Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 9.1s

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: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   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 99.5

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

4. Applied rewrites 99.5%

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

Alternative 2: 30.5% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 0.0)
     (fma (/ 1.0 s) 0.25 0.0)
     (/ (+ (/ (/ (* -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 / ((t_1 * s) * t_1)) <= 0.0f) {
		tmp = fmaf((1.0f / s), 0.25f, 0.0f);
	} else {
		tmp = ((((-0.0625f * (x * x)) / s) / s) + 0.25f) / s;
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. lift-/.f32 N/A

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

      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. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f32 N/A

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

      6. lower-*.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. exp-sum N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

         \[\leadsto e^{\left(-1 \cdot \log \left(\frac{1}{s}\right)\right) \cdot -1} \cdot \frac{1}{4} + -1 \cdot \frac{e^{-1 \cdot \left(\log 4 + -1 \cdot \log \left(\frac{1}{s}\right)\right)} \cdot \left(\color{blue}{0} \cdot \left|x\right|\right)}{s} \]

      10. mul0-lft N/A

          \[\leadsto e^{\left(-1 \cdot \log \left(\frac{1}{s}\right)\right) \cdot -1} \cdot \frac{1}{4} + -1 \cdot \frac{e^{-1 \cdot \left(\log 4 + -1 \cdot \log \left(\frac{1}{s}\right)\right)} \cdot \color{blue}{0}}{s} \]

      11. *-commutative N/A

          \[\leadsto e^{\left(-1 \cdot \log \left(\frac{1}{s}\right)\right) \cdot -1} \cdot \frac{1}{4} + -1 \cdot \frac{\color{blue}{0 \cdot e^{-1 \cdot \left(\log 4 + -1 \cdot \log \left(\frac{1}{s}\right)\right)}}}{s} \]

   7. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{s}, 0.25, 0\right)} \]

   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 99.3%

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

   3. 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 92.7%

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

4. Final simplification 98.0%

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

Alternative 3: 30.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 0.0)
     (fma (/ 1.0 s) 0.25 0.0)
     (/ 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 / ((t_1 * s) * t_1)) <= 0.0f) {
		tmp = fmaf((1.0f / s), 0.25f, 0.0f);
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. lift-/.f32 N/A

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

      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. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f32 N/A

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

      6. lower-*.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. exp-sum N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

         \[\leadsto e^{\left(-1 \cdot \log \left(\frac{1}{s}\right)\right) \cdot -1} \cdot \frac{1}{4} + -1 \cdot \frac{e^{-1 \cdot \left(\log 4 + -1 \cdot \log \left(\frac{1}{s}\right)\right)} \cdot \left(\color{blue}{0} \cdot \left|x\right|\right)}{s} \]

      10. mul0-lft N/A

          \[\leadsto e^{\left(-1 \cdot \log \left(\frac{1}{s}\right)\right) \cdot -1} \cdot \frac{1}{4} + -1 \cdot \frac{e^{-1 \cdot \left(\log 4 + -1 \cdot \log \left(\frac{1}{s}\right)\right)} \cdot \color{blue}{0}}{s} \]

      11. *-commutative N/A

          \[\leadsto e^{\left(-1 \cdot \log \left(\frac{1}{s}\right)\right) \cdot -1} \cdot \frac{1}{4} + -1 \cdot \frac{\color{blue}{0 \cdot e^{-1 \cdot \left(\log 4 + -1 \cdot \log \left(\frac{1}{s}\right)\right)}}}{s} \]

   7. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{s}, 0.25, 0\right)} \]

   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 99.3%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 97.3%

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

Alternative 4: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   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 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

6. Applied rewrites 99.1%

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

Alternative 5: 96.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / ((2.0f - ((fabsf(x) - (0.5f * ((x * x) / s))) / s)) * ((1.0f + t_0) * 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 / ((2.0e0 - ((abs(x) - (0.5e0 * ((x * x) / s))) / s)) * ((1.0e0 + t_0) * s))
end function

Julia

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

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	tmp = t_0 / ((single(2.0) - ((abs(x) - (single(0.5) * ((x * x) / s))) / s)) * ((single(1.0) + t_0) * s));
end

Derivation

1. Initial program 99.5%

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

3. 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 97.8%

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

6. Final simplification 97.8%

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

Alternative 6: 96.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   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 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 97.2

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

7. Applied rewrites 97.2%

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

   1. lift-pow.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 97.2

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

9. Applied rewrites 97.2%

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

Alternative 7: 94.9% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 (/ (/ 1.0 (* 4.0 s)) (exp (/ (fabs x) s))))

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / (single(4.0) * s)) / exp((abs(x) / s));
end

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   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 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 96.0%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. distribute-frac-neg N/A

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

   8. lift-/.f32 N/A

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

   9. exp-neg N/A

      \[\leadsto \frac{1}{4 \cdot s} \cdot \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} \]

   10. remove-double-neg N/A

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

   11. lift-/.f32 N/A

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

   12. distribute-frac-neg N/A

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

   13. lift-neg.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. rec-exp N/A

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

   16. lift-exp.f32 N/A

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

9. Applied rewrites 96.0%

   \[\leadsto \color{blue}{\frac{\frac{1}{4 \cdot s}}{e^{\frac{\left|x\right|}{s}}}} \]
10. Add Preprocessing

Alternative 8: 94.9% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (* (* (exp (/ (fabs x) s)) 4.0) s)))

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((exp((abs(x) / s)) * single(4.0)) * s);
end

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   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 99.5%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 96.0%

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

8. Final simplification 96.0%

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

Alternative 9: 94.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. 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 96.0

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

5. Applied rewrites 96.0%

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

Alternative 10: 94.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   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 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

6. Applied rewrites 99.1%

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

7. Taylor expanded in s around inf

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

9. Applied rewrites 96.0%

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

Alternative 11: 76.5% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(\frac{\frac{x \cdot x}{s}}{s} + 4\right) \cdot s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (* (+ (/ (/ (* x x) s) s) 4.0) s)))

C

float code(float x, float s) {
	return 1.0f / (((((x * x) / s) / s) + 4.0f) * s);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (((((x * x) / s) / s) + 4.0e0) * s)
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(x * x) / s) / s) + Float32(4.0)) * s))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (((((x * x) / s) / s) + single(4.0)) * s);
end

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   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 99.5%

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 76.8%

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

8. Final simplification 76.8%

   \[\leadsto \frac{1}{\left(\frac{\frac{x \cdot x}{s}}{s} + 4\right) \cdot s} \]
9. Add Preprocessing

Alternative 12: 27.1% accurate, 31.1× speedup

Math

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

FPCore

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

C

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

Fortran

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 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 23.7

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

5. Applied rewrites 23.7%

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

6. Final simplification 23.7%

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

Reproduce

herbie shell --seed 2024271 
(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))))))