Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 4.8s

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-fabs.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. exp-neg N/A

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

   7. lower-/.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-fabs.f32 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s}\\ t_2 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_2\right) \cdot t\_2} \leq 0:\\ \;\;\;\;\frac{t\_0}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.0625 - t\_1 \cdot t\_1}{\mathsf{fma}\left(\frac{x}{s} \cdot \frac{x}{s}, 0.0625, 0.25\right)}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s)))
        (t_1 (/ (/ (fma (* (* x x) 3.0) -0.0625 (* (* x x) 0.125)) s) s))
        (t_2 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_2) t_2)) 0.0)
     (/ t_0 (* 4.0 s))
     (/ (/ (- 0.0625 (* t_1 t_1)) (fma (* (/ x s) (/ x s)) 0.0625 0.25)) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = (fmaf(((x * x) * 3.0f), -0.0625f, ((x * x) * 0.125f)) / s) / s;
	float t_2 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_2) * t_2)) <= 0.0f) {
		tmp = t_0 / (4.0f * s);
	} else {
		tmp = ((0.0625f - (t_1 * t_1)) / fmaf(((x / s) * (x / s)), 0.0625f, 0.25f)) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(fma(Float32(Float32(x * x) * Float32(3.0)), Float32(-0.0625), Float32(Float32(x * x) * Float32(0.125))) / s) / s)
	t_2 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(s * t_2) * t_2)) <= Float32(0.0))
		tmp = Float32(t_0 / Float32(Float32(4.0) * s));
	else
		tmp = Float32(Float32(Float32(Float32(0.0625) - Float32(t_1 * t_1)) / fma(Float32(Float32(x / s) * Float32(x / s)), Float32(0.0625), Float32(0.25))) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0

   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

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

      1. lower-*.f32 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]

   8. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\mathsf{fma}\left(0.125, x \cdot x, -0.0625 \cdot \left(\left(x \cdot x\right) \cdot 3\right)\right)}{s \cdot s}}{s}} \]

   10. Applied rewrites 91.5%

       \[\leadsto \frac{\frac{0.0625 - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s} \cdot \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s}}{0.25 - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s}}}{s} \]
   11. Step-by-step derivation

       1. lift--.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{16} - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s} \cdot \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s}}{\frac{1}{4} - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s}}}{s} \]

       2. lift-/.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{16} - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s} \cdot \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s}}{\frac{1}{4} - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s}}}{s} \]

       3. lift-/.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{16} - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s} \cdot \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s}}{\frac{1}{4} - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s}}}{s} \]

       4. lift-fma.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{16} - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s} \cdot \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s}}{\frac{1}{4} - \frac{\frac{\left(\left(x \cdot x\right) \cdot 3\right) \cdot \frac{-1}{16} + \left(x \cdot x\right) \cdot \frac{1}{8}}{s}}{s}}}{s} \]

       5. lift-*.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{16} - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s} \cdot \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s}}{\frac{1}{4} - \frac{\frac{\left(\left(x \cdot x\right) \cdot 3\right) \cdot \frac{-1}{16} + \left(x \cdot x\right) \cdot \frac{1}{8}}{s}}{s}}}{s} \]

       6. lift-*.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{16} - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s} \cdot \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s}}{\frac{1}{4} - \frac{\frac{\left(\left(x \cdot x\right) \cdot 3\right) \cdot \frac{-1}{16} + \left(x \cdot x\right) \cdot \frac{1}{8}}{s}}{s}}}{s} \]

       7. lift-*.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{16} - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s} \cdot \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s}}{\frac{1}{4} - \frac{\frac{\left(\left(x \cdot x\right) \cdot 3\right) \cdot \frac{-1}{16} + \left(x \cdot x\right) \cdot \frac{1}{8}}{s}}{s}}}{s} \]

       8. lift-*.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{16} - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s} \cdot \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s}}{\frac{1}{4} - \frac{\frac{\left(\left(x \cdot x\right) \cdot 3\right) \cdot \frac{-1}{16} + \left(x \cdot x\right) \cdot \frac{1}{8}}{s}}{s}}}{s} \]

       9. associate-/l/ N/A

          \[\leadsto \frac{\frac{\frac{1}{16} - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s} \cdot \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, \frac{-1}{16}, \left(x \cdot x\right) \cdot \frac{1}{8}\right)}{s}}{s}}{\frac{1}{4} - \frac{\left(\left(x \cdot x\right) \cdot 3\right) \cdot \frac{-1}{16} + \left(x \cdot x\right) \cdot \frac{1}{8}}{s \cdot s}}}{s} \]

   12. Applied rewrites 93.0%

       \[\leadsto \frac{\frac{0.0625 - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s} \cdot \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s}}{\mathsf{fma}\left(\frac{x}{s} \cdot \frac{x}{s}, 0.0625, 0.25\right)}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 0:\\ \;\;\;\;\frac{t\_0}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s} \cdot \frac{x}{s}, -0.0625, 0.25\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 0.0)
     (/ t_0 (* 4.0 s))
     (/ (fma (* (/ x s) (/ x s)) -0.0625 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 0.0f) {
		tmp = t_0 / (4.0f * s);
	} else {
		tmp = fmaf(((x / s) * (x / s)), -0.0625f, 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(s * t_1) * t_1)) <= Float32(0.0))
		tmp = Float32(t_0 / Float32(Float32(4.0) * s));
	else
		tmp = Float32(fma(Float32(Float32(x / s) * Float32(x / s)), Float32(-0.0625), Float32(0.25)) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0

   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

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

      1. lower-*.f32 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]

   8. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\mathsf{fma}\left(0.125, x \cdot x, -0.0625 \cdot \left(\left(x \cdot x\right) \cdot 3\right)\right)}{s \cdot s}}{s}} \]

   10. Applied rewrites 91.5%

       \[\leadsto \frac{\frac{0.0625 - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s} \cdot \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s}}{0.25 - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s}}}{s} \]

   12. Applied rewrites 92.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x}{s} \cdot \frac{x}{s}, -0.0625, 0.25\right)}{s}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 0:\\ \;\;\;\;\frac{1}{\left(\frac{\mathsf{fma}\left(\left|x\right| \cdot s, 4, -3 \cdot \left(x \cdot x\right)\right)}{s \cdot s} - 4\right) \cdot \left(-s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s} \cdot \frac{x}{s}, -0.0625, 0.25\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 0.0)
     (/
      1.0
      (* (- (/ (fma (* (fabs x) s) 4.0 (* -3.0 (* x x))) (* s s)) 4.0) (- s)))
     (/ (fma (* (/ x s) (/ x s)) -0.0625 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 0.0f) {
		tmp = 1.0f / (((fmaf((fabsf(x) * s), 4.0f, (-3.0f * (x * x))) / (s * s)) - 4.0f) * -s);
	} else {
		tmp = fmaf(((x / s) * (x / s)), -0.0625f, 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(s * t_1) * t_1)) <= Float32(0.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(fma(Float32(abs(x) * s), Float32(4.0), Float32(Float32(-3.0) * Float32(x * x))) / Float32(s * s)) - Float32(4.0)) * Float32(-s)));
	else
		tmp = Float32(fma(Float32(Float32(x / s) * Float32(x / s)), Float32(-0.0625), Float32(0.25)) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0

   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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in s around 0

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{4 \cdot \left(s \cdot \left|x\right|\right) - 3 \cdot {x}^{2}}{{s}^{2}} - 4\right) \cdot s} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{4 \cdot \left(s \cdot \left|x\right|\right) + \left(\mathsf{neg}\left(3\right)\right) \cdot {x}^{2}}{{s}^{2}} - 4\right) \cdot s} \]

      2. metadata-eval N/A

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{4 \cdot \left(s \cdot \left|x\right|\right) + -3 \cdot {x}^{2}}{{s}^{2}} - 4\right) \cdot s} \]

      3. +-commutative N/A

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{-3 \cdot {x}^{2} + 4 \cdot \left(s \cdot \left|x\right|\right)}{{s}^{2}} - 4\right) \cdot s} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{-3 \cdot {x}^{2} + 4 \cdot \left(s \cdot \left|x\right|\right)}{{s}^{2}} - 4\right) \cdot s} \]

      5. +-commutative N/A

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{4 \cdot \left(s \cdot \left|x\right|\right) + -3 \cdot {x}^{2}}{{s}^{2}} - 4\right) \cdot s} \]

      6. *-commutative N/A

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{\left(s \cdot \left|x\right|\right) \cdot 4 + -3 \cdot {x}^{2}}{{s}^{2}} - 4\right) \cdot s} \]

      7. lower-fma.f32 N/A

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{\mathsf{fma}\left(s \cdot \left|x\right|, 4, -3 \cdot {x}^{2}\right)}{{s}^{2}} - 4\right) \cdot s} \]

      8. *-commutative N/A

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{\mathsf{fma}\left(\left|x\right| \cdot s, 4, -3 \cdot {x}^{2}\right)}{{s}^{2}} - 4\right) \cdot s} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{\mathsf{fma}\left(\left|x\right| \cdot s, 4, -3 \cdot {x}^{2}\right)}{{s}^{2}} - 4\right) \cdot s} \]

      10. lift-fabs.f32 N/A

          \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{\mathsf{fma}\left(\left|x\right| \cdot s, 4, -3 \cdot {x}^{2}\right)}{{s}^{2}} - 4\right) \cdot s} \]

      11. lower-*.f32 N/A

          \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{\mathsf{fma}\left(\left|x\right| \cdot s, 4, -3 \cdot {x}^{2}\right)}{{s}^{2}} - 4\right) \cdot s} \]

      12. pow2 N/A

          \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{\mathsf{fma}\left(\left|x\right| \cdot s, 4, -3 \cdot \left(x \cdot x\right)\right)}{{s}^{2}} - 4\right) \cdot s} \]

      13. lift-*.f32 N/A

          \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{\mathsf{fma}\left(\left|x\right| \cdot s, 4, -3 \cdot \left(x \cdot x\right)\right)}{{s}^{2}} - 4\right) \cdot s} \]

      14. unpow2 N/A

          \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{\mathsf{fma}\left(\left|x\right| \cdot s, 4, -3 \cdot \left(x \cdot x\right)\right)}{s \cdot s} - 4\right) \cdot s} \]

      15. lower-*.f32 96.8

          \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{\mathsf{fma}\left(\left|x\right| \cdot s, 4, -3 \cdot \left(x \cdot x\right)\right)}{s \cdot s} - 4\right) \cdot s} \]

   8. Applied rewrites 96.8%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\frac{\mathsf{fma}\left(\left|x\right| \cdot s, 4, -3 \cdot \left(x \cdot x\right)\right)}{s \cdot s} - 4\right) \cdot s} \]

   9. Taylor expanded in s around inf

      \[\leadsto \frac{\color{blue}{1}}{-\left(\frac{\mathsf{fma}\left(\left|x\right| \cdot s, 4, -3 \cdot \left(x \cdot x\right)\right)}{s \cdot s} - 4\right) \cdot s} \]
   10. Step-by-step derivation

       1. distribute-frac-neg 79.4

          \[\leadsto \frac{1}{-\left(\frac{\mathsf{fma}\left(\left|x\right| \cdot s, 4, -3 \cdot \left(x \cdot x\right)\right)}{s \cdot s} - 4\right) \cdot s} \]

   11. Applied rewrites 79.4%

       \[\leadsto \frac{\color{blue}{1}}{-\left(\frac{\mathsf{fma}\left(\left|x\right| \cdot s, 4, -3 \cdot \left(x \cdot x\right)\right)}{s \cdot s} - 4\right) \cdot s} \]

   if 0.0 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]

   8. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\mathsf{fma}\left(0.125, x \cdot x, -0.0625 \cdot \left(\left(x \cdot x\right) \cdot 3\right)\right)}{s \cdot s}}{s}} \]

   10. Applied rewrites 91.5%

       \[\leadsto \frac{\frac{0.0625 - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s} \cdot \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s}}{0.25 - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s}}}{s} \]

   12. Applied rewrites 92.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x}{s} \cdot \frac{x}{s}, -0.0625, 0.25\right)}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 0:\\ \;\;\;\;\frac{1}{\left(\frac{\mathsf{fma}\left(\left|x\right| \cdot s, 4, -3 \cdot \left(x \cdot x\right)\right)}{s \cdot s} - 4\right) \cdot \left(-s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s} \cdot \frac{x}{s}, -0.0625, 0.25\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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

Derivation

1. Initial program 99.8%

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

Alternative 6: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{t\_0}{{\left(t\_0 + 1\right)}^{2} \cdot s} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s)))) (/ t_0 (* (pow (+ t_0 1.0) 2.0) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / (powf((t_0 + 1.0f), 2.0f) * 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
    real(4) :: t_0
    t_0 = exp((-abs(x) / s))
    code = t_0 / (((t_0 + 1.0e0) ** 2.0e0) * s)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(t_0 / Float32((Float32(t_0 + Float32(1.0)) ^ Float32(2.0)) * s))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	tmp = t_0 / (((t_0 + single(1.0)) ^ single(2.0)) * s);
end

Derivation

1. Initial program 99.8%

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

   1. lift-*.f32 N/A

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

   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

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

   4. lift-exp.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-fabs.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-neg.f32 N/A

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

   12. lift-fabs.f32 N/A

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

   13. associate-*l* N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 7: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\left|x\right|\\ t_1 := e^{\frac{t\_0}{s}}\\ \frac{t\_1}{\left(s \cdot \left(1 + \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x \cdot \frac{x}{s}, -0.5, t\_0\right)}{s}, -1, 1\right)}\right)\right) \cdot \left(1 + t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- (fabs x))) (t_1 (exp (/ t_0 s))))
   (/
    t_1
    (*
     (* s (+ 1.0 (/ 1.0 (fma (/ (fma (* x (/ x s)) -0.5 t_0) s) -1.0 1.0))))
     (+ 1.0 t_1)))))

C

float code(float x, float s) {
	float t_0 = -fabsf(x);
	float t_1 = expf((t_0 / s));
	return t_1 / ((s * (1.0f + (1.0f / fmaf((fmaf((x * (x / s)), -0.5f, t_0) / s), -1.0f, 1.0f)))) * (1.0f + t_1));
}

Julia

function code(x, s)
	t_0 = Float32(-abs(x))
	t_1 = exp(Float32(t_0 / s))
	return Float32(t_1 / Float32(Float32(s * Float32(Float32(1.0) + Float32(Float32(1.0) / fma(Float32(fma(Float32(x * Float32(x / s)), Float32(-0.5), t_0) / s), Float32(-1.0), Float32(1.0))))) * Float32(Float32(1.0) + t_1)))
end

Derivation

1. Initial program 99.8%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-fabs.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. exp-neg N/A

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

   7. lower-/.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-fabs.f32 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in s around -inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

7. Applied rewrites 98.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.3

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

9. Applied rewrites 98.3%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x \cdot \frac{x}{s}, -0.5, -\left|x\right|\right)}{s}, -1, 1\right)}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
10. Add Preprocessing

Alternative 8: 97.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{t\_0}{\left(s \cdot \left(1 + \frac{1}{\frac{\mathsf{fma}\left(\frac{x \cdot x}{s}, 0.5, \left|x\right|\right)}{s} + 1}\right)\right) \cdot \left(1 + t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))))
   (/
    t_0
    (*
     (* s (+ 1.0 (/ 1.0 (+ (/ (fma (/ (* x x) s) 0.5 (fabs x)) s) 1.0))))
     (+ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / ((s * (1.0f + (1.0f / ((fmaf(((x * x) / s), 0.5f, fabsf(x)) / s) + 1.0f)))) * (1.0f + t_0));
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(t_0 / Float32(Float32(s * Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(fma(Float32(Float32(x * x) / s), Float32(0.5), abs(x)) / s) + Float32(1.0))))) * Float32(Float32(1.0) + t_0)))
end

Derivation

1. Initial program 99.8%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-fabs.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. exp-neg N/A

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

   7. lower-/.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-fabs.f32 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in s around -inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

7. Applied rewrites 98.0%

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

8. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

10. Applied rewrites 98.0%

    \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\frac{x \cdot x}{s}, 0.5, \left|x\right|\right)}{s} + 1}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
11. Add Preprocessing

Alternative 9: 97.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{t\_0}{\left(s \cdot \left(1 + t\_0\right)\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(x \cdot \frac{x}{s}, -0.5, \left|x\right|\right)}{s}, -1, 2\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))))
   (/
    t_0
    (*
     (* s (+ 1.0 t_0))
     (fma (/ (fma (* x (/ x s)) -0.5 (fabs x)) s) -1.0 2.0)))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / ((s * (1.0f + t_0)) * fmaf((fmaf((x * (x / s)), -0.5f, fabsf(x)) / s), -1.0f, 2.0f));
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(t_0 / Float32(Float32(s * Float32(Float32(1.0) + t_0)) * fma(Float32(fma(Float32(x * Float32(x / s)), Float32(-0.5), abs(x)) / s), Float32(-1.0), Float32(2.0))))
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around -inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. unpow2 N/A

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

   10. sqr-abs-rev N/A

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

   11. lower-*.f32 N/A

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

   12. lift-fabs.f32 97.7

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

5. Applied rewrites 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 97.9

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

7. Applied rewrites 97.9%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(x \cdot \frac{x}{s}, -0.5, \left|x\right|\right)}{s}, -1, 2\right)} \]
8. Add Preprocessing

Alternative 10: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{t\_0}{\left(s \cdot \left(1 + t\_0\right)\right) \cdot \left(\frac{\mathsf{fma}\left(\frac{x \cdot x}{s}, -0.5, \left|x\right|\right)}{-s} + 2\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))))
   (/
    t_0
    (*
     (* s (+ 1.0 t_0))
     (+ (/ (fma (/ (* x x) s) -0.5 (fabs x)) (- s)) 2.0)))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / ((s * (1.0f + t_0)) * ((fmaf(((x * x) / s), -0.5f, fabsf(x)) / -s) + 2.0f));
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(t_0 / Float32(Float32(s * Float32(Float32(1.0) + t_0)) * Float32(Float32(fma(Float32(Float32(x * x) / s), Float32(-0.5), abs(x)) / Float32(-s)) + Float32(2.0))))
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around -inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. unpow2 N/A

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

   10. sqr-abs-rev N/A

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

   11. lower-*.f32 N/A

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

   12. lift-fabs.f32 97.7

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

5. Applied rewrites 97.7%

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

   1. lift-fma.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. sqr-abs-rev N/A

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

   6. unpow2 N/A

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

   7. lift-fabs.f32 N/A

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

   8. lower-fma.f32 N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-+.f32 N/A

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

7. Applied rewrites 97.7%

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

8. Final simplification 97.7%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\frac{\mathsf{fma}\left(\frac{x \cdot x}{s}, -0.5, \left|x\right|\right)}{-s} + 2\right)} \]
9. Add Preprocessing

Alternative 11: 96.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 97.3%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(-4, \left|x\right|, 3 \cdot \frac{x \cdot x}{s}\right)}{s}\right) - 4\right) \cdot s}} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\left(-\frac{\mathsf{fma}\left(-4, \left|x\right|, 3 \cdot \frac{x \cdot x}{s}\right)}{s}\right) - 4\right) \cdot s} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\left(-\frac{\mathsf{fma}\left(-4, \left|x\right|, 3 \cdot \frac{x \cdot x}{s}\right)}{s}\right) - 4\right) \cdot s} \]

   3. associate-/l* N/A

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\left(-\frac{\mathsf{fma}\left(-4, \left|x\right|, 3 \cdot \left(x \cdot \frac{x}{s}\right)\right)}{s}\right) - 4\right) \cdot s} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\left(-\frac{\mathsf{fma}\left(-4, \left|x\right|, 3 \cdot \left(x \cdot \frac{x}{s}\right)\right)}{s}\right) - 4\right) \cdot s} \]

   5. lower-/.f32 97.6

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\left(-\frac{\mathsf{fma}\left(-4, \left|x\right|, 3 \cdot \left(x \cdot \frac{x}{s}\right)\right)}{s}\right) - 4\right) \cdot s} \]

7. Applied rewrites 97.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{-\left(\left(-\frac{\mathsf{fma}\left(-4, \left|x\right|, 3 \cdot \left(x \cdot \frac{x}{s}\right)\right)}{s}\right) - 4\right) \cdot s} \]

8. Final simplification 97.6%

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

Alternative 12: 26.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{x}{s} \cdot \frac{x}{s}, -0.0625, 0.25\right)}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (fma (* (/ x s) (/ x s)) -0.0625 0.25) s))

C

float code(float x, float s) {
	return fmaf(((x / s) * (x / s)), -0.0625f, 0.25f) / s;
}

Julia

function code(x, s)
	return Float32(fma(Float32(Float32(x / s) * Float32(x / s)), Float32(-0.0625), Float32(0.25)) / s)
end

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 26.5

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

5. Applied rewrites 26.5%

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

6. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]

8. Applied rewrites 21.8%

   \[\leadsto \color{blue}{\frac{0.25 + \frac{\mathsf{fma}\left(0.125, x \cdot x, -0.0625 \cdot \left(\left(x \cdot x\right) \cdot 3\right)\right)}{s \cdot s}}{s}} \]

10. Applied rewrites 24.8%

    \[\leadsto \frac{\frac{0.0625 - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s} \cdot \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s}}{0.25 - \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 3, -0.0625, \left(x \cdot x\right) \cdot 0.125\right)}{s}}{s}}}{s} \]

12. Applied rewrites 26.7%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x}{s} \cdot \frac{x}{s}, -0.0625, 0.25\right)}{s}} \]
13. Add Preprocessing

Alternative 13: 26.8% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 s))

C

float code(float x, float s) {
	return 0.25f / 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 = 0.25e0 / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 26.5

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

5. Applied rewrites 26.5%

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

6. Final simplification 26.5%

   \[\leadsto \frac{0.25}{s} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025051 
(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))))))