Logistic distribution

Percentage Accurate: 99.5% → 99.4%

Time: 18.5s

Alternatives: 12

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + e^{\frac{-x}{s}}\right) \cdot \left(s \cdot e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right)}\right)} \end{array} \]

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) * Float32(s * exp(log1p(exp(Float32(x / s)))))))
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.9%

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

   1. fma-udef 98.9%

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

   2. +-commutative 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

   5. add-sqr-sqrt 98.9%

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

   6. sqrt-unprod 93.5%

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

   7. sqr-neg 93.5%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 23.1%

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

   10. distribute-rgt-in 23.2%

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

   11. add-exp-log 22.3%

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

   12. add-exp-log 22.3%

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

   13. log1p-udef 22.3%

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

   14. prod-exp 22.2%

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

6. Applied egg-rr 60.9%

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

   1. exp-sum 61.1%

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

   2. rem-exp-log 62.3%

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

8. Simplified 62.3%

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

   1. distribute-frac-neg 62.3%

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

   2. rec-exp 62.3%

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

   3. pow1 62.3%

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

   4. pow1 62.3%

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

   5. frac-2neg 62.3%

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

   6. frac-2neg 62.3%

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

   7. add-sqr-sqrt 62.3%

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

   8. sqrt-unprod 61.6%

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

   9. sqr-neg 61.6%

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

   10. sqrt-unprod -0.0%

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

   11. add-sqr-sqrt 95.0%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 61.6%

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

   14. sqr-neg 61.6%

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

   15. sqrt-unprod 62.3%

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

   16. add-sqr-sqrt 62.3%

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

   17. add-sqr-sqrt 51.0%

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

   18. fabs-sqr 51.0%

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

   19. add-sqr-sqrt 99.0%

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

10. Applied egg-rr 99.0%

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

    1. rec-exp 99.0%

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

    2. distribute-neg-frac 99.0%

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

12. Simplified 99.0%

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

13. Final simplification 99.0%

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

Alternative 2: 99.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.9%

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

   1. fma-udef 98.9%

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

   2. +-commutative 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

   5. add-sqr-sqrt 98.9%

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

   6. sqrt-unprod 93.5%

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

   7. sqr-neg 93.5%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 23.1%

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

   10. distribute-rgt-in 23.2%

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

   11. *-commutative 23.2%

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

   12. *-un-lft-identity 23.2%

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

6. Applied egg-rr 62.3%

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

   1. distribute-frac-neg 62.3%

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

   2. rec-exp 62.3%

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

   3. pow1 62.3%

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

   4. pow1 62.3%

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

   5. frac-2neg 62.3%

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

   6. frac-2neg 62.3%

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

   7. add-sqr-sqrt 62.3%

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

   8. sqrt-unprod 61.6%

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

   9. sqr-neg 61.6%

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

   10. sqrt-unprod -0.0%

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

   11. add-sqr-sqrt 95.0%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 61.6%

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

   14. sqr-neg 61.6%

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

   15. sqrt-unprod 62.3%

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

   16. add-sqr-sqrt 62.3%

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

   17. add-sqr-sqrt 51.0%

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

   18. fabs-sqr 51.0%

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

   19. add-sqr-sqrt 99.0%

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

8. Applied egg-rr 98.9%

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

   1. rec-exp 99.0%

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

   2. distribute-neg-frac 99.0%

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

10. Simplified 98.9%

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

11. Final simplification 98.9%

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

Alternative 3: 99.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.9%

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

   1. fma-udef 98.9%

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

   2. +-commutative 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

   5. add-sqr-sqrt 98.9%

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

   6. sqrt-unprod 93.5%

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

   7. sqr-neg 93.5%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 23.1%

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

   10. distribute-rgt-in 23.2%

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

   11. *-commutative 23.2%

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

   12. *-un-lft-identity 23.2%

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

6. Applied egg-rr 62.3%

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

   1. distribute-frac-neg 62.3%

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

   2. rec-exp 62.3%

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

   3. pow1 62.3%

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

   4. pow1 62.3%

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

   5. frac-2neg 62.3%

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

   6. frac-2neg 62.3%

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

   7. add-sqr-sqrt 62.3%

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

   8. sqrt-unprod 61.6%

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

   9. sqr-neg 61.6%

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

   10. sqrt-unprod -0.0%

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

   11. add-sqr-sqrt 95.0%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 61.6%

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

   14. sqr-neg 61.6%

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

   15. sqrt-unprod 62.3%

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

   16. add-sqr-sqrt 62.3%

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

   17. add-sqr-sqrt 51.0%

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

   18. fabs-sqr 51.0%

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

   19. add-sqr-sqrt 99.0%

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

8. Applied egg-rr 98.9%

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

   1. rec-exp 99.0%

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

   2. distribute-neg-frac 99.0%

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

10. Simplified 98.9%

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

    1. distribute-lft1-in 72.6%

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

    2. *-commutative 72.6%

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

12. Applied egg-rr 98.9%

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

13. Final simplification 98.9%

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

Alternative 4: 74.0% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.9%

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

   1. fma-udef 98.9%

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

   2. +-commutative 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

   5. add-sqr-sqrt 98.9%

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

   6. sqrt-unprod 93.5%

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

   7. sqr-neg 93.5%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 23.1%

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

   10. distribute-rgt-in 23.2%

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

   11. *-commutative 23.2%

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

   12. *-un-lft-identity 23.2%

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

6. Applied egg-rr 62.3%

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

   1. distribute-frac-neg 62.3%

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

   2. rec-exp 62.3%

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

   3. pow1 62.3%

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

   4. pow1 62.3%

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

   5. frac-2neg 62.3%

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

   6. frac-2neg 62.3%

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

   7. add-sqr-sqrt 62.3%

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

   8. sqrt-unprod 61.6%

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

   9. sqr-neg 61.6%

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

   10. sqrt-unprod -0.0%

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

   11. add-sqr-sqrt 95.0%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 61.6%

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

   14. sqr-neg 61.6%

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

   15. sqrt-unprod 62.3%

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

   16. add-sqr-sqrt 62.3%

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

   17. add-sqr-sqrt 51.0%

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

   18. fabs-sqr 51.0%

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

   19. add-sqr-sqrt 99.0%

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

8. Applied egg-rr 98.9%

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

   1. rec-exp 99.0%

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

   2. distribute-neg-frac 99.0%

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

10. Simplified 98.9%

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

11. Taylor expanded in x around 0 72.6%

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

    1. mul-1-neg 72.6%

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

    2. unsub-neg 72.6%

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

13. Simplified 72.6%

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

    1. distribute-lft1-in 72.6%

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

    2. *-commutative 72.6%

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

15. Applied egg-rr 72.6%

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

16. Final simplification 72.6%

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

Alternative 5: 62.7% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.9%

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

   1. fma-udef 98.9%

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

   2. +-commutative 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

   5. add-sqr-sqrt 98.9%

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

   6. sqrt-unprod 93.5%

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

   7. sqr-neg 93.5%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 23.1%

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

   10. distribute-rgt-in 23.2%

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

   11. *-commutative 23.2%

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

   12. *-un-lft-identity 23.2%

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

6. Applied egg-rr 62.3%

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

   1. distribute-frac-neg 62.3%

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

   2. rec-exp 62.3%

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

   3. pow1 62.3%

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

   4. pow1 62.3%

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

   5. frac-2neg 62.3%

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

   6. frac-2neg 62.3%

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

   7. add-sqr-sqrt 62.3%

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

   8. sqrt-unprod 61.6%

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

   9. sqr-neg 61.6%

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

   10. sqrt-unprod -0.0%

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

   11. add-sqr-sqrt 95.0%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 61.6%

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

   14. sqr-neg 61.6%

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

   15. sqrt-unprod 62.3%

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

   16. add-sqr-sqrt 62.3%

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

   17. add-sqr-sqrt 51.0%

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

   18. fabs-sqr 51.0%

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

   19. add-sqr-sqrt 99.0%

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

8. Applied egg-rr 98.9%

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

   1. rec-exp 99.0%

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

   2. distribute-neg-frac 99.0%

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

10. Simplified 98.9%

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

11. Taylor expanded in x around 0 72.6%

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

    1. mul-1-neg 72.6%

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

    2. unsub-neg 72.6%

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

13. Simplified 72.6%

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

14. Taylor expanded in x around inf 25.7%

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

    1. associate-*r* 25.7%

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

    2. neg-mul-1 25.7%

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

    3. associate-*r* 25.7%

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

    4. distribute-rgt-out 62.5%

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

    5. +-commutative 62.5%

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

    6. sub-neg 62.5%

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

    7. *-commutative 62.5%

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

16. Simplified 62.5%

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

17. Final simplification 62.5%

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

Alternative 6: 59.7% accurate, 5.5× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (* 2.0 (* s (+ 1.0 (pow E (/ x s)))))))

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(2.0) * Float32(s * Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(x / s))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(2.0) * (s * (single(1.0) + (single(2.71828182845904523536) ^ (x / s)))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.9%

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

   1. fma-udef 98.9%

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

   2. +-commutative 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

   5. add-sqr-sqrt 98.9%

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

   6. sqrt-unprod 93.5%

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

   7. sqr-neg 93.5%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 23.1%

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

   10. distribute-rgt-in 23.2%

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

   11. *-commutative 23.2%

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

   12. *-un-lft-identity 23.2%

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

6. Applied egg-rr 62.3%

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

7. Taylor expanded in s around inf 59.1%

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

   1. *-un-lft-identity 59.1%

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

   2. exp-prod 59.1%

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

9. Applied egg-rr 59.1%

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

    1. exp-1-e 59.1%

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

11. Simplified 59.1%

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

12. Final simplification 59.1%

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

Alternative 7: 59.7% 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 98.8%

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

3. Simplified 98.9%

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

   1. fma-udef 98.9%

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

   2. +-commutative 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

   5. add-sqr-sqrt 98.9%

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

   6. sqrt-unprod 93.5%

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

   7. sqr-neg 93.5%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 23.1%

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

   10. distribute-rgt-in 23.2%

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

   11. *-commutative 23.2%

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

   12. *-un-lft-identity 23.2%

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

6. Applied egg-rr 62.3%

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

7. Taylor expanded in s around inf 59.1%

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

8. Taylor expanded in x around inf 59.1%

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

9. Final simplification 59.1%

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

Alternative 8: 63.9% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.9%

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

   1. fma-udef 98.9%

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

   2. +-commutative 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

   5. add-sqr-sqrt 98.9%

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

   6. sqrt-unprod 93.5%

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

   7. sqr-neg 93.5%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 23.1%

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

   10. distribute-rgt-in 23.2%

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

   11. *-commutative 23.2%

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

   12. *-un-lft-identity 23.2%

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

6. Applied egg-rr 62.3%

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

   1. distribute-frac-neg 62.3%

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

   2. rec-exp 62.3%

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

   3. pow1 62.3%

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

   4. pow1 62.3%

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

   5. frac-2neg 62.3%

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

   6. frac-2neg 62.3%

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

   7. add-sqr-sqrt 62.3%

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

   8. sqrt-unprod 61.6%

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

   9. sqr-neg 61.6%

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

   10. sqrt-unprod -0.0%

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

   11. add-sqr-sqrt 95.0%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 61.6%

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

   14. sqr-neg 61.6%

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

   15. sqrt-unprod 62.3%

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

   16. add-sqr-sqrt 62.3%

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

   17. add-sqr-sqrt 51.0%

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

   18. fabs-sqr 51.0%

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

   19. add-sqr-sqrt 99.0%

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

8. Applied egg-rr 98.9%

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

   1. rec-exp 99.0%

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

   2. distribute-neg-frac 99.0%

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

10. Simplified 98.9%

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

11. Taylor expanded in x around 0 72.6%

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

    1. mul-1-neg 72.6%

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

    2. unsub-neg 72.6%

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

13. Simplified 72.6%

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

14. Taylor expanded in x around 0 64.0%

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

    1. *-commutative 27.4%

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

16. Simplified 64.0%

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

17. Final simplification 64.0%

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

Alternative 9: 50.6% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.9%

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

   1. fma-udef 98.9%

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

   2. +-commutative 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

   5. add-sqr-sqrt 98.9%

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

   6. sqrt-unprod 93.5%

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

   7. sqr-neg 93.5%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 23.1%

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

   10. distribute-rgt-in 23.2%

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

   11. *-commutative 23.2%

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

   12. *-un-lft-identity 23.2%

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

6. Applied egg-rr 62.3%

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

7. Taylor expanded in s around inf 59.1%

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

8. Taylor expanded in x around 0 50.0%

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

   1. +-commutative 50.0%

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

10. Simplified 50.0%

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

11. Final simplification 50.0%

    \[\leadsto \frac{1}{2 \cdot \left(s \cdot \left(\frac{x}{s} + 2\right)\right)} \]
12. Add Preprocessing

Alternative 10: 28.9% accurate, 68.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.9%

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

   1. fma-udef 98.9%

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

   2. +-commutative 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

   5. add-sqr-sqrt 98.9%

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

   6. sqrt-unprod 93.5%

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

   7. sqr-neg 93.5%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 23.1%

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

   10. distribute-rgt-in 23.2%

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

   11. *-commutative 23.2%

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

   12. *-un-lft-identity 23.2%

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

6. Applied egg-rr 62.3%

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

7. Taylor expanded in s around inf 59.1%

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

8. Taylor expanded in x around 0 27.4%

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

   1. *-commutative 27.4%

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

10. Simplified 27.4%

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

11. Final simplification 27.4%

    \[\leadsto \frac{1}{2 \cdot \left(x + s \cdot 2\right)} \]
12. Add Preprocessing

Alternative 11: 28.4% accurate, 77.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.999999974752427e-7) (/ 0.25 s) (/ 0.5 x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.999999974752427e-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 <= 9.999999974752427e-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(9.999999974752427e-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(9.999999974752427e-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 < 9.99999997e-7

   1. Initial program 98.5%

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

      1. *-commutative 98.5%

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

      2. distribute-lft-in 98.6%

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

      3. *-rgt-identity 98.6%

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

      4. fabs-neg 98.6%

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

      5. distribute-frac-neg 98.6%

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

      6. exp-neg 98.6%

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

      7. associate-*r/ 98.6%

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

      8. *-rgt-identity 98.6%

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

      9. *-lft-identity 98.6%

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

      10. metadata-eval 98.6%

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

      11. times-frac 98.6%

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

      12. neg-mul-1 98.6%

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

      13. neg-mul-1 98.6%

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

      14. fabs-neg 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in s around inf 35.2%

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

   if 9.99999997e-7 < x

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. +-commutative 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. *-commutative 99.8%

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

      5. add-sqr-sqrt 99.8%

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

      6. sqrt-unprod 99.8%

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

      7. sqr-neg 99.8%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt 4.6%

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

      10. distribute-rgt-in 4.6%

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

      11. *-commutative 4.6%

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

      12. *-un-lft-identity 4.6%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in s around inf 97.2%

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

   8. Taylor expanded in x around 0 11.6%

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

      1. *-commutative 11.6%

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

   10. Simplified 11.6%

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

   11. Taylor expanded in x around inf 10.8%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.8% 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 98.8%

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

   1. *-commutative 98.8%

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

   2. distribute-lft-in 98.9%

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

   3. *-rgt-identity 98.9%

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

   4. fabs-neg 98.9%

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

   5. distribute-frac-neg 98.9%

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

   6. exp-neg 98.9%

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

   7. associate-*r/ 98.9%

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

   8. *-rgt-identity 98.9%

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

   9. *-lft-identity 98.9%

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

   10. metadata-eval 98.9%

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

   11. times-frac 98.9%

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

   12. neg-mul-1 98.9%

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

   13. neg-mul-1 98.9%

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

   14. fabs-neg 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 25.9%

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

6. Final simplification 25.9%

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

Reproduce

herbie shell --seed 2024027 
(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))))))