Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 16.7s

Alternatives: 13

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s)))) (t_1 (+ t_0 1.0)))
   (/ t_0 (* s (* t_1 t_1)))))

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	float t_1 = t_0 + 1.0f;
	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 = t_0 + 1.0e0
    code = t_0 / (s * (t_1 * t_1))
end function

Julia

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

MATLAB

function tmp = code(x, s)
	t_0 = exp((abs(x) / -s));
	t_1 = t_0 + single(1.0);
	tmp = t_0 / (s * (t_1 * t_1));
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. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{e^{\mathsf{fma}\left(x, \frac{1}{s}, -\mathsf{log1p}\left(t\_0\right)\right)}}{s + s \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (/ (exp (fma x (/ 1.0 s) (- (log1p t_0)))) (+ s (* s t_0)))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	return expf(fmaf(x, (1.0f / s), -log1pf(t_0))) / (s + (s * t_0));
}

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	return Float32(exp(fma(x, Float32(Float32(1.0) / s), Float32(-log1p(t_0)))) / Float32(s + Float32(s * 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. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.9%

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

6. Applied egg-rr 88.2%

   \[\leadsto \color{blue}{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot \frac{1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ 88.1%

      \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot 1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]

   2. *-rgt-identity 88.1%

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

8. Simplified 88.1%

   \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]
9. Step-by-step derivation

   1. div-inv 88.1%

      \[\leadsto \frac{e^{\color{blue}{x \cdot \frac{1}{s}} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

   2. fma-neg 99.8%

      \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(x, \frac{1}{s}, -\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

10. Applied egg-rr 99.8%

    \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(x, \frac{1}{s}, -\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]
11. Step-by-step derivation

    1. fma-undefine 99.9%

       \[\leadsto \frac{e^{\mathsf{fma}\left(x, \frac{1}{s}, -\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}}{\color{blue}{s \cdot e^{\frac{x}{s}} + s}} \]

12. Applied egg-rr 99.9%

    \[\leadsto \frac{e^{\mathsf{fma}\left(x, \frac{1}{s}, -\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}}{\color{blue}{s \cdot e^{\frac{x}{s}} + s}} \]

13. Final simplification 99.9%

    \[\leadsto \frac{e^{\mathsf{fma}\left(x, \frac{1}{s}, -\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}}{s + s \cdot e^{\frac{x}{s}}} \]
14. Add Preprocessing

Alternative 3: 59.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{s}}{1 + e^{\frac{x}{s}}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = (single(0.5) / s) / (single(1.0) + exp((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. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.9%

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

6. Applied egg-rr 88.2%

   \[\leadsto \color{blue}{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot \frac{1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ 88.1%

      \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot 1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]

   2. *-rgt-identity 88.1%

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

8. Simplified 88.1%

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

9. Taylor expanded in x around 0 63.2%

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

    1. neg-mul-1 63.2%

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

    2. *-commutative 63.2%

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

    3. exp-to-pow 63.2%

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

    4. metadata-eval 63.2%

       \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

11. Simplified 63.2%

    \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

12. Taylor expanded in s around 0 63.2%

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

    1. associate-/r* 63.2%

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

14. Simplified 63.2%

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

15. Final simplification 63.2%

    \[\leadsto \frac{\frac{0.5}{s}}{1 + e^{\frac{x}{s}}} \]
16. Add Preprocessing

Alternative 4: 76.1% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 60000000:\\ \;\;\;\;\frac{\left(0.25 + \frac{0.25 \cdot \left(x \cdot 0.5\right)}{s}\right) + \frac{0.25 \cdot \left(x \cdot -0.5\right)}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{1}{\frac{s}{x} \cdot \frac{s}{x}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 60000000.0)
   (/ (+ (+ 0.25 (/ (* 0.25 (* x 0.5)) s)) (/ (* 0.25 (* x -0.5)) s)) s)
   (/ 1.0 (* s (+ 4.0 (/ 1.0 (* (/ s x) (/ s x))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 60000000.0f) {
		tmp = ((0.25f + ((0.25f * (x * 0.5f)) / s)) + ((0.25f * (x * -0.5f)) / s)) / s;
	} else {
		tmp = 1.0f / (s * (4.0f + (1.0f / ((s / x) * (s / x)))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 60000000.0e0) then
        tmp = ((0.25e0 + ((0.25e0 * (x * 0.5e0)) / s)) + ((0.25e0 * (x * (-0.5e0))) / s)) / s
    else
        tmp = 1.0e0 / (s * (4.0e0 + (1.0e0 / ((s / x) * (s / x)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(60000000.0))
		tmp = Float32(Float32(Float32(Float32(0.25) + Float32(Float32(Float32(0.25) * Float32(x * Float32(0.5))) / s)) + Float32(Float32(Float32(0.25) * Float32(x * Float32(-0.5))) / s)) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(Float32(1.0) / Float32(Float32(s / x) * Float32(s / x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(60000000.0))
		tmp = ((single(0.25) + ((single(0.25) * (x * single(0.5))) / s)) + ((single(0.25) * (x * single(-0.5))) / s)) / s;
	else
		tmp = single(1.0) / (s * (single(4.0) + (single(1.0) / ((s / x) * (s / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 6e7

   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. fabs-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-frac-neg2 99.8%

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

      4. fabs-neg 99.8%

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

      5. *-commutative 99.8%

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

      6. fabs-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. fabs-neg 99.8%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot \frac{1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 99.3%

         \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot 1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]

      2. *-rgt-identity 99.3%

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

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]
   9. Step-by-step derivation

      1. div-inv 99.3%

         \[\leadsto \frac{e^{\color{blue}{x \cdot \frac{1}{s}} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

      2. fma-neg 99.8%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(x, \frac{1}{s}, -\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

   10. Applied egg-rr 99.8%

       \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(x, \frac{1}{s}, -\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]
   11. Step-by-step derivation

       1. fma-undefine 99.8%

          \[\leadsto \frac{e^{\mathsf{fma}\left(x, \frac{1}{s}, -\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}}{\color{blue}{s \cdot e^{\frac{x}{s}} + s}} \]

   12. Applied egg-rr 99.8%

       \[\leadsto \frac{e^{\mathsf{fma}\left(x, \frac{1}{s}, -\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}}{\color{blue}{s \cdot e^{\frac{x}{s}} + s}} \]

   13. Taylor expanded in s around inf 74.6%

       \[\leadsto \color{blue}{\frac{\left(0.5 \cdot e^{-\log 2} + 0.5 \cdot \frac{e^{-\log 2} \cdot \left(x - 0.5 \cdot x\right)}{s}\right) - 0.25 \cdot \frac{x \cdot e^{-\log 2}}{s}}{s}} \]
   14. Step-by-step derivation

   15. Simplified 74.6%

       \[\leadsto \color{blue}{\frac{\left(0.25 + \frac{\left(0.5 \cdot x\right) \cdot 0.25}{s}\right) + \frac{0.25 \cdot \left(x \cdot -0.5\right)}{s}}{s}} \]

   if 6e7 < x

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   6. Applied egg-rr -0.0%

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

      1. unpow-1 -0.0%

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

   8. Simplified -0.0%

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

   9. Taylor expanded in x around 0 98.4%

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

       1. unpow2 98.4%

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

       2. unpow2 98.4%

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

       3. times-frac 98.4%

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

       4. unpow2 98.4%

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

   11. Simplified 98.4%

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

       1. unpow2 98.4%

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

       2. clear-num 98.4%

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

       3. clear-num 98.4%

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

       4. frac-times 98.4%

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

       5. metadata-eval 98.4%

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

   13. Applied egg-rr 98.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 60000000:\\ \;\;\;\;\frac{\left(0.25 + \frac{0.25 \cdot \left(x \cdot 0.5\right)}{s}\right) + \frac{0.25 \cdot \left(x \cdot -0.5\right)}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{1}{\frac{s}{x} \cdot \frac{s}{x}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.1% accurate, 41.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * (single(4.0) + (x * ((single(1.0) / s) * (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. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.9%

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

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

6. Applied egg-rr 60.0%

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

   1. unpow-1 60.0%

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

8. Simplified 60.0%

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

9. Taylor expanded in x around 0 77.6%

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

    1. unpow2 77.6%

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

    2. unpow2 77.6%

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

    3. times-frac 75.2%

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

    4. unpow2 75.2%

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

11. Simplified 75.2%

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

    1. unpow2 75.2%

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

    2. div-inv 75.2%

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

    3. associate-*l* 81.9%

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

13. Applied egg-rr 81.9%

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

14. Final simplification 81.9%

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

Alternative 6: 64.9% accurate, 47.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s \cdot 2 + x \cdot \left(x \cdot \frac{0.5}{s}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (+ (* s 2.0) (* x (* x (/ 0.5 s))))))

C

float code(float x, float s) {
	return 0.5f / ((s * 2.0f) + (x * (x * (0.5f / s))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / ((s * 2.0e0) + (x * (x * (0.5e0 / s))))
end function

Julia

function code(x, s)
	return Float32(Float32(0.5) / Float32(Float32(s * Float32(2.0)) + Float32(x * Float32(x * Float32(Float32(0.5) / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / ((s * single(2.0)) + (x * (x * (single(0.5) / 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. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.9%

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

6. Applied egg-rr 88.2%

   \[\leadsto \color{blue}{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot \frac{1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ 88.1%

      \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot 1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]

   2. *-rgt-identity 88.1%

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

8. Simplified 88.1%

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

9. Taylor expanded in x around 0 63.2%

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

    1. neg-mul-1 63.2%

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

    2. *-commutative 63.2%

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

    3. exp-to-pow 63.2%

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

    4. metadata-eval 63.2%

       \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

11. Simplified 63.2%

    \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

12. Taylor expanded in x around 0 61.3%

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

13. Taylor expanded in x around inf 63.3%

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

    1. associate-*r/ 63.3%

       \[\leadsto \frac{0.5}{2 \cdot s + x \cdot \color{blue}{\frac{0.5 \cdot x}{s}}} \]

    2. metadata-eval 63.3%

       \[\leadsto \frac{0.5}{2 \cdot s + x \cdot \frac{\color{blue}{\frac{1}{2}} \cdot x}{s}} \]

    3. rem-exp-log 63.3%

       \[\leadsto \frac{0.5}{2 \cdot s + x \cdot \frac{\frac{1}{\color{blue}{e^{\log 2}}} \cdot x}{s}} \]

    4. exp-neg 63.3%

       \[\leadsto \frac{0.5}{2 \cdot s + x \cdot \frac{\color{blue}{e^{-\log 2}} \cdot x}{s}} \]

    5. *-commutative 63.3%

       \[\leadsto \frac{0.5}{2 \cdot s + x \cdot \frac{\color{blue}{x \cdot e^{-\log 2}}}{s}} \]

    6. associate-/l* 63.3%

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

    7. exp-neg 63.3%

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

    8. rem-exp-log 63.3%

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

    9. metadata-eval 63.3%

       \[\leadsto \frac{0.5}{2 \cdot s + x \cdot \left(x \cdot \frac{\color{blue}{0.5}}{s}\right)} \]

15. Simplified 63.3%

    \[\leadsto \frac{0.5}{2 \cdot s + x \cdot \color{blue}{\left(x \cdot \frac{0.5}{s}\right)}} \]

16. Final simplification 63.3%

    \[\leadsto \frac{0.5}{s \cdot 2 + x \cdot \left(x \cdot \frac{0.5}{s}\right)} \]
17. Add Preprocessing

Alternative 7: 64.8% accurate, 47.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s \cdot 2 + x \cdot \frac{x \cdot 0.5}{s}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (+ (* s 2.0) (* x (/ (* x 0.5) s)))))

C

float code(float x, float s) {
	return 0.5f / ((s * 2.0f) + (x * ((x * 0.5f) / s)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / ((s * 2.0e0) + (x * ((x * 0.5e0) / s)))
end function

Julia

function code(x, s)
	return Float32(Float32(0.5) / Float32(Float32(s * Float32(2.0)) + Float32(x * Float32(Float32(x * Float32(0.5)) / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / ((s * single(2.0)) + (x * ((x * single(0.5)) / 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. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.9%

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

6. Applied egg-rr 88.2%

   \[\leadsto \color{blue}{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot \frac{1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ 88.1%

      \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot 1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]

   2. *-rgt-identity 88.1%

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

8. Simplified 88.1%

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

9. Taylor expanded in x around 0 63.2%

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

    1. neg-mul-1 63.2%

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

    2. *-commutative 63.2%

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

    3. exp-to-pow 63.2%

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

    4. metadata-eval 63.2%

       \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

11. Simplified 63.2%

    \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

12. Taylor expanded in x around 0 61.3%

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

13. Taylor expanded in x around inf 63.3%

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

    1. associate-*r/ 63.3%

       \[\leadsto \frac{0.5}{2 \cdot s + x \cdot \color{blue}{\frac{0.5 \cdot x}{s}}} \]

15. Simplified 63.3%

    \[\leadsto \frac{0.5}{2 \cdot s + x \cdot \color{blue}{\frac{0.5 \cdot x}{s}}} \]

16. Final simplification 63.3%

    \[\leadsto \frac{0.5}{s \cdot 2 + x \cdot \frac{x \cdot 0.5}{s}} \]
17. Add Preprocessing

Alternative 8: 76.4% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * (single(4.0) + ((x / s) * (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. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.9%

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

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

6. Applied egg-rr 60.0%

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

   1. unpow-1 60.0%

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

8. Simplified 60.0%

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

9. Taylor expanded in x around 0 77.6%

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

    1. unpow2 77.6%

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

    2. unpow2 77.6%

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

    3. times-frac 75.2%

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

    4. unpow2 75.2%

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

11. Simplified 75.2%

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

    1. unpow2 75.2%

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

13. Applied egg-rr 75.2%

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

14. Final simplification 75.2%

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

Alternative 9: 78.7% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * (single(4.0) + (x / (s * (s / x)))));
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. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.9%

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

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

6. Applied egg-rr 60.0%

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

   1. unpow-1 60.0%

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

8. Simplified 60.0%

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

9. Taylor expanded in x around 0 77.6%

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

    1. unpow2 77.6%

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

    2. unpow2 77.6%

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

    3. times-frac 75.2%

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

    4. unpow2 75.2%

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

11. Simplified 75.2%

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

    1. unpow2 75.2%

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

    2. clear-num 75.2%

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

    3. frac-times 77.8%

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

    4. *-un-lft-identity 77.8%

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

13. Applied egg-rr 77.8%

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

14. Final simplification 77.8%

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

Alternative 10: 49.6% accurate, 68.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s \cdot \left(\frac{x}{s} + 2\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (* s (+ (/ x s) 2.0))))

C

float code(float x, float s) {
	return 0.5f / (s * ((x / s) + 2.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / (s * ((x / s) + 2.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(0.5) / Float32(s * Float32(Float32(x / s) + Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / (s * ((x / s) + 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. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.9%

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

6. Applied egg-rr 88.2%

   \[\leadsto \color{blue}{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot \frac{1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ 88.1%

      \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot 1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]

   2. *-rgt-identity 88.1%

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

8. Simplified 88.1%

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

9. Taylor expanded in x around 0 63.2%

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

    1. neg-mul-1 63.2%

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

    2. *-commutative 63.2%

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

    3. exp-to-pow 63.2%

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

    4. metadata-eval 63.2%

       \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

11. Simplified 63.2%

    \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

12. Taylor expanded in s around inf 47.4%

    \[\leadsto \frac{0.5}{\color{blue}{s \cdot \left(2 + \frac{x}{s}\right)}} \]

13. Final simplification 47.4%

    \[\leadsto \frac{0.5}{s \cdot \left(\frac{x}{s} + 2\right)} \]
14. Add Preprocessing

Alternative 11: 28.2% accurate, 77.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999987376214e-7) (/ 0.25 s) (/ 0.5 x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999987376214e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.5f / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.999999987376214e-7) then
        tmp = 0.25e0 / s
    else
        tmp = 0.5e0 / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.999999987376214e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999987376214e-7))
		tmp = single(0.25) / s;
	else
		tmp = single(0.5) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999e-7

   1. Initial program 99.7%

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

      1. fabs-neg 99.7%

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

      2. distribute-frac-neg 99.7%

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

      3. distribute-frac-neg2 99.7%

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

      4. fabs-neg 99.7%

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

      5. *-commutative 99.7%

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

      6. fabs-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. fabs-neg 99.7%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in s around inf 35.6%

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

   if 4.99999999e-7 < x

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 60.5%

      \[\leadsto \color{blue}{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot \frac{1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 60.5%

         \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot 1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]

      2. *-rgt-identity 60.5%

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

   8. Simplified 60.5%

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

   9. Taylor expanded in x around 0 100.0%

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

       1. neg-mul-1 100.0%

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

       2. *-commutative 100.0%

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

       3. exp-to-pow 100.0%

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

       4. metadata-eval 100.0%

          \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

   11. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

   12. Taylor expanded in x around 0 11.1%

       \[\leadsto \frac{0.5}{\color{blue}{x + 2 \cdot s}} \]

   13. Taylor expanded in x around inf 11.1%

       \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 28.6% accurate, 88.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x + s \cdot 2} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (+ x (* s 2.0))))

C

float code(float x, float s) {
	return 0.5f / (x + (s * 2.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / (x + (s * 2.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(0.5) / Float32(x + Float32(s * Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / (x + (s * 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. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.9%

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

6. Applied egg-rr 88.2%

   \[\leadsto \color{blue}{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot \frac{1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ 88.1%

      \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot 1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}} \]

   2. *-rgt-identity 88.1%

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

8. Simplified 88.1%

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

9. Taylor expanded in x around 0 63.2%

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

    1. neg-mul-1 63.2%

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

    2. *-commutative 63.2%

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

    3. exp-to-pow 63.2%

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

    4. metadata-eval 63.2%

       \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

11. Simplified 63.2%

    \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

12. Taylor expanded in x around 0 28.8%

    \[\leadsto \frac{0.5}{\color{blue}{x + 2 \cdot s}} \]

13. Final simplification 28.8%

    \[\leadsto \frac{0.5}{x + s \cdot 2} \]
14. Add Preprocessing

Alternative 13: 26.7% 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. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in s around inf 26.4%

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

6. Final simplification 26.4%

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

Reproduce

herbie shell --seed 2024066 
(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))))))