Logistic distribution

Percentage Accurate: 99.6% → 99.7%

Time: 4.5s

Alternatives: 11

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: 99.6% 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: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-fabs.f32 N/A

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

   6. distribute-frac-neg N/A

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

   7. mul-1-neg N/A

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

   8. lift-*.f32 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-fabs.f32 N/A

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

   7. lift-pow.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-fabs.f32 N/A

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

   13. associate-/l/ N/A

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

6. Applied rewrites 99.7%

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

8. Applied rewrites 61.1%

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

Alternative 2: 97.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{\left|x\_m\right|}{-s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 0.019999999552965164:\\ \;\;\;\;\frac{e^{\frac{x\_m}{-s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x\_m}{s} \cdot \frac{x\_m}{s}, -0.0625, 0.25\right)}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x_m) (- s)))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 0.019999999552965164)
     (/ (exp (/ x_m (- s))) s)
     (/ (fma (* (/ x_m s) (/ x_m s)) -0.0625 0.25) s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((fabsf(x_m) / -s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 0.019999999552965164f) {
		tmp = expf((x_m / -s)) / s;
	} else {
		tmp = fmaf(((x_m / s) * (x_m / s)), -0.0625f, 0.25f) / s;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(abs(x_m) / Float32(-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(0.019999999552965164))
		tmp = Float32(exp(Float32(x_m / Float32(-s))) / s);
	else
		tmp = Float32(fma(Float32(Float32(x_m / s) * Float32(x_m / 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))))) < 0.0199999996

   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

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

      2. lift-exp.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-neg.f32 N/A

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

      5. lift-fabs.f32 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f32 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-exp.f32 N/A

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

      4. lift-neg.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift-fabs.f32 N/A

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

      7. lift-pow.f32 N/A

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

      8. lift-+.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. lift-neg.f32 N/A

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

      11. lift-/.f32 N/A

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

      12. lift-fabs.f32 N/A

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

      13. associate-/l/ N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in s around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{s} \]

      2. distribute-frac-neg2 N/A

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

      3. lower-/.f32 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. pow2 N/A

         \[\leadsto \frac{e^{\frac{\sqrt{{x}^{2}}}{\mathsf{neg}\left(s\right)}}}{s} \]

      6. sqrt-pow1 N/A

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

      7. metadata-eval N/A

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

      8. unpow1 N/A

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

      9. lift-neg.f32 49.4

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

   9. Applied rewrites 49.4%

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

   if 0.0199999996 < (/.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 99.3%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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-exp.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-neg.f32 N/A

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

      5. lift-fabs.f32 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f32 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in s around inf

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. distribute-frac-neg N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 78.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{4} + \frac{-1}{16} \cdot \frac{{x}^{2}}{{s}^{2}}}{s} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \frac{{x}^{2}}{{s}^{2}} + \frac{1}{4}}{s} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{{x}^{2}}{{s}^{2}} \cdot \frac{-1}{16} + \frac{1}{4}}{s} \]

      3. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot x}{{s}^{2}} \cdot \frac{-1}{16} + \frac{1}{4}}{s} \]

      4. sqr-abs-rev N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left|x\right| \cdot \left|x\right|}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      8. sqr-abs-rev N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{x \cdot x}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      9. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{{x}^{2}}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      10. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{{x}^{2}}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x \cdot x}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      12. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x \cdot x}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

      13. pow2 N/A

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

      14. lift-*.f32 78.2

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

   10. Applied rewrites 78.2%

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

       1. lift-*.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. lift-/.f32 N/A

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

       4. times-frac N/A

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

       5. lower-*.f32 N/A

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

       6. lift-/.f32 N/A

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

       7. lift-/.f32 93.3

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

   12. Applied rewrites 93.3%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\left|x\right|}{-s}}}{\left(s \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)} \leq 0.019999999552965164:\\ \;\;\;\;\frac{e^{\frac{x}{-s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s} \cdot \frac{x}{s}, -0.0625, 0.25\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-fabs.f32 N/A

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

   6. distribute-frac-neg N/A

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

   7. mul-1-neg N/A

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

   8. lift-*.f32 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-fabs.f32 N/A

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

   7. lift-pow.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-fabs.f32 N/A

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

   13. associate-/l/ N/A

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

6. Applied rewrites 99.7%

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

   1. lift-fabs.f32 N/A

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

   2. rem-sqrt-square-rev N/A

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

   3. pow2 N/A

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

   4. sqrt-pow1 N/A

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

   5. metadata-eval N/A

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

   6. unpow1 59.9

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

   7. lift-fabs.f32 N/A

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

   8. rem-sqrt-square-rev N/A

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

   9. pow2 N/A

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

   10. sqrt-pow1 N/A

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

   11. metadata-eval N/A

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

   12. unpow1 85.6

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

8. Applied rewrites 85.6%

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

Alternative 4: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{\frac{\left|x\_m\right|}{-s} - \mathsf{fma}\left(\frac{x\_m \cdot \left(1 - 0.25 \cdot \frac{x\_m}{s}\right)}{s}, -1, \log 4\right)}}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/
  (exp
   (-
    (/ (fabs x_m) (- s))
    (fma (/ (* x_m (- 1.0 (* 0.25 (/ x_m s)))) s) -1.0 (log 4.0))))
  s))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf(((fabsf(x_m) / -s) - fmaf(((x_m * (1.0f - (0.25f * (x_m / s)))) / s), -1.0f, logf(4.0f)))) / s;
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(exp(Float32(Float32(abs(x_m) / Float32(-s)) - fma(Float32(Float32(x_m * Float32(Float32(1.0) - Float32(Float32(0.25) * Float32(x_m / s)))) / s), Float32(-1.0), log(Float32(4.0))))) / s)
end

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-fabs.f32 N/A

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

   6. distribute-frac-neg N/A

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

   7. mul-1-neg N/A

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

   8. lift-*.f32 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-fabs.f32 N/A

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

   7. lift-pow.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-fabs.f32 N/A

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

   13. associate-/l/ N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in s around -inf

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

9. Applied rewrites 94.7%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{e^{\frac{\left|x\right|}{-s} - \mathsf{fma}\left(\frac{x \cdot \left(1 + \frac{-1}{4} \cdot \frac{x}{s}\right)}{s}, -1, \log 4\right)}}{s} \]
11. Step-by-step derivation

    1. lower-*.f32 N/A

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

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

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

    3. metadata-eval N/A

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

    4. lower--.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. lift-/.f32 96.7

       \[\leadsto \frac{e^{\frac{\left|x\right|}{-s} - \mathsf{fma}\left(\frac{x \cdot \left(1 - 0.25 \cdot \frac{x}{s}\right)}{s}, -1, \log 4\right)}}{s} \]

12. Applied rewrites 96.7%

    \[\leadsto \frac{e^{\frac{\left|x\right|}{-s} - \mathsf{fma}\left(\frac{x \cdot \left(1 - 0.25 \cdot \frac{x}{s}\right)}{s}, -1, \log 4\right)}}{s} \]
13. Add Preprocessing

Alternative 5: 97.1% accurate, 1.7× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (fma (/ (fma (* x_m (/ x_m s)) -0.5 x_m) s) -1.0 2.0)))
   (/ (exp (/ (fabs x_m) (- s))) (* (* t_0 t_0) s))))

C

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

Julia

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

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 99.7%

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

5. Taylor expanded in s around -inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. sqr-abs-rev N/A

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

   10. lower-*.f32 N/A

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

   11. lift-fabs.f32 97.3

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

7. Applied rewrites 97.3%

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

9. Applied rewrites 96.5%

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

10. Final simplification 96.5%

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

Alternative 6: 97.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 95.7%

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

6. Taylor expanded in s around -inf

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

8. Applied rewrites 96.5%

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

9. Final simplification 96.5%

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

Alternative 7: 94.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

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 = exp((abs(x_m) / -s)) / ((-s * ((x_m / s) - 2.0e0)) * 2.0e0)
end function

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = exp((abs(x_m) / -s)) / ((-s * ((x_m / s) - single(2.0))) * single(2.0));
end

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 95.7%

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

6. Taylor expanded in s around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. rem-sqrt-square-rev N/A

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

   8. pow2 N/A

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

   9. sqrt-pow1 N/A

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

   10. metadata-eval N/A

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

   11. unpow1 95.4

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

8. Applied rewrites 95.4%

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

9. Final simplification 95.4%

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

Alternative 8: 95.1% 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(abs(x_m) / Float32(-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 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 95.4

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

5. Applied rewrites 95.4%

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

6. Final simplification 95.4%

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

Alternative 9: 77.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 65000001503232:\\ \;\;\;\;\frac{\left(0.25 + -0.25 \cdot \frac{\left|x\_m\right|}{s}\right) - 0.0625 \cdot \frac{-2 \cdot \left(x\_m + \left|x\_m\right|\right)}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(-s\right) \cdot \left(\frac{x\_m}{s} - 2\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 65000001503232.0)
   (/
    (-
     (+ 0.25 (* -0.25 (/ (fabs x_m) s)))
     (* 0.0625 (/ (* -2.0 (+ x_m (fabs x_m))) s)))
    s)
   (/ 1.0 (* (* (- s) (- (/ x_m s) 2.0)) 2.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 65000001503232.0f) {
		tmp = ((0.25f + (-0.25f * (fabsf(x_m) / s))) - (0.0625f * ((-2.0f * (x_m + fabsf(x_m))) / s))) / s;
	} else {
		tmp = 1.0f / ((-s * ((x_m / s) - 2.0f)) * 2.0f);
	}
	return tmp;
}

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
    real(4) :: tmp
    if (x_m <= 65000001503232.0e0) then
        tmp = ((0.25e0 + ((-0.25e0) * (abs(x_m) / s))) - (0.0625e0 * (((-2.0e0) * (x_m + abs(x_m))) / s))) / s
    else
        tmp = 1.0e0 / ((-s * ((x_m / s) - 2.0e0)) * 2.0e0)
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(65000001503232.0))
		tmp = Float32(Float32(Float32(Float32(0.25) + Float32(Float32(-0.25) * Float32(abs(x_m) / s))) - Float32(Float32(0.0625) * Float32(Float32(Float32(-2.0) * Float32(x_m + abs(x_m))) / s))) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(-s) * Float32(Float32(x_m / s) - Float32(2.0))) * Float32(2.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(65000001503232.0))
		tmp = ((single(0.25) + (single(-0.25) * (abs(x_m) / s))) - (single(0.0625) * ((single(-2.0) * (x_m + abs(x_m))) / s))) / s;
	else
		tmp = single(1.0) / ((-s * ((x_m / s) - single(2.0))) * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 6.50000015e13

   1. Initial program 99.7%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 96.9

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

   5. Applied rewrites 96.9%

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

   7. Applied rewrites 96.0%

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

   8. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{-1}{4} \cdot \frac{\left|x\right|}{s}\right) - \frac{1}{16} \cdot \frac{-2 \cdot x + -2 \cdot \left|x\right|}{s}}{s}} \]
   9. Step-by-step derivation

      1. lower-/.f32 N/A

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

   10. Applied rewrites 44.8%

       \[\leadsto \color{blue}{\frac{\left(0.25 + -0.25 \cdot \frac{\left|x\right|}{s}\right) - 0.0625 \cdot \frac{-2 \cdot \left(x + \left|x\right|\right)}{s}}{s}} \]

   if 6.50000015e13 < x

   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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in s around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f32 N/A

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

      4. lift-neg.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. pow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in s around inf

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

   11. Applied rewrites 70.5%

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

Alternative 10: 51.1% accurate, 9.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\left(\left(-s\right) \cdot \left(\frac{x\_m}{s} - 2\right)\right) \cdot 2} \end{array} \]

FPCore

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

C

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

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 = 1.0e0 / ((-s * ((x_m / s) - 2.0e0)) * 2.0e0)
end function

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(1.0) / ((-s * ((x_m / s) - single(2.0))) * single(2.0));
end

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 95.7%

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

6. Taylor expanded in s around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. rem-sqrt-square-rev N/A

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

   8. pow2 N/A

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

   9. sqrt-pow1 N/A

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

   10. metadata-eval N/A

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

   11. unpow1 95.4

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

8. Applied rewrites 95.4%

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

9. Taylor expanded in s around inf

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

11. Applied rewrites 51.8%

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

Alternative 11: 27.0% 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 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 26.7

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

5. Applied rewrites 26.7%

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

Reproduce

herbie shell --seed 2025061 
(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))))))