Logistic distribution

Percentage Accurate: 99.5% → 99.4%

Time: 9.0s

Alternatives: 10

Speedup: N/A×

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: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f32 N/A

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

   2. 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. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. inv-pow N/A

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

   7. lower-/.f32 N/A

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

6. Applied rewrites 99.7%

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

Alternative 2: 83.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 0.0f) {
		tmp = 1.0f / (((x * x) * ((1.0f / s) / s)) * s);
	} else {
		tmp = -1.0f / ((-4.0f - ((x / s) * (x / s))) * 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 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 0.0e0) then
        tmp = 1.0e0 / (((x * x) * ((1.0e0 / s) / s)) * s)
    else
        tmp = (-1.0e0) / (((-4.0e0) - ((x / s) * (x / s))) * 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(1.0) / Float32(Float32(Float32(x * x) * Float32(Float32(Float32(1.0) / s) / s)) * s));
	else
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(-4.0) - Float32(Float32(x / s) * Float32(x / s))) * 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(1.0) / (((x * x) * ((single(1.0) / s) / s)) * s);
	else
		tmp = single(-1.0) / ((single(-4.0) - ((x / s) * (x / s))) * 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 100.0%

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

      1. lift-/.f32 N/A

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

      2. 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. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. distribute-lft-out N/A

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

      5. lower-fma.f32 N/A

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

   7. Applied rewrites 40.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-4, \frac{\left|x\right|}{s} + \frac{x}{s} \cdot \frac{x}{s}, \mathsf{fma}\left(5, \frac{x}{s} \cdot \frac{x}{s}, 4 \cdot \frac{\left|x\right|}{s}\right) + 4\right)} \cdot s} \]

   8. Taylor expanded in s around 0

      \[\leadsto \frac{1}{\frac{-4 \cdot {x}^{2} + 5 \cdot {x}^{2}}{\color{blue}{{s}^{2}}} \cdot s} \]
   9. Step-by-step derivation

   10. Applied rewrites 67.1%

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

   12. Applied rewrites 81.0%

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

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

   1. Initial program 99.0%

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

      1. lift-/.f32 N/A

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

      2. 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. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-exp.f32 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 93.8%

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

Alternative 3: 85.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 0.0f) {
		tmp = 1.0f / ((x * (x / (s * s))) * s);
	} else {
		tmp = -1.0f / ((-4.0f - ((x / s) * (x / s))) * 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 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 0.0e0) then
        tmp = 1.0e0 / ((x * (x / (s * s))) * s)
    else
        tmp = (-1.0e0) / (((-4.0e0) - ((x / s) * (x / s))) * 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(1.0) / Float32(Float32(x * Float32(x / Float32(s * s))) * s));
	else
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(-4.0) - Float32(Float32(x / s) * Float32(x / s))) * 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(1.0) / ((x * (x / (s * s))) * s);
	else
		tmp = single(-1.0) / ((single(-4.0) - ((x / s) * (x / s))) * 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 100.0%

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

      1. lift-/.f32 N/A

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

      2. 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. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. distribute-lft-out N/A

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

      5. lower-fma.f32 N/A

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

   7. Applied rewrites 40.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-4, \frac{\left|x\right|}{s} + \frac{x}{s} \cdot \frac{x}{s}, \mathsf{fma}\left(5, \frac{x}{s} \cdot \frac{x}{s}, 4 \cdot \frac{\left|x\right|}{s}\right) + 4\right)} \cdot s} \]

   8. Taylor expanded in s around 0

      \[\leadsto \frac{1}{\frac{-4 \cdot {x}^{2} + 5 \cdot {x}^{2}}{\color{blue}{{s}^{2}}} \cdot s} \]
   9. Step-by-step derivation

   10. Applied rewrites 67.1%

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

   12. Applied rewrites 80.6%

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

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

   1. Initial program 99.0%

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

      1. lift-/.f32 N/A

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

      2. 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. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-exp.f32 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 93.8%

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

Alternative 4: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 0.20000000298023224:\\ \;\;\;\;\frac{1}{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.0625 \cdot \left(x \cdot x\right)}{s}}{s} + 0.25}{s}\\ \end{array} \end{array} \]

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)) 0.20000000298023224)
     (/ 1.0 (* (* x (/ x (* s s))) s))
     (/ (+ (/ (/ (* -0.0625 (* 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)) <= 0.20000000298023224f) {
		tmp = 1.0f / ((x * (x / (s * s))) * s);
	} else {
		tmp = ((((-0.0625f * (x * x)) / s) / 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 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 0.20000000298023224e0) then
        tmp = 1.0e0 / ((x * (x / (s * s))) * s)
    else
        tmp = (((((-0.0625e0) * (x * x)) / s) / 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(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(s * t_1) * t_1)) <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(x / Float32(s * s))) * s));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(-0.0625) * Float32(x * x)) / s) / s) + 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(0.20000000298023224))
		tmp = single(1.0) / ((x * (x / (s * s))) * s);
	else
		tmp = ((((single(-0.0625) * (x * x)) / s) / 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))))) < 0.200000003

   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. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. distribute-lft-out N/A

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

      5. lower-fma.f32 N/A

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

   7. Applied rewrites 40.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-4, \frac{\left|x\right|}{s} + \frac{x}{s} \cdot \frac{x}{s}, \mathsf{fma}\left(5, \frac{x}{s} \cdot \frac{x}{s}, 4 \cdot \frac{\left|x\right|}{s}\right) + 4\right)} \cdot s} \]

   8. Taylor expanded in s around 0

      \[\leadsto \frac{1}{\frac{-4 \cdot {x}^{2} + 5 \cdot {x}^{2}}{\color{blue}{{s}^{2}}} \cdot s} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.5%

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

   12. Applied rewrites 79.9%

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

   if 0.200000003 < (/.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

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

      1. lower-/.f32 N/A

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

   5. Applied rewrites 94.1%

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

Alternative 5: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 0.20000000298023224:\\ \;\;\;\;\frac{1}{\left(x \cdot \frac{x}{s \cdot s}\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

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)) 0.20000000298023224)
     (/ 1.0 (* (* x (/ x (* s 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)) <= 0.20000000298023224f) {
		tmp = 1.0f / ((x * (x / (s * s))) * s);
	} else {
		tmp = 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 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 0.20000000298023224e0) then
        tmp = 1.0e0 / ((x * (x / (s * 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(0.20000000298023224))
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(x / Float32(s * 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(0.20000000298023224))
		tmp = single(1.0) / ((x * (x / (s * 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))))) < 0.200000003

   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. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. distribute-lft-out N/A

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

      5. lower-fma.f32 N/A

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

   7. Applied rewrites 40.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-4, \frac{\left|x\right|}{s} + \frac{x}{s} \cdot \frac{x}{s}, \mathsf{fma}\left(5, \frac{x}{s} \cdot \frac{x}{s}, 4 \cdot \frac{\left|x\right|}{s}\right) + 4\right)} \cdot s} \]

   8. Taylor expanded in s around 0

      \[\leadsto \frac{1}{\frac{-4 \cdot {x}^{2} + 5 \cdot {x}^{2}}{\color{blue}{{s}^{2}}} \cdot s} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.5%

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

   12. Applied rewrites 79.9%

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

   if 0.200000003 < (/.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 93.3

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

   5. Applied rewrites 93.3%

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

Alternative 6: 96.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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 / ((s * (1.0e0 + t_0)) * ((((((x * x) / s) * 0.5e0) - abs(x)) / s) + 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(Float32(Float32(Float32(Float32(Float32(x * x) / s) * Float32(0.5)) - abs(x)) / s) + 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)) * ((((((x * x) / s) * single(0.5)) - abs(x)) / s) + single(2.0)));
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

5. Applied rewrites 97.3%

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

Alternative 7: 95.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	tmp = t_0 / ((s * (single(1.0) + (single(1.0) - (abs(x) / s)))) * (single(1.0) + 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. 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 96.9

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

5. Applied rewrites 96.9%

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

Alternative 8: 94.8% 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

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 / ((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 99.7%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 96.1%

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

Alternative 9: 94.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{e^{\frac{\left|x\right|}{s}} \cdot s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 (* (exp (/ (fabs x) s)) s)))

C

float code(float x, float s) {
	return 0.25f / (expf((fabsf(x) / s)) * s);
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f32 N/A

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

   2. 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. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. inv-pow N/A

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

   7. lower-/.f32 N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in s around inf

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

9. Applied rewrites 95.8%

   \[\leadsto \frac{\color{blue}{0.25}}{e^{\frac{\left|x\right|}{s}} \cdot s} \]
10. Add Preprocessing

Alternative 10: 27.6% 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 31.9

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

5. Applied rewrites 31.9%

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

Reproduce

herbie shell --seed 2024322 
(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))))))