Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 5.4s

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: 99.5% 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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 99.5

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. distribute-neg-frac2 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-neg.f32 99.5

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. metadata-eval N/A

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

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

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 29.9% accurate, 0.9× speedup

Math

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

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)
     (/ (* (/ -0.0625 s) (/ (* x x) s)) s)
     (/ (fma 0.0625 (* (/ x s) (/ x s)) -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 = ((-0.0625f / s) * ((x * x) / s)) / s;
	} else {
		tmp = fmaf(0.0625f, ((x / s) * (x / s)), -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(Float32(Float32(-0.0625) / s) * Float32(Float32(x * x) / s)) / s);
	else
		tmp = Float32(fma(Float32(0.0625), Float32(Float32(x / s) * Float32(x / s)), Float32(-0.25)) / Float32(-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 99.5%

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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f32 99.5

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

      7. lift-/.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-neg.f32 99.5

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 99.5%

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

   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. lower-/.f32 N/A

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

   7. Applied rewrites 2.1%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\mathsf{fma}\left(0.125, x \cdot x, -0.1875 \cdot \left(x \cdot x\right)\right)}{s \cdot s}}{s}} \]

   8. Taylor expanded in s around 0

      \[\leadsto \frac{\frac{\frac{-3}{16} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}}{s} \]
   9. Step-by-step derivation

   10. Applied rewrites 9.1%

       \[\leadsto \frac{\frac{-0.0625}{s} \cdot \frac{x \cdot x}{s}}{s} \]

   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. 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. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   7. Applied rewrites 93.1%

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

Alternative 3: 29.9% accurate, 0.9× speedup

Math

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

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)
     (/ (* (/ -0.0625 s) (/ (* x x) s)) s)
     (/ (fma (* (/ x s) x) (/ -0.0625 s) 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 = ((-0.0625f / s) * ((x * x) / s)) / s;
	} else {
		tmp = fmaf(((x / s) * x), (-0.0625f / s), 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(Float32(Float32(-0.0625) / s) * Float32(Float32(x * x) / s)) / s);
	else
		tmp = Float32(fma(Float32(Float32(x / s) * x), Float32(Float32(-0.0625) / s), 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 99.5%

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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f32 99.5

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

      7. lift-/.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-neg.f32 99.5

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 99.5%

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

   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. lower-/.f32 N/A

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

   7. Applied rewrites 2.1%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\mathsf{fma}\left(0.125, x \cdot x, -0.1875 \cdot \left(x \cdot x\right)\right)}{s \cdot s}}{s}} \]

   8. Taylor expanded in s around 0

      \[\leadsto \frac{\frac{\frac{-3}{16} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}}{s} \]
   9. Step-by-step derivation

   10. Applied rewrites 9.1%

       \[\leadsto \frac{\frac{-0.0625}{s} \cdot \frac{x \cdot x}{s}}{s} \]

   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. 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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f32 99.4

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

      7. lift-/.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-neg.f32 99.4

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 99.4%

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

   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. lower-/.f32 N/A

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

   7. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\mathsf{fma}\left(0.125, x \cdot x, -0.1875 \cdot \left(x \cdot x\right)\right)}{s \cdot s}}{s}} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.5%

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

   11. Applied rewrites 93.1%

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

Alternative 4: 29.3% accurate, 0.9× speedup

Math

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

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)
     (/ (* (/ -0.0625 s) (/ (* x x) s)) s)
     (/ 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 = ((-0.0625f / s) * ((x * x) / s)) / s;
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 0.0e0) then
        tmp = (((-0.0625e0) / s) * ((x * x) / s)) / s
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

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(Float32(Float32(-0.0625) / s) * Float32(Float32(x * x) / s)) / s);
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((s * t_1) * t_1)) <= single(0.0))
		tmp = ((single(-0.0625) / s) * ((x * x) / s)) / s;
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = 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 99.5%

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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f32 99.5

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

      7. lift-/.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-neg.f32 99.5

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 99.5%

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

   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. lower-/.f32 N/A

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

   7. Applied rewrites 2.1%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\mathsf{fma}\left(0.125, x \cdot x, -0.1875 \cdot \left(x \cdot x\right)\right)}{s \cdot s}}{s}} \]

   8. Taylor expanded in s around 0

      \[\leadsto \frac{\frac{\frac{-3}{16} \cdot {x}^{2} + \frac{1}{8} \cdot {x}^{2}}{{s}^{2}}}{s} \]
   9. Step-by-step derivation

   10. Applied rewrites 9.1%

       \[\leadsto \frac{\frac{-0.0625}{s} \cdot \frac{x \cdot x}{s}}{s} \]

   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 90.4

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

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f32 N/A

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

   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. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 6: 94.6% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s)))) (/ t_0 (* (* s (+ 1.0 t_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)) * 2.0f);
}

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 + t_0)) * 2.0e0)
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) + t_0)) * Float32(2.0)))
end

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. 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 95.8%

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

Alternative 7: 94.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 99.5

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. distribute-neg-frac2 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-neg.f32 99.5

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. metadata-eval N/A

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

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

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 95.8%

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

8. Final simplification 95.8%

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

Alternative 8: 48.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{\sqrt{x} \cdot \sqrt{x}}{-s}}}{4 \cdot s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (exp (/ (* (sqrt x) (sqrt x)) (- s))) (* 4.0 s)))

C

float code(float x, float s) {
	return expf(((sqrtf(x) * sqrtf(x)) / -s)) / (4.0f * 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
    code = exp(((sqrt(x) * sqrt(x)) / -s)) / (4.0e0 * s)
end function

Julia

function code(x, s)
	return Float32(exp(Float32(Float32(sqrt(x) * sqrt(x)) / Float32(-s))) / Float32(Float32(4.0) * s))
end

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. 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 95.5

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

5. Applied rewrites 95.5%

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

   1. lift-fabs.f32 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-sqrt.f32 44.2

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

7. Applied rewrites 44.2%

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

8. Final simplification 44.2%

   \[\leadsto \frac{e^{\frac{\sqrt{x} \cdot \sqrt{x}}{-s}}}{4 \cdot s} \]
9. Add Preprocessing

Alternative 9: 94.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return expf((-fabsf(x) / s)) / (4.0f * 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
    code = exp((-abs(x) / s)) / (4.0e0 * s)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. 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 95.5

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

5. Applied rewrites 95.5%

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

Alternative 10: 55.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, \frac{x}{s} \cdot \frac{x}{s}, -0.25\right)}{-s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{x \cdot x}{s \cdot s} - -4\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.0000000390829628e-24)
   (/ (fma 0.0625 (* (/ x s) (/ x s)) -0.25) (- s))
   (/ 1.0 (* (- (/ (* x x) (* s s)) -4.0) s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.0000000390829628e-24f) {
		tmp = fmaf(0.0625f, ((x / s) * (x / s)), -0.25f) / -s;
	} else {
		tmp = 1.0f / ((((x * x) / (s * s)) - -4.0f) * s);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.0000000390829628e-24))
		tmp = Float32(fma(Float32(0.0625), Float32(Float32(x / s) * Float32(x / s)), Float32(-0.25)) / Float32(-s));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x * x) / Float32(s * s)) - Float32(-4.0)) * s));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 2.00000004e-24

   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. 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. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   7. Applied rewrites 35.1%

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

   if 2.00000004e-24 < x

   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

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f32 99.9

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

      7. lift-/.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-neg.f32 99.9

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in s around inf

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

   9. Applied rewrites 82.6%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(0 + \frac{\left(x \cdot x\right) \cdot 1}{s \cdot s}\right) - -4\right) \cdot s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, \frac{x}{s} \cdot \frac{x}{s}, -0.25\right)}{-s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{x \cdot x}{s \cdot s} - -4\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.1% 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 99.4%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 28.3

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

5. Applied rewrites 28.3%

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

Reproduce

herbie shell --seed 2025017 
(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))))))