Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 13.5s

Alternatives: 12

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

real(4) function code(x, s)
    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

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{-2}\right)}^{\left(\frac{\left|x\right|}{s} \cdot 0.5\right)}}{{\left(e^{\frac{-\left|x\right|}{s}} - -1\right)}^{2} \cdot s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  (pow (exp -2.0) (* (/ (fabs x) s) 0.5))
  (* (pow (- (exp (/ (- (fabs x)) s)) -1.0) 2.0) s)))

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (exp((-2.0e0)) ** ((abs(x) / s) * 0.5e0)) / (((exp((-abs(x) / s)) - (-1.0e0)) ** 2.0e0) * s)
end function

Julia

function code(x, s)
	return Float32((exp(Float32(-2.0)) ^ Float32(Float32(abs(x) / s) * Float32(0.5))) / Float32((Float32(exp(Float32(Float32(-abs(x)) / s)) - Float32(-1.0)) ^ Float32(2.0)) * s))
end

MATLAB

function tmp = code(x, s)
	tmp = (exp(single(-2.0)) ^ ((abs(x) / s) * single(0.5))) / (((exp((-abs(x) / s)) - single(-1.0)) ^ single(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

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

   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

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

   6. pow2 N/A

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

   7. lower-pow.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-/.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. pow-exp N/A

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

   8. lift-exp.f32 N/A

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

   9. sqr-pow N/A

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

   10. pow-prod-down N/A

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

   11. lower-pow.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-exp.f32 N/A

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

   14. prod-exp N/A

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

   15. metadata-eval N/A

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

   16. lower-exp.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. associate-/l/ N/A

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

   19. *-lft-identity N/A

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

   20. times-frac N/A

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

   21. lift-/.f32 N/A

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

   22. lower-*.f32 N/A

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

   23. metadata-eval 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 60:\\ \;\;\;\;\frac{t\_0}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (- t_0 -1.0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 60.0)
     (/ t_0 (* 4.0 s))
     (/ (+ (/ (* (/ x s) (* -0.0625 x)) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = t_0 - -1.0f;
	float tmp;
	if ((t_0 / ((t_1 * s) * t_1)) <= 60.0f) {
		tmp = t_0 / (4.0f * s);
	} else {
		tmp = ((((x / s) * (-0.0625f * x)) / s) + 0.25f) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    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 = t_0 - (-1.0e0)
    if ((t_0 / ((t_1 * s) * t_1)) <= 60.0e0) then
        tmp = t_0 / (4.0e0 * s)
    else
        tmp = ((((x / s) * ((-0.0625e0) * x)) / s) + 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(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(60.0))
		tmp = Float32(t_0 / Float32(Float32(4.0) * s));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(x / s) * Float32(Float32(-0.0625) * x)) / s) + Float32(0.25)) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = t_0 - single(-1.0);
	tmp = single(0.0);
	if ((t_0 / ((t_1 * s) * t_1)) <= single(60.0))
		tmp = t_0 / (single(4.0) * s);
	else
		tmp = ((((x / s) * (single(-0.0625) * x)) / s) + 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))))) < 60

   1. Initial program 99.9%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

   if 60 < (/.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 99.0%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. lower-/.f32 98.8

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

   4. Applied rewrites 98.8%

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

   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 89.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.0625 \cdot \left(x \cdot x\right)}{s}}{s} + 0.25}{s}} \]
   8. Step-by-step derivation

   9. Applied rewrites 90.5%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 60:\\ \;\;\;\;\frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 60:\\ \;\;\;\;\frac{0.25}{s} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (- t_0 -1.0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 60.0)
     (* (/ 0.25 s) t_0)
     (/ (+ (/ (* (/ x s) (* -0.0625 x)) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = t_0 - -1.0f;
	float tmp;
	if ((t_0 / ((t_1 * s) * t_1)) <= 60.0f) {
		tmp = (0.25f / s) * t_0;
	} else {
		tmp = ((((x / s) * (-0.0625f * x)) / s) + 0.25f) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    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 = t_0 - (-1.0e0)
    if ((t_0 / ((t_1 * s) * t_1)) <= 60.0e0) then
        tmp = (0.25e0 / s) * t_0
    else
        tmp = ((((x / s) * ((-0.0625e0) * x)) / s) + 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(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(60.0))
		tmp = Float32(Float32(Float32(0.25) / s) * t_0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(x / s) * Float32(Float32(-0.0625) * x)) / s) + Float32(0.25)) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = t_0 - single(-1.0);
	tmp = single(0.0);
	if ((t_0 / ((t_1 * s) * t_1)) <= single(60.0))
		tmp = (single(0.25) / s) * t_0;
	else
		tmp = ((((x / s) * (single(-0.0625) * x)) / s) + 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))))) < 60

   1. Initial program 99.9%

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

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

      3. associate-/r/ N/A

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

      4. lower-*.f32 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 98.7%

      \[\leadsto \frac{\color{blue}{0.25}}{s} \cdot e^{\frac{-\left|x\right|}{s}} \]

   if 60 < (/.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 99.0%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. lower-/.f32 98.8

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

   4. Applied rewrites 98.8%

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

   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 89.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.0625 \cdot \left(x \cdot x\right)}{s}}{s} + 0.25}{s}} \]
   8. Step-by-step derivation

   9. Applied rewrites 90.5%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 60:\\ \;\;\;\;\frac{0.25}{s} \cdot e^{\frac{-\left|x\right|}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{-4}\right)}^{\left(0.25 \cdot \frac{\left|x\right|}{s}\right)}}{{\left(e^{\frac{-\left|x\right|}{s}} - -1\right)}^{2} \cdot s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  (pow (exp -4.0) (* 0.25 (/ (fabs x) s)))
  (* (pow (- (exp (/ (- (fabs x)) s)) -1.0) 2.0) s)))

C

float code(float x, float s) {
	return powf(expf(-4.0f), (0.25f * (fabsf(x) / s))) / (powf((expf((-fabsf(x) / s)) - -1.0f), 2.0f) * s);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (exp((-4.0e0)) ** (0.25e0 * (abs(x) / s))) / (((exp((-abs(x) / s)) - (-1.0e0)) ** 2.0e0) * s)
end function

Julia

function code(x, s)
	return Float32((exp(Float32(-4.0)) ^ Float32(Float32(0.25) * Float32(abs(x) / s))) / Float32((Float32(exp(Float32(Float32(-abs(x)) / s)) - Float32(-1.0)) ^ Float32(2.0)) * s))
end

MATLAB

function tmp = code(x, s)
	tmp = (exp(single(-4.0)) ^ (single(0.25) * (abs(x) / s))) / (((exp((-abs(x) / s)) - single(-1.0)) ^ single(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

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

   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

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

   6. pow2 N/A

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

   7. lower-pow.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-/.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. pow-exp N/A

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

   8. lift-exp.f32 N/A

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

   9. sqr-pow N/A

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

   10. pow-prod-down N/A

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

   11. lower-pow.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-exp.f32 N/A

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

   14. prod-exp N/A

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

   15. metadata-eval N/A

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

   16. lower-exp.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. associate-/l/ N/A

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

   19. *-lft-identity N/A

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

   20. times-frac N/A

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

   21. lift-/.f32 N/A

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

   22. lower-*.f32 N/A

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

   23. metadata-eval 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-pow.f32 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f32 N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-exp.f32 N/A

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

   7. prod-exp N/A

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

   8. metadata-eval N/A

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

   9. lower-exp.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f32 99.7

       \[\leadsto \frac{{\left(e^{-4}\right)}^{\color{blue}{\left(\frac{\left|x\right|}{s} \cdot 0.25\right)}}}{{\left(1 + e^{\frac{-\left|x\right|}{s}}\right)}^{2} \cdot s} \]

8. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{{\left(e^{-4}\right)}^{\left(\frac{\left|x\right|}{s} \cdot 0.25\right)}}}{{\left(1 + e^{\frac{-\left|x\right|}{s}}\right)}^{2} \cdot s} \]

9. Final simplification 99.7%

   \[\leadsto \frac{{\left(e^{-4}\right)}^{\left(0.25 \cdot \frac{\left|x\right|}{s}\right)}}{{\left(e^{\frac{-\left|x\right|}{s}} - -1\right)}^{2} \cdot s} \]
10. Add Preprocessing

Alternative 5: 51.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\left(\frac{\mathsf{fma}\left(\frac{x \cdot x}{s}, 3, -4 \cdot \left|x\right|\right)}{s} - -4\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (- t_0 -1.0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 1.999999987845058e-8)
     (/ 1.0 (* (- (/ (fma (/ (* x x) s) 3.0 (* -4.0 (fabs x))) s) -4.0) s))
     (/ (+ (/ (* (/ x s) (* -0.0625 x)) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = t_0 - -1.0f;
	float tmp;
	if ((t_0 / ((t_1 * s) * t_1)) <= 1.999999987845058e-8f) {
		tmp = 1.0f / (((fmaf(((x * x) / s), 3.0f, (-4.0f * fabsf(x))) / s) - -4.0f) * s);
	} else {
		tmp = ((((x / s) * (-0.0625f * x)) / s) + 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 - Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(1.999999987845058e-8))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(fma(Float32(Float32(x * x) / s), Float32(3.0), Float32(Float32(-4.0) * abs(x))) / s) - Float32(-4.0)) * s));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(x / s) * Float32(Float32(-0.0625) * x)) / 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))))) < 1.99999999e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f32 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 36.0%

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

   if 1.99999999e-8 < (/.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 98.9%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. lower-/.f32 98.6

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

   4. Applied rewrites 98.6%

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

   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 87.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.0625 \cdot \left(x \cdot x\right)}{s}}{s} + 0.25}{s}} \]
   8. Step-by-step derivation

   9. Applied rewrites 88.7%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(e^{\frac{-\left|x\right|}{s}} - -1\right) \cdot s\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\left(\frac{\mathsf{fma}\left(\frac{x \cdot x}{s}, 3, -4 \cdot \left|x\right|\right)}{s} - -4\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{s} \cdot \left(-0.0625 \cdot x\right)}{s} + 0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((-abs(x) / s))
    code = t_0 / (((t_0 - (-1.0e0)) ** 2.0e0) * s)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

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

   6. pow2 N/A

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

   7. lower-pow.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 7: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((-abs(x) / s))
    code = (((t_0 - (-1.0e0)) ** (-2.0e0)) / s) * t_0
end function

Julia

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

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	tmp = (((t_0 - single(-1.0)) ^ single(-2.0)) / s) * t_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. 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. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 8: 80.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(exp(Float32(Float32(-abs(x)) / s)) / Float32(Float32(1.0) / Float32((Float32(Float32(1.0) + fma(Float32(Float32(Float32(x * x) / s) / s), Float32(0.5), Float32(Float32(1.0) - Float32(abs(x) / s)))) ^ 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. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 N/A

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

   14. mul-1-neg N/A

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

   15. unsub-neg N/A

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

   16. lower--.f32 N/A

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

   17. lower-/.f32 N/A

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

   18. lower-fabs.f32 78.0

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

7. Applied rewrites 78.0%

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

Alternative 9: 95.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((-abs(x) / s))
    code = t_0 / ((((1.0e0 - (abs(x) / s)) - (-1.0e0)) * s) * (t_0 - (-1.0e0)))
end function

Julia

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

MATLAB

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

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 95.3

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

5. Applied rewrites 95.3%

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

6. Final simplification 95.3%

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

Alternative 10: 75.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(exp(Float32(Float32(-abs(x)) / s)) / Float32(Float32(Float32(fma(Float32(Float32(x * x) / s), Float32(3.0), Float32(Float32(-4.0) * abs(x))) / s) - 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. Taylor expanded in s around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f32 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. lower--.f32 N/A

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

5. Applied rewrites 74.7%

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

6. Final simplification 75.0%

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

Alternative 11: 54.4% accurate, 5.7× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (* (- (/ (/ (fma (* x x) 3.0 (* (* -4.0 (fabs x)) s)) s) s) -4.0) s)))

C

float code(float x, float s) {
	return 1.0f / ((((fmaf((x * x), 3.0f, ((-4.0f * fabsf(x)) * s)) / s) / s) - -4.0f) * s);
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(Float32(Float32(fma(Float32(x * x), Float32(3.0), Float32(Float32(Float32(-4.0) * abs(x)) * s)) / s) / s) - 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. Taylor expanded in s around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f32 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. lower--.f32 N/A

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

5. Applied rewrites 74.7%

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

6. Taylor expanded in s around 0

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

8. Applied rewrites 74.1%

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

9. Taylor expanded in s around inf

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

11. Applied rewrites 48.2%

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

12. Final simplification 48.2%

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

Alternative 12: 26.6% 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

real(4) function code(x, s)
    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 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 27.6

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

5. Applied rewrites 27.6%

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

Reproduce

herbie shell --seed 2024268 
(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))))))