5. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \frac{\color{blue}{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}}{s} \]

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

      \[\leadsto \frac{\color{blue}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) + \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}{s} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) \cdot \cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) \cdot \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}}{s} \]

   4. sinh-cosh N/A

      \[\leadsto \frac{\frac{\color{blue}{1}}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}{s} \]

   5. sinh---cosh-rev N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)\right)}}}}{s} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)\right)}}}}{s} \]

6. Applied rewrites 88.7%

   \[\leadsto \frac{\color{blue}{\frac{1}{e^{-\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{\frac{-x}{s}}\right), \frac{-x}{s}\right)}}}}{s} \]
7. Add Preprocessing

Alternative 3: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{1}{s}}{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{-x\_m}{s}}\right), 2, \frac{x\_m}{s}\right)}} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (/ 1.0 s) (exp (fma (log1p (exp (/ (- x_m) s))) 2.0 (/ x_m s)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (1.0f / s) / expf(fmaf(log1pf(expf((-x_m / s))), 2.0f, (x_m / s)));
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(1.0) / s) / exp(fma(log1p(exp(Float32(Float32(-x_m) / s))), Float32(2.0), Float32(x_m / s))))
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 88.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}{s}} \]
5. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \frac{\color{blue}{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}}{s} \]

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

      \[\leadsto \frac{\color{blue}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) + \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}{s} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) \cdot \cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) \cdot \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}}{s} \]

   4. sinh-cosh N/A

      \[\leadsto \frac{\frac{\color{blue}{1}}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}{s} \]

   5. sinh---cosh-rev N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)\right)}}}}{s} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)\right)}}}}{s} \]

6. Applied rewrites 88.7%

   \[\leadsto \frac{\color{blue}{\frac{1}{e^{-\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{\frac{-x}{s}}\right), \frac{-x}{s}\right)}}}}{s} \]

7. Taylor expanded in x around inf

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

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

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. lower-exp.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f32 N/A

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

9. Applied rewrites 88.7%

   \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{-x}{s}}\right), 2, \frac{x}{s}\right)}}} \]
10. Add Preprocessing

Alternative 4: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{-x\_m}{s}\\ \frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{t\_0}\right), -2, t\_0\right)}}{s} \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (/ (- x_m) s))) (/ (exp (fma (log1p (exp t_0)) -2.0 t_0)) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = -x_m / s;
	return expf(fmaf(log1pf(expf(t_0)), -2.0f, t_0)) / s;
}

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = Float32(Float32(-x_m) / s)
	return Float32(exp(fma(log1p(exp(t_0)), Float32(-2.0), t_0)) / s)
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 88.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}{s}} \]
5. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}}{s} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{e^{\frac{-x}{s} - \color{blue}{\mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}}{s} \]

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

      \[\leadsto \frac{e^{\color{blue}{\frac{-x}{s} + \left(\mathsf{neg}\left(\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)\right)\right) \cdot 2}}}{s} \]

   4. +-commutative N/A

      \[\leadsto \frac{e^{\color{blue}{\left(\mathsf{neg}\left(\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)\right)\right) \cdot 2 + \frac{-x}{s}}}}{s} \]

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

      \[\leadsto \frac{e^{\color{blue}{\left(\mathsf{neg}\left(\mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)\right)} + \frac{-x}{s}}}{s} \]

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

      \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot \left(\mathsf{neg}\left(2\right)\right)} + \frac{-x}{s}}}{s} \]

   7. lower-fma.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{-x}{s}}\right), \mathsf{neg}\left(2\right), \frac{-x}{s}\right)}}}{s} \]

   8. metadata-eval 88.7

      \[\leadsto \frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{-x}{s}}\right), \color{blue}{-2}, \frac{-x}{s}\right)}}{s} \]

6. Applied rewrites 88.7%

   \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(e^{\frac{-x}{s}}\right), -2, \frac{-x}{s}\right)}}}{s} \]
7. Add Preprocessing

Alternative 5: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{\frac{-x\_m}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{0.125}{s}, \frac{x\_m}{s}, \frac{-0.5}{s}\right), x\_m, \log 2\right) \cdot 2}}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/
  (exp
   (-
    (/ (- x_m) s)
    (* (fma (fma (/ 0.125 s) (/ x_m s) (/ -0.5 s)) x_m (log 2.0)) 2.0)))
  s))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf(((-x_m / s) - (fmaf(fmaf((0.125f / s), (x_m / s), (-0.5f / s)), x_m, logf(2.0f)) * 2.0f))) / s;
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(exp(Float32(Float32(Float32(-x_m) / s) - Float32(fma(fma(Float32(Float32(0.125) / s), Float32(x_m / s), Float32(Float32(-0.5) / s)), x_m, log(Float32(2.0))) * Float32(2.0)))) / s)
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 88.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}{s}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{e^{\frac{-x}{s} - \color{blue}{\left(\log 2 + x \cdot \left(\frac{1}{8} \cdot \frac{x}{{s}^{2}} - \frac{1}{2} \cdot \frac{1}{s}\right)\right)} \cdot 2}}{s} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

      \[\leadsto \frac{e^{\frac{-x}{s} - \color{blue}{\mathsf{fma}\left(\frac{1}{8} \cdot \frac{x}{{s}^{2}} - \frac{1}{2} \cdot \frac{1}{s}, x, \log 2\right)} \cdot 2}}{s} \]

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

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. lower-fma.f32 N/A

      \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{8}}{s}, \frac{x}{s}, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{s}\right)}, x, \log 2\right) \cdot 2}}{s} \]

   9. lower-/.f32 N/A

      \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{8}}{s}}, \frac{x}{s}, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{s}\right), x, \log 2\right) \cdot 2}}{s} \]

   10. lower-/.f32 N/A

       \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{8}}{s}, \color{blue}{\frac{x}{s}}, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{s}\right), x, \log 2\right) \cdot 2}}{s} \]

   11. metadata-eval N/A

       \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{8}}{s}, \frac{x}{s}, \color{blue}{\frac{-1}{2}} \cdot \frac{1}{s}\right), x, \log 2\right) \cdot 2}}{s} \]

   12. associate-*r/ N/A

       \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{8}}{s}, \frac{x}{s}, \color{blue}{\frac{\frac{-1}{2} \cdot 1}{s}}\right), x, \log 2\right) \cdot 2}}{s} \]

   13. metadata-eval N/A

       \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{8}}{s}, \frac{x}{s}, \frac{\color{blue}{\frac{-1}{2}}}{s}\right), x, \log 2\right) \cdot 2}}{s} \]

   14. lower-/.f32 N/A

       \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{8}}{s}, \frac{x}{s}, \color{blue}{\frac{\frac{-1}{2}}{s}}\right), x, \log 2\right) \cdot 2}}{s} \]

   15. lower-log.f32 86.3

       \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{0.125}{s}, \frac{x}{s}, \frac{-0.5}{s}\right), x, \color{blue}{\log 2}\right) \cdot 2}}{s} \]

7. Applied rewrites 86.3%

   \[\leadsto \frac{e^{\frac{-x}{s} - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.125}{s}, \frac{x}{s}, \frac{-0.5}{s}\right), x, \log 2\right)} \cdot 2}}{s} \]
8. Add Preprocessing

Alternative 6: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{\frac{-x\_m}{s} - \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.125}{s}, x\_m, -0.5\right)}{s}, x\_m, \log 2\right) \cdot 2}}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/
  (exp
   (-
    (/ (- x_m) s)
    (* (fma (/ (fma (/ 0.125 s) x_m -0.5) s) x_m (log 2.0)) 2.0)))
  s))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf(((-x_m / s) - (fmaf((fmaf((0.125f / s), x_m, -0.5f) / s), x_m, logf(2.0f)) * 2.0f))) / s;
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(exp(Float32(Float32(Float32(-x_m) / s) - Float32(fma(Float32(fma(Float32(Float32(0.125) / s), x_m, Float32(-0.5)) / s), x_m, log(Float32(2.0))) * Float32(2.0)))) / s)
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 88.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}{s}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{e^{\frac{-x}{s} - \color{blue}{\left(\log 2 + x \cdot \left(\frac{1}{8} \cdot \frac{x}{{s}^{2}} - \frac{1}{2} \cdot \frac{1}{s}\right)\right)} \cdot 2}}{s} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

      \[\leadsto \frac{e^{\frac{-x}{s} - \color{blue}{\mathsf{fma}\left(\frac{1}{8} \cdot \frac{x}{{s}^{2}} - \frac{1}{2} \cdot \frac{1}{s}, x, \log 2\right)} \cdot 2}}{s} \]

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

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. lower-fma.f32 N/A

      \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{8}}{s}, \frac{x}{s}, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{s}\right)}, x, \log 2\right) \cdot 2}}{s} \]

   9. lower-/.f32 N/A

      \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{8}}{s}}, \frac{x}{s}, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{s}\right), x, \log 2\right) \cdot 2}}{s} \]

   10. lower-/.f32 N/A

       \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{8}}{s}, \color{blue}{\frac{x}{s}}, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{s}\right), x, \log 2\right) \cdot 2}}{s} \]

   11. metadata-eval N/A

       \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{8}}{s}, \frac{x}{s}, \color{blue}{\frac{-1}{2}} \cdot \frac{1}{s}\right), x, \log 2\right) \cdot 2}}{s} \]

   12. associate-*r/ N/A

       \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{8}}{s}, \frac{x}{s}, \color{blue}{\frac{\frac{-1}{2} \cdot 1}{s}}\right), x, \log 2\right) \cdot 2}}{s} \]

   13. metadata-eval N/A

       \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{8}}{s}, \frac{x}{s}, \frac{\color{blue}{\frac{-1}{2}}}{s}\right), x, \log 2\right) \cdot 2}}{s} \]

   14. lower-/.f32 N/A

       \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{8}}{s}, \frac{x}{s}, \color{blue}{\frac{\frac{-1}{2}}{s}}\right), x, \log 2\right) \cdot 2}}{s} \]

   15. lower-log.f32 86.3

       \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{0.125}{s}, \frac{x}{s}, \frac{-0.5}{s}\right), x, \color{blue}{\log 2}\right) \cdot 2}}{s} \]

7. Applied rewrites 86.3%

   \[\leadsto \frac{e^{\frac{-x}{s} - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.125}{s}, \frac{x}{s}, \frac{-0.5}{s}\right), x, \log 2\right)} \cdot 2}}{s} \]
8. Step-by-step derivation

9. Applied rewrites 86.3%

   \[\leadsto \frac{e^{\frac{-x}{s} - \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.125}{s}, x, -0.5\right)}{s}, x, \log 2\right) \cdot 2}}{s} \]
10. Add Preprocessing

Alternative 7: 97.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{\mathsf{fma}\left(\frac{-0.25}{s}, x\_m \cdot \frac{x\_m}{s}, \log 2 \cdot -2\right)}}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (exp (fma (/ -0.25 s) (* x_m (/ x_m s)) (* (log 2.0) -2.0))) s))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf(fmaf((-0.25f / s), (x_m * (x_m / s)), (logf(2.0f) * -2.0f))) / s;
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(exp(fma(Float32(Float32(-0.25) / s), Float32(x_m * Float32(x_m / s)), Float32(log(Float32(2.0)) * Float32(-2.0)))) / s)
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 88.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}{s}} \]

5. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. unpow2 N/A

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

   5. times-frac N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. exp-to-pow N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

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

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

   12. mul-1-neg N/A

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

   13. lower-fma.f32 N/A

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

7. Applied rewrites 97.2%

   \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(\frac{-0.25}{s}, x \cdot \frac{x}{s}, \log 2 \cdot -2\right)}}}{s} \]
8. Add Preprocessing

Alternative 8: 95.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{\frac{-\left|x\_m\right|}{s}}}{\left(\frac{\mathsf{fma}\left(x\_m \cdot \frac{x\_m}{s}, 3, -4 \cdot \left|x\_m\right|\right)}{s} + 4\right) \cdot s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/
  (exp (/ (- (fabs x_m)) s))
  (* (+ (/ (fma (* x_m (/ x_m s)) 3.0 (* -4.0 (fabs x_m))) s) 4.0) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf((-fabsf(x_m) / s)) / (((fmaf((x_m * (x_m / s)), 3.0f, (-4.0f * fabsf(x_m))) / s) + 4.0f) * s);
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(exp(Float32(Float32(-abs(x_m)) / s)) / Float32(Float32(Float32(fma(Float32(x_m * Float32(x_m / s)), Float32(3.0), Float32(Float32(-4.0) * abs(x_m))) / s) + Float32(4.0)) * s))
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

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

5. Applied rewrites 94.9%

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

6. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

8. Applied rewrites 95.5%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(\frac{\mathsf{fma}\left(x \cdot \frac{x}{s}, 3, -4 \cdot \left|x\right|\right)}{s} + 4\right) \cdot s}} \]
9. Add Preprocessing

Alternative 9: 94.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{\frac{-\left|x\_m\right|}{s}}}{4 \cdot s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ (exp (/ (- (fabs x_m)) s)) (* 4.0 s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf((-fabsf(x_m) / s)) / (4.0f * s);
}

Fortran

x_m =     private
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_m, s)
use fmin_fmax_functions
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = exp((-abs(x_m) / s)) / (4.0e0 * s)
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(exp(Float32(Float32(-abs(x_m)) / s)) / Float32(Float32(4.0) * s))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = exp((-abs(x_m) / s)) / (single(4.0) * s);
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{4 \cdot s}} \]
4. Step-by-step derivation

   1. lower-*.f32 94.6

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{4 \cdot s}} \]

5. Applied rewrites 94.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{4 \cdot s}} \]
6. Add Preprocessing

Alternative 10: 86.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;-\left|x\_m\right| \leq -3999999983616:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(\frac{0.25}{s}, \frac{4 \cdot \left(x\_m \cdot x\_m\right)}{s}, 4\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.25, \frac{x\_m}{s}, \frac{\frac{x\_m}{s} + 1}{4}\right)}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (- (fabs x_m)) -3999999983616.0)
   (/ (/ 1.0 (fma (/ 0.25 s) (/ (* 4.0 (* x_m x_m)) s) 4.0)) s)
   (/ (fma -0.25 (/ x_m s) (/ (+ (/ x_m s) 1.0) 4.0)) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (-fabsf(x_m) <= -3999999983616.0f) {
		tmp = (1.0f / fmaf((0.25f / s), ((4.0f * (x_m * x_m)) / s), 4.0f)) / s;
	} else {
		tmp = fmaf(-0.25f, (x_m / s), (((x_m / s) + 1.0f) / 4.0f)) / s;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (Float32(-abs(x_m)) <= Float32(-3999999983616.0))
		tmp = Float32(Float32(Float32(1.0) / fma(Float32(Float32(0.25) / s), Float32(Float32(Float32(4.0) * Float32(x_m * x_m)) / s), Float32(4.0))) / s);
	else
		tmp = Float32(fma(Float32(-0.25), Float32(x_m / s), Float32(Float32(Float32(x_m / s) + Float32(1.0)) / Float32(4.0))) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (fabs.f32 x)) < -3999999980000

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}{s}} \]
   5. Step-by-step derivation

      1. lift-exp.f32 N/A

         \[\leadsto \frac{\color{blue}{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}}{s} \]

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

         \[\leadsto \frac{\color{blue}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) + \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}{s} \]

      3. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) \cdot \cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) \cdot \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}}{s} \]

      4. sinh-cosh N/A

         \[\leadsto \frac{\frac{\color{blue}{1}}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}{s} \]

      5. sinh---cosh-rev N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)\right)}}}}{s} \]

      6. lower-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)\right)}}}}{s} \]

   6. Applied rewrites 71.1%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{-\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{\frac{-x}{s}}\right), \frac{-x}{s}\right)}}}}{s} \]

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. exp-to-pow N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. lower-fma.f32 N/A

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

   9. Applied rewrites 100.0%

      \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{0.25}{s}, \frac{4 \cdot \left(x \cdot x\right)}{s}, 4\right)}}}{s} \]

   if -3999999980000 < (neg.f32 (fabs.f32 x))

   1. Initial program 99.4%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}{s}} \]
   5. Step-by-step derivation

      1. lift-exp.f32 N/A

         \[\leadsto \frac{\color{blue}{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}}{s} \]

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

         \[\leadsto \frac{\color{blue}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) + \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}{s} \]

      3. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) \cdot \cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) \cdot \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}}{s} \]

      4. sinh-cosh N/A

         \[\leadsto \frac{\frac{\color{blue}{1}}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}{s} \]

      5. sinh---cosh-rev N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)\right)}}}}{s} \]

      6. lower-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)\right)}}}}{s} \]

   6. Applied rewrites 98.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{-\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{\frac{-x}{s}}\right), \frac{-x}{s}\right)}}}}{s} \]

   7. Taylor expanded in s around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. exp-to-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. associate-/r* N/A

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

   9. Applied rewrites 84.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.25, \frac{x}{s}, \frac{\frac{x}{s} + 1}{4}\right)}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 77.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;-\left|x\_m\right| \leq -2499999995126612000:\\ \;\;\;\;\frac{\frac{0.5}{s}}{\left(1 - \frac{x\_m}{s}\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.25, \frac{x\_m}{s}, \frac{\frac{x\_m}{s} + 1}{4}\right)}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (- (fabs x_m)) -2499999995126612000.0)
   (/ (/ 0.5 s) (+ (- 1.0 (/ x_m s)) 1.0))
   (/ (fma -0.25 (/ x_m s) (/ (+ (/ x_m s) 1.0) 4.0)) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (-fabsf(x_m) <= -2499999995126612000.0f) {
		tmp = (0.5f / s) / ((1.0f - (x_m / s)) + 1.0f);
	} else {
		tmp = fmaf(-0.25f, (x_m / s), (((x_m / s) + 1.0f) / 4.0f)) / s;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (Float32(-abs(x_m)) <= Float32(-2499999995126612000.0))
		tmp = Float32(Float32(Float32(0.5) / s) / Float32(Float32(Float32(1.0) - Float32(x_m / s)) + Float32(1.0)));
	else
		tmp = Float32(fma(Float32(-0.25), Float32(x_m / s), Float32(Float32(Float32(x_m / s) + Float32(1.0)) / Float32(4.0))) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (fabs.f32 x)) < -2.5e18

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fabs.f32 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in s around inf

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

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

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

      2. *-inverses N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. div-sub N/A

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

      6. lower-/.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-fabs.f32 1.5

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

   8. Applied rewrites 1.5%

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

   10. Applied rewrites 17.5%

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

   11. Taylor expanded in s around inf

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

       1. lower-/.f32 76.8

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

   13. Applied rewrites 76.8%

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

   if -2.5e18 < (neg.f32 (fabs.f32 x))

   1. Initial program 99.4%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\frac{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}{s}} \]
   5. Step-by-step derivation

      1. lift-exp.f32 N/A

         \[\leadsto \frac{\color{blue}{e^{\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2}}}{s} \]

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

         \[\leadsto \frac{\color{blue}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) + \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}{s} \]

      3. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) \cdot \cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) \cdot \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}}{s} \]

      4. sinh-cosh N/A

         \[\leadsto \frac{\frac{\color{blue}{1}}{\cosh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right) - \sinh \left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)}}{s} \]

      5. sinh---cosh-rev N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(\left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)\right)}}}}{s} \]

      6. lower-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\left(\frac{-x}{s} - \mathsf{log1p}\left(e^{\frac{-x}{s}}\right) \cdot 2\right)\right)}}}}{s} \]

   6. Applied rewrites 97.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{-\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{\frac{-x}{s}}\right), \frac{-x}{s}\right)}}}}{s} \]

   7. Taylor expanded in s around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. exp-to-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. associate-/r* N/A

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

   9. Applied rewrites 83.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.25, \frac{x}{s}, \frac{\frac{x}{s} + 1}{4}\right)}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 49.9% accurate, 9.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{0.5}{s}}{\left(1 - \frac{x\_m}{s}\right) + 1} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ (/ 0.5 s) (+ (- 1.0 (/ x_m s)) 1.0)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (0.5f / s) / ((1.0f - (x_m / s)) + 1.0f);
}

Fortran

x_m =     private
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_m, s)
use fmin_fmax_functions
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = (0.5e0 / s) / ((1.0e0 - (x_m / s)) + 1.0e0)
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(0.5) / s) / Float32(Float32(Float32(1.0) - Float32(x_m / s)) + Float32(1.0)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (single(0.5) / s) / ((single(1.0) - (x_m / s)) + single(1.0));
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fabs.f32 27.2

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

5. Applied rewrites 27.2%

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

6. Taylor expanded in s around inf

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

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

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

   2. *-inverses N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. div-sub N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-fabs.f32 27.3

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

8. Applied rewrites 27.3%

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

10. Applied rewrites 47.2%

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

11. Taylor expanded in s around inf

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

    1. lower-/.f32 52.0

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

13. Applied rewrites 52.0%

    \[\leadsto \frac{\color{blue}{\frac{0.5}{s}}}{\left(1 - \frac{x}{s}\right) + 1} \]
14. Add Preprocessing

Alternative 13: 27.4% accurate, 31.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{0.25}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 0.25 s))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 0.25f / s;
}

Fortran

x_m =     private
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_m, s)
use fmin_fmax_functions
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(0.25) / s)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.25) / s;
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

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

   1. lower-/.f32 28.7

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

5. Applied rewrites 28.7%

   \[\leadsto \color{blue}{\frac{0.25}{s}} \]
6. Add Preprocessing

Reproduce

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