Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 16.9s

Alternatives: 16

Speedup: 2.9×

Specification

\[0 \leq s \land s \leq 1.0651631\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. fabs-neg 99.4%

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

   2. *-commutative 99.4%

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

   3. fabs-neg 99.4%

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

   4. fabs-neg 99.4%

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

   5. associate-*r* 99.4%

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

   6. /-rgt-identity 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 97.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. neg-mul-1 99.4%

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

   3. times-frac 99.4%

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

   4. metadata-eval 99.4%

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

   5. metadata-eval 99.4%

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

   6. times-frac 99.4%

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

   7. neg-mul-1 99.4%

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

   8. *-un-lft-identity 99.4%

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

   9. distribute-frac-neg 99.4%

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

   10. rec-exp 99.4%

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

   11. add-sqr-sqrt 51.1%

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

   12. fabs-sqr 51.1%

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

   13. add-sqr-sqrt 96.7%

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

5. Applied egg-rr 96.7%

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

6. Final simplification 96.7%

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

Alternative 3: 99.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. neg-mul-1 99.4%

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

   3. times-frac 99.4%

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

   4. metadata-eval 99.4%

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

   5. metadata-eval 99.4%

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

   6. times-frac 99.4%

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

   7. neg-mul-1 99.4%

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

   8. *-un-lft-identity 99.4%

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

   9. distribute-frac-neg 99.4%

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

   10. rec-exp 99.4%

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

   11. add-sqr-sqrt 51.1%

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

   12. fabs-sqr 51.1%

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

   13. add-sqr-sqrt 96.7%

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

5. Applied egg-rr 96.7%

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

   1. associate-/r* 96.7%

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

   2. div-inv 96.7%

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

   3. fma-udef 96.7%

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

   4. +-commutative 96.7%

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

   5. un-div-inv 96.7%

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

   6. add-sqr-sqrt 51.1%

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

   7. fabs-sqr 51.1%

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

   8. add-sqr-sqrt 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in s around 0 99.3%

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

   1. associate-/r* 99.4%

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

   2. rec-exp 99.4%

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

   3. distribute-neg-frac 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 4: 99.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. neg-mul-1 99.4%

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

   3. times-frac 99.4%

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

   4. metadata-eval 99.4%

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

   5. metadata-eval 99.4%

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

   6. times-frac 99.4%

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

   7. neg-mul-1 99.4%

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

   8. *-un-lft-identity 99.4%

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

   9. distribute-frac-neg 99.4%

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

   10. rec-exp 99.4%

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

   11. add-sqr-sqrt 51.1%

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

   12. fabs-sqr 51.1%

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

   13. add-sqr-sqrt 96.7%

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

5. Applied egg-rr 96.7%

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

   1. expm1-log1p-u 95.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{x}{s}}}, s\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}\right)\right)} \]

   2. expm1-udef 94.3%

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

   3. fma-udef 94.3%

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

   4. +-commutative 94.3%

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

   5. un-div-inv 94.3%

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

   6. add-sqr-sqrt 49.4%

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

   7. fabs-sqr 49.4%

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

   8. add-sqr-sqrt 96.6%

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

7. Applied egg-rr 96.6%

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

   1. expm1-def 97.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left(s + \frac{s}{e^{\frac{x}{s}}}\right) \cdot \left(1 + e^{\frac{x}{s}}\right)}\right)\right)} \]

   2. expm1-log1p 99.4%

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

9. Simplified 99.4%

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

10. Final simplification 99.4%

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

Alternative 5: 99.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{\frac{1}{1 + t_0}}{s + \frac{s}{t_0}} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. neg-mul-1 99.4%

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

   3. times-frac 99.4%

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

   4. metadata-eval 99.4%

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

   5. metadata-eval 99.4%

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

   6. times-frac 99.4%

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

   7. neg-mul-1 99.4%

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

   8. *-un-lft-identity 99.4%

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

   9. distribute-frac-neg 99.4%

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

   10. rec-exp 99.4%

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

   11. add-sqr-sqrt 51.1%

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

   12. fabs-sqr 51.1%

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

   13. add-sqr-sqrt 96.7%

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

5. Applied egg-rr 96.7%

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

   1. associate-/r* 96.7%

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

   2. div-inv 96.7%

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

   3. fma-udef 96.7%

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

   4. +-commutative 96.7%

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

   5. un-div-inv 96.7%

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

   6. add-sqr-sqrt 51.1%

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

   7. fabs-sqr 51.1%

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

   8. add-sqr-sqrt 99.4%

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

7. Applied egg-rr 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-un-lft-identity 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 6: 99.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{\frac{1}{s + \frac{s}{t_0}}}{1 + t_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. neg-mul-1 99.4%

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

   3. times-frac 99.4%

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

   4. metadata-eval 99.4%

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

   5. metadata-eval 99.4%

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

   6. times-frac 99.4%

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

   7. neg-mul-1 99.4%

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

   8. *-un-lft-identity 99.4%

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

   9. distribute-frac-neg 99.4%

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

   10. rec-exp 99.4%

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

   11. add-sqr-sqrt 51.1%

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

   12. fabs-sqr 51.1%

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

   13. add-sqr-sqrt 96.7%

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

5. Applied egg-rr 96.7%

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

   1. associate-/r* 96.7%

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

   2. div-inv 96.7%

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

   3. fma-udef 96.7%

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

   4. +-commutative 96.7%

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

   5. un-div-inv 96.7%

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

   6. add-sqr-sqrt 51.1%

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

   7. fabs-sqr 51.1%

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

   8. add-sqr-sqrt 99.4%

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

7. Applied egg-rr 99.4%

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

   1. un-div-inv 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 7: 62.4% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. neg-mul-1 99.4%

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

   3. times-frac 99.4%

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

   4. metadata-eval 99.4%

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

   5. metadata-eval 99.4%

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

   6. times-frac 99.4%

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

   7. neg-mul-1 99.4%

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

   8. *-un-lft-identity 99.4%

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

   9. distribute-frac-neg 99.4%

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

   10. rec-exp 99.4%

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

   11. add-sqr-sqrt 51.1%

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

   12. fabs-sqr 51.1%

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

   13. add-sqr-sqrt 96.7%

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

5. Applied egg-rr 96.7%

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

   1. associate-/r* 96.7%

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

   2. div-inv 96.7%

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

   3. fma-udef 96.7%

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

   4. +-commutative 96.7%

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

   5. un-div-inv 96.7%

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

   6. add-sqr-sqrt 51.1%

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

   7. fabs-sqr 51.1%

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

   8. add-sqr-sqrt 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in x around 0 63.0%

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

9. Final simplification 63.0%

   \[\leadsto \frac{1}{1 + e^{\frac{x}{s}}} \cdot \frac{1}{s + \frac{s}{1 + \frac{x}{s}}} \]

Alternative 8: 63.8% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in s around inf 95.1%

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

   1. +-commutative 95.1%

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

   2. mul-1-neg 95.1%

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

   3. unsub-neg 95.1%

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

   4. *-commutative 95.1%

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

6. Simplified 95.1%

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

   1. distribute-rgt-in 95.1%

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

   2. *-un-lft-identity 95.1%

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

   3. add-sqr-sqrt 48.9%

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

   4. fabs-sqr 48.9%

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

   5. add-sqr-sqrt 94.6%

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

   6. div-inv 94.6%

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

   7. exp-prod 80.1%

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

   8. add-sqr-sqrt 42.1%

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

   9. fabs-sqr 42.1%

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

   10. add-sqr-sqrt 56.9%

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

   11. exp-prod 63.2%

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

   12. div-inv 63.2%

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

   13. add-sqr-sqrt 48.9%

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

   14. fabs-sqr 48.9%

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

   15. add-sqr-sqrt 63.8%

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

8. Applied egg-rr 63.8%

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

9. Final simplification 63.8%

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

Alternative 9: 62.5% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. neg-mul-1 99.4%

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

   3. times-frac 99.4%

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

   4. metadata-eval 99.4%

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

   5. metadata-eval 99.4%

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

   6. times-frac 99.4%

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

   7. neg-mul-1 99.4%

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

   8. *-un-lft-identity 99.4%

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

   9. distribute-frac-neg 99.4%

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

   10. rec-exp 99.4%

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

   11. add-sqr-sqrt 51.1%

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

   12. fabs-sqr 51.1%

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

   13. add-sqr-sqrt 96.7%

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

5. Applied egg-rr 96.7%

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

   1. associate-/r* 96.7%

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

   2. div-inv 96.7%

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

   3. fma-udef 96.7%

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

   4. +-commutative 96.7%

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

   5. un-div-inv 96.7%

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

   6. add-sqr-sqrt 51.1%

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

   7. fabs-sqr 51.1%

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

   8. add-sqr-sqrt 99.4%

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

7. Applied egg-rr 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-un-lft-identity 99.4%

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

9. Applied egg-rr 99.4%

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

10. Taylor expanded in x around 0 63.0%

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

11. Final simplification 63.0%

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

Alternative 10: 63.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{\frac{x}{s}}} \cdot \frac{1}{s \cdot 2 - x} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in s around inf 95.1%

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

   1. +-commutative 95.1%

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

   2. mul-1-neg 95.1%

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

   3. unsub-neg 95.1%

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

   4. *-commutative 95.1%

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

6. Simplified 95.1%

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

   1. inv-pow 95.1%

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

   2. unpow-prod-down 95.1%

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

   3. inv-pow 95.1%

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

   4. add-sqr-sqrt 48.9%

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

   5. fabs-sqr 48.9%

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

   6. add-sqr-sqrt 94.7%

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

   7. inv-pow 94.7%

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

   8. frac-2neg 94.7%

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

   9. add-log-exp 94.5%

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

   10. neg-log 94.5%

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

   11. exp-sum 93.9%

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

   12. div-inv 93.9%

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

   13. exp-prod 79.2%

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

   14. add-sqr-sqrt 41.9%

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

   15. fabs-sqr 41.9%

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

   16. add-sqr-sqrt 56.0%

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

8. Applied egg-rr 63.8%

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

9. Final simplification 63.8%

   \[\leadsto \frac{1}{1 + e^{\frac{x}{s}}} \cdot \frac{1}{s \cdot 2 - x} \]

Alternative 11: 60.9% accurate, 5.7× speedup

Math

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

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(0.5) / Float32(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 99.4%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. neg-mul-1 99.4%

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

   3. times-frac 99.4%

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

   4. metadata-eval 99.4%

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

   5. metadata-eval 99.4%

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

   6. times-frac 99.4%

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

   7. neg-mul-1 99.4%

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

   8. *-un-lft-identity 99.4%

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

   9. distribute-frac-neg 99.4%

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

   10. rec-exp 99.4%

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

   11. add-sqr-sqrt 51.1%

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

   12. fabs-sqr 51.1%

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

   13. add-sqr-sqrt 96.7%

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

5. Applied egg-rr 96.7%

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

6. Taylor expanded in x around 0 94.7%

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

   1. associate-/r* 94.7%

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

8. Simplified 94.7%

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

   1. expm1-log1p-u 93.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.5}{s}}{1 + e^{\frac{\left|x\right|}{s}}}\right)\right)} \]

   2. expm1-udef 93.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0.5}{s}}{1 + e^{\frac{\left|x\right|}{s}}}\right)} - 1} \]

   3. associate-/l/ 93.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.5}{\left(1 + e^{\frac{\left|x\right|}{s}}\right) \cdot s}}\right)} - 1 \]

   4. div-inv 93.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{\left(1 + e^{\color{blue}{\left|x\right| \cdot \frac{1}{s}}}\right) \cdot s}\right)} - 1 \]

   5. exp-prod 79.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{\left(1 + \color{blue}{{\left(e^{\left|x\right|}\right)}^{\left(\frac{1}{s}\right)}}\right) \cdot s}\right)} - 1 \]

   6. add-sqr-sqrt 42.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{\left(1 + {\left(e^{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)}^{\left(\frac{1}{s}\right)}\right) \cdot s}\right)} - 1 \]

   7. fabs-sqr 42.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{\left(1 + {\left(e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}^{\left(\frac{1}{s}\right)}\right) \cdot s}\right)} - 1 \]

   8. add-sqr-sqrt 55.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{\left(1 + {\left(e^{\color{blue}{x}}\right)}^{\left(\frac{1}{s}\right)}\right) \cdot s}\right)} - 1 \]

   9. exp-prod 60.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{\left(1 + \color{blue}{e^{x \cdot \frac{1}{s}}}\right) \cdot s}\right)} - 1 \]

   10. div-inv 60.3%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{\left(1 + e^{\color{blue}{\frac{x}{s}}}\right) \cdot s}\right)} - 1 \]

10. Applied egg-rr 60.3%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5}{\left(1 + e^{\frac{x}{s}}\right) \cdot s}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 60.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{\left(1 + e^{\frac{x}{s}}\right) \cdot s}\right)\right)} \]

    2. expm1-log1p 61.6%

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

    3. *-commutative 61.6%

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

12. Simplified 61.6%

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

13. Final simplification 61.6%

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

Alternative 12: 60.9% 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 99.4%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. neg-mul-1 99.4%

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

   3. times-frac 99.4%

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

   4. metadata-eval 99.4%

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

   5. metadata-eval 99.4%

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

   6. times-frac 99.4%

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

   7. neg-mul-1 99.4%

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

   8. *-un-lft-identity 99.4%

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

   9. distribute-frac-neg 99.4%

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

   10. rec-exp 99.4%

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

   11. add-sqr-sqrt 51.1%

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

   12. fabs-sqr 51.1%

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

   13. add-sqr-sqrt 96.7%

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

5. Applied egg-rr 96.7%

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

6. Taylor expanded in x around 0 94.7%

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

   1. associate-/r* 94.7%

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

8. Simplified 94.7%

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

   1. div-inv 94.7%

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

   2. frac-2neg 94.7%

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

   3. add-log-exp 94.0%

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

   4. neg-log 94.0%

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

   5. exp-sum 93.4%

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

   6. div-inv 93.4%

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

   7. exp-prod 80.3%

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

   8. add-sqr-sqrt 42.6%

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

   9. fabs-sqr 42.6%

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

   10. add-sqr-sqrt 55.7%

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

   11. exp-prod 60.7%

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

   12. div-inv 60.7%

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

   13. exp-sum 61.2%

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

10. Applied egg-rr 61.6%

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

    1. associate-*r/ 61.6%

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

    2. *-rgt-identity 61.6%

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

12. Simplified 61.6%

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

13. Final simplification 61.6%

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

Alternative 13: 64.6% accurate, 56.4× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / ((s * 4.0f) + ((x * 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 * x) / s))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in s around -inf 38.9%

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

   1. +-commutative 38.9%

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

   2. mul-1-neg 38.9%

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

   3. distribute-lft1-in 63.6%

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

   4. metadata-eval 63.6%

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

   5. associate-*r/ 63.6%

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

   6. mul-1-neg 63.6%

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

   7. remove-double-neg 63.6%

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

   8. associate-+r+ 63.6%

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

6. Simplified 63.6%

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

7. Final simplification 63.6%

   \[\leadsto \frac{1}{s \cdot 4 + \frac{x \cdot x}{s}} \]

Alternative 14: 29.2% accurate, 68.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot 2 - x} \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in s around inf 95.1%

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

   1. +-commutative 95.1%

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

   2. mul-1-neg 95.1%

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

   3. unsub-neg 95.1%

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

   4. *-commutative 95.1%

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

6. Simplified 95.1%

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

   1. inv-pow 95.1%

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

   2. unpow-prod-down 95.1%

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

   3. inv-pow 95.1%

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

   4. add-sqr-sqrt 48.9%

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

   5. fabs-sqr 48.9%

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

   6. add-sqr-sqrt 94.7%

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

   7. inv-pow 94.7%

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

   8. frac-2neg 94.7%

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

   9. add-log-exp 94.5%

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

   10. neg-log 94.5%

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

   11. exp-sum 93.9%

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

   12. div-inv 93.9%

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

   13. exp-prod 79.2%

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

   14. add-sqr-sqrt 41.9%

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

   15. fabs-sqr 41.9%

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

   16. add-sqr-sqrt 56.0%

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

8. Applied egg-rr 63.8%

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

9. Taylor expanded in x around 0 25.8%

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

10. Final simplification 25.8%

    \[\leadsto \frac{1}{s \cdot 2 - x} \cdot 0.5 \]

Alternative 15: 27.4% 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 99.4%

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

   1. fabs-neg 99.4%

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

   2. *-commutative 99.4%

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

   3. fabs-neg 99.4%

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

   4. fabs-neg 99.4%

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

   5. associate-*r* 99.4%

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

   6. /-rgt-identity 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in s around inf 23.6%

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

5. Final simplification 23.6%

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

Alternative 16: 8.3% accurate, 620.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 1.0)

C

float code(float x, float s) {
	return 1.0f;
}

Fortran

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

Julia

function code(x, s)
	return Float32(1.0)
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.4%

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

   1. add-exp-log 97.7%

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

   2. log-div 97.7%

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

   3. add-log-exp 97.9%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 19.9%

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

   6. sqr-neg 19.9%

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

   7. sqrt-unprod 19.8%

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

   8. add-sqr-sqrt 19.8%

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

   9. add-sqr-sqrt 8.2%

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

   10. fabs-sqr 8.2%

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

   11. add-sqr-sqrt 55.5%

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

   12. associate-*l* 55.5%

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

3. Applied egg-rr 83.4%

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

4. Taylor expanded in x around inf 38.8%

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

5. Taylor expanded in x around 0 8.1%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 8.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023336 
(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))))))