Logistic distribution

Percentage Accurate: 99.5% → 100.0%

Time: 23.8s

Alternatives: 9

Speedup: 1.2×

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: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \langle \left( \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ \frac{t_0}{s \cdot {\left(t_0 + 1\right)}^{2}} \end{array} \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (let* ((t_0 (exp (/ (fabs x) (- s)))))
     (/ t_0 (* s (pow (+ t_0 1.0) 2.0)))))))

C

float code(float x, float s) {
	double t_0_1 = exp((fabs(x) / -((double) s)));
	double tmp = t_0_1 / (((double) s) * pow((t_0_1 + 1.0), 2.0));
	return (float) tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_0_1
    real(8) :: tmp
    t_0_1 = exp((abs(x) / -real(s, 8)))
    tmp = t_0_1 / (real(s, 8) * ((t_0_1 + 1.0d0) ** 2.0d0))
    code = real(tmp, 4)
end function

Julia

function code(x, s)
	t_0_1 = exp(Float64(abs(x) / Float64(-Float64(s))))
	tmp = Float64(t_0_1 / Float64(Float64(s) * (Float64(t_0_1 + 1.0) ^ 2.0)))
	return Float32(tmp)
end

MATLAB

function tmp_3 = code(x, s)
	t_0_2 = exp((abs(x) / -s));
	tmp = t_0_2 / (double(s) * ((t_0_2 + 1.0) ^ 2.0));
	tmp_3 = single(tmp);
end

Derivation

1. Initial program 100.0%

   \[\langle \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)}}{e^{\frac{-\left|x\right|}{s}} + 1} \rangle_{\text{binary64}} \]
2. Step-by-step derivation

   1. clear-num 100.0%

      \[\leadsto \langle \frac{\color{blue}{1}}{\frac{e^{\frac{-\left|x\right|}{s}} + 1}{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)}}} \rangle_{\text{binary64}} \]

   2. div-inv 100.0%

      \[\leadsto \langle \color{blue}{1} \cdot \frac{1}{\frac{e^{\frac{-\left|x\right|}{s}} + 1}{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)}}} \rangle_{\text{binary64}} \]

   3. clear-num 100.0%

      \[\leadsto \langle 1 \cdot \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)}}{e^{\frac{-\left|x\right|}{s}} + 1} \rangle_{\text{binary64}} \]

   4. associate-/l/ 100.0%

      \[\leadsto \langle 1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)} \rangle_{\text{binary64}} \]

3. Applied egg-rr 100.0%

   \[\leadsto \langle \color{blue}{1} \cdot \frac{e^{\frac{\left|x\right|}{-s}}}{{\left(e^{\frac{\left|x\right|}{-s}} + 1\right)}^{2} \cdot s} \rangle_{\text{binary64}} \]
4. Step-by-step derivation

   1. *-lft-identity 100.0%

      \[\leadsto \langle \frac{\color{blue}{e^{\frac{\left|x\right|}{-s}}}}{{\left(e^{\frac{\left|x\right|}{-s}} + 1\right)}^{2} \cdot s} \rangle_{\text{binary64}} \]

   2. *-commutative 100.0%

      \[\leadsto \langle \frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot {\left(e^{\frac{\left|x\right|}{-s}} + 1\right)}^{2}} \rangle_{\text{binary64}} \]

   3. +-commutative 100.0%

      \[\leadsto \langle \frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot {\left(1 + e^{\frac{\left|x\right|}{-s}}\right)}^{2}} \rangle_{\text{binary64}} \]

5. Simplified 100.0%

   \[\leadsto \langle \frac{\color{blue}{e^{\frac{\left|x\right|}{-s}}}}{s \cdot {\left(1 + e^{\frac{\left|x\right|}{-s}}\right)}^{2}} \rangle_{\text{binary64}} \]

6. Final simplification 100.0%

   \[\leadsto \langle \frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot {\left(e^{\frac{\left|x\right|}{-s}} + 1\right)}^{2}} \rangle_{\text{binary64}} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-/r* 99.8%

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

   3. *-lft-identity 99.8%

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

   4. associate-*l* 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

3. Simplified 99.8%

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

   1. clear-num 99.8%

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

   2. div-inv 99.8%

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

   3. clear-num 99.8%

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

   4. associate-/r* 99.8%

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

   5. div-inv 99.8%

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

   6. frac-2neg 99.8%

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

   7. remove-double-neg 99.8%

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

   8. pow2 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

7. Simplified 99.8%

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

   1. pow-to-exp 99.8%

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

   2. *-commutative 99.8%

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

   3. log1p-def 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 3: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / (s * powf((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(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 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-/r* 99.8%

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

   3. *-lft-identity 99.8%

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

   4. associate-*l* 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 99.8%

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

   1. unpow2 99.8%

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

   2. mul-1-neg 99.8%

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

   3. distribute-frac-neg 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. rec-exp 99.8%

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

   6. mul-1-neg 99.8%

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

   7. distribute-frac-neg 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-/r* 99.8%

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

   3. *-lft-identity 99.8%

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

   4. associate-*l* 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

3. Simplified 99.8%

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

   1. associate-/r* 99.8%

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

   2. frac-2neg 99.8%

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

   3. distribute-frac-neg 99.8%

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

   4. frac-2neg 99.8%

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

   5. remove-double-neg 99.8%

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

   6. pow2 99.8%

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

5. Applied egg-rr 99.8%

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

   1. distribute-neg-frac 99.8%

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

7. Simplified 99.8%

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

   1. frac-2neg 99.8%

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

   2. div-inv 99.8%

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

   3. clear-num 99.8%

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

   4. pow-flip 99.7%

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

   5. metadata-eval 99.7%

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

   6. associate-*l/ 99.8%

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

   7. div-inv 99.8%

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

   8. frac-2neg 99.8%

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

   9. remove-double-neg 99.8%

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

   10. times-frac 99.4%

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

9. Applied egg-rr 99.5%

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

    1. add-exp-log_binary32 97.7%

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

11. Applied rewrite-once 97.7%

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

    1. rem-exp-log 99.5%

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

    2. associate-*l/ 99.8%

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

    3. *-lft-identity 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 5: 95.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. associate-/r* 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 95.8%

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

   1. clear-num 95.8%

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

   2. div-inv 95.8%

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

   3. associate-/r/ 95.4%

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

   4. associate-/r* 95.4%

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

   5. distribute-frac-neg 95.4%

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

   6. exp-neg 95.4%

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

   7. frac-times 95.8%

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

   8. metadata-eval 95.8%

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

   9. metadata-eval 95.8%

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

6. Applied egg-rr 95.8%

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

   1. *-lft-identity 95.8%

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

   2. associate-/r* 95.4%

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

8. Simplified 95.4%

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

9. Taylor expanded in s around 0 95.8%

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

10. Final simplification 95.8%

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

Alternative 6: 95.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-/r* 99.8%

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

   3. *-lft-identity 99.8%

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

   4. associate-*l* 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 95.7%

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

5. Final simplification 95.7%

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

Alternative 7: 94.7% accurate, 3.0× speedup

Math

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

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(abs(x) / s)))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-/r* 99.8%

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

   3. *-lft-identity 99.8%

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

   4. associate-*l* 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 95.5%

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

   1. *-lft-identity 95.5%

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

   2. *-commutative 95.5%

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

   3. div-inv 95.5%

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

   4. *-commutative 95.5%

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

   5. *-commutative 95.5%

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

   6. associate-/r* 95.5%

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

   7. metadata-eval 95.5%

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

   8. distribute-frac-neg 95.5%

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

   9. exp-neg 95.5%

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

   10. un-div-inv 95.5%

       \[\leadsto \color{blue}{\frac{\frac{0.25}{s}}{e^{\frac{\left|x\right|}{s}}}} \cdot 1 \]

6. Applied egg-rr 95.5%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{s}}{e^{\frac{\left|x\right|}{s}}} \cdot 1} \]
7. Step-by-step derivation

   1. *-rgt-identity 95.5%

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

8. Simplified 95.5%

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

9. Final simplification 95.5%

   \[\leadsto \frac{\frac{0.25}{s}}{e^{\frac{\left|x\right|}{s}}} \]

Alternative 8: 50.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{s}}{1 + \frac{\left|x\right|}{s}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ 0.25 s) (+ 1.0 (/ (fabs x) s))))

C

float code(float x, float s) {
	return (0.25f / s) / (1.0f + (fabsf(x) / s));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (0.25e0 / s) / (1.0e0 + (abs(x) / s))
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(0.25) / s) / Float32(Float32(1.0) + Float32(abs(x) / s)))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(0.25) / s) / (single(1.0) + (abs(x) / s));
end

Derivation

1. Initial program 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-/r* 99.8%

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

   3. *-lft-identity 99.8%

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

   4. associate-*l* 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 95.5%

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

   1. *-lft-identity 95.5%

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

   2. *-commutative 95.5%

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

   3. div-inv 95.5%

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

   4. *-commutative 95.5%

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

   5. *-commutative 95.5%

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

   6. associate-/r* 95.5%

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

   7. metadata-eval 95.5%

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

   8. distribute-frac-neg 95.5%

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

   9. exp-neg 95.5%

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

   10. un-div-inv 95.5%

       \[\leadsto \color{blue}{\frac{\frac{0.25}{s}}{e^{\frac{\left|x\right|}{s}}}} \cdot 1 \]

6. Applied egg-rr 95.5%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{s}}{e^{\frac{\left|x\right|}{s}}} \cdot 1} \]
7. Step-by-step derivation

   1. *-rgt-identity 95.5%

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

8. Simplified 95.5%

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

9. Taylor expanded in s around inf 51.0%

   \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{1 + \frac{\left|x\right|}{s}}} \]
10. Step-by-step derivation

    1. +-commutative 51.0%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\frac{\left|x\right|}{s} + 1}} \]

11. Simplified 51.0%

    \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\frac{\left|x\right|}{s} + 1}} \]

12. Final simplification 51.0%

    \[\leadsto \frac{\frac{0.25}{s}}{1 + \frac{\left|x\right|}{s}} \]

Alternative 9: 27.1% accurate, 206.7× 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 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-/r* 99.8%

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

   3. *-lft-identity 99.8%

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

   4. associate-*l* 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 25.7%

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

5. Final simplification 25.7%

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

Reproduce

herbie shell --seed 2023297 
(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))))))