Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 12.3s

Alternatives: 7

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

   1. distribute-frac-neg 99.9%

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

   2. rec-exp 99.8%

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

   3. pow1 99.8%

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

   4. pow1 99.8%

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

   5. frac-2neg 99.8%

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

   6. add-sqr-sqrt -0.0%

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

   7. sqrt-unprod 96.4%

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

   8. sqr-neg 96.4%

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

   9. sqrt-unprod 96.3%

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

   10. add-sqr-sqrt 96.3%

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

   11. remove-double-neg 96.3%

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

   12. frac-2neg 96.3%

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

   13. add-sqr-sqrt -0.0%

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

   14. sqrt-unprod 97.8%

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

   15. sqr-neg 97.8%

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

   16. sqrt-unprod 99.8%

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

   17. add-sqr-sqrt 99.8%

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

   18. add-sqr-sqrt 51.5%

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

   19. fabs-sqr 51.5%

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

   20. add-sqr-sqrt 99.0%

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

5. Applied egg-rr 99.0%

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

   1. rec-exp 99.0%

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

7. Simplified 99.0%

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

   1. add-sqr-sqrt 99.0%

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

   2. add-sqr-sqrt 99.0%

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

   3. add-sqr-sqrt 51.5%

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

   4. fabs-sqr 51.5%

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

   5. add-sqr-sqrt 99.9%

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

   6. expm1-log1p-u 99.8%

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

   7. +-commutative 99.8%

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

   8. expm1-udef 99.7%

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

   9. +-commutative 99.7%

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

9. Applied egg-rr 99.7%

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

    1. expm1-def 99.8%

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

    2. expm1-log1p 99.9%

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

11. Simplified 99.9%

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

12. Taylor expanded in s around 0 99.8%

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

    1. associate-*r* 99.9%

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

14. Simplified 99.9%

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

15. Final simplification 99.9%

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

Alternative 2: 58.7% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

4. Taylor expanded in s around inf 96.6%

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

   1. /-rgt-identity 96.6%

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

   2. *-commutative 96.6%

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

   3. associate-/l* 96.6%

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

   4. add-sqr-sqrt 96.6%

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

   5. add-sqr-sqrt 96.6%

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

   6. add-sqr-sqrt 49.0%

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

   7. fabs-sqr 49.0%

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

   8. add-sqr-sqrt 62.5%

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

   9. fma-udef 62.5%

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

   10. *-rgt-identity 62.5%

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

6. Applied egg-rr 62.5%

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

7. Final simplification 62.5%

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

Alternative 3: 58.7% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

4. Taylor expanded in s around inf 96.6%

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

   1. expm1-log1p-u 95.5%

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

   2. expm1-udef 95.5%

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

   3. associate-/r* 95.5%

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

   4. fma-udef 95.5%

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

   5. *-rgt-identity 95.5%

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

   6. add-sqr-sqrt 95.5%

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

   7. add-sqr-sqrt 95.5%

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

   8. add-sqr-sqrt 48.6%

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

   9. fabs-sqr 48.6%

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

   10. add-sqr-sqrt 61.4%

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

6. Applied egg-rr 61.4%

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

   1. expm1-def 61.4%

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

   2. expm1-log1p 62.5%

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

   3. count-2 62.5%

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

8. Simplified 62.5%

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

9. Final simplification 62.5%

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

Alternative 4: 65.1% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

   1. distribute-frac-neg 99.9%

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

   2. rec-exp 99.8%

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

   3. pow1 99.8%

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

   4. pow1 99.8%

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

   5. frac-2neg 99.8%

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

   6. add-sqr-sqrt -0.0%

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

   7. sqrt-unprod 96.4%

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

   8. sqr-neg 96.4%

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

   9. sqrt-unprod 96.3%

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

   10. add-sqr-sqrt 96.3%

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

   11. remove-double-neg 96.3%

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

   12. frac-2neg 96.3%

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

   13. add-sqr-sqrt -0.0%

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

   14. sqrt-unprod 97.8%

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

   15. sqr-neg 97.8%

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

   16. sqrt-unprod 99.8%

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

   17. add-sqr-sqrt 99.8%

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

   18. add-sqr-sqrt 51.5%

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

   19. fabs-sqr 51.5%

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

   20. add-sqr-sqrt 99.0%

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

5. Applied egg-rr 99.0%

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

   1. rec-exp 99.0%

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

7. Simplified 99.0%

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

   1. add-sqr-sqrt 99.0%

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

   2. add-sqr-sqrt 99.0%

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

   3. add-sqr-sqrt 51.5%

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

   4. fabs-sqr 51.5%

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

   5. add-sqr-sqrt 99.9%

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

   6. expm1-log1p-u 99.8%

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

   7. +-commutative 99.8%

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

   8. expm1-udef 99.7%

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

   9. +-commutative 99.7%

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

9. Applied egg-rr 99.7%

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

    1. expm1-def 99.8%

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

    2. expm1-log1p 99.9%

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

11. Simplified 99.9%

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

12. Taylor expanded in x around 0 65.8%

    \[\leadsto \frac{1}{\color{blue}{4 \cdot s + \frac{{x}^{2}}{s}}} \]
13. Step-by-step derivation

    1. unpow2 65.8%

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

    2. *-un-lft-identity 65.8%

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

    3. times-frac 66.5%

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

14. Applied egg-rr 66.5%

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

15. Final simplification 66.5%

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

Alternative 5: 28.8% accurate, 68.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot 4 + x \cdot 2} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ (* s 4.0) (* x 2.0))))

C

float code(float x, float s) {
	return 1.0f / ((s * 4.0f) + (x * 2.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((s * 4.0e0) + (x * 2.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(s * Float32(4.0)) + Float32(x * Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((s * single(4.0)) + (x * single(2.0)));
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

4. Taylor expanded in s around inf 96.6%

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

   1. distribute-lft-in 96.6%

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

   2. *-rgt-identity 96.6%

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

   3. fma-udef 96.6%

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

   4. *-rgt-identity 96.6%

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

   5. fma-udef 96.6%

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

   6. *-rgt-identity 96.6%

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

   7. add-sqr-sqrt 96.6%

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

   8. add-sqr-sqrt 96.6%

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

   9. add-sqr-sqrt 49.0%

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

   10. fabs-sqr 49.0%

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

   11. add-sqr-sqrt 62.5%

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

6. Applied egg-rr 62.5%

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

7. Taylor expanded in s around inf 28.9%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot x + 4 \cdot s}} \]

8. Final simplification 28.9%

   \[\leadsto \frac{1}{s \cdot 4 + x \cdot 2} \]

Alternative 6: 28.2% accurate, 121.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.004999999888241291:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.004999999888241291) (/ 0.25 s) (/ 0.5 x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.004999999888241291f) {
		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 <= 0.004999999888241291e0) 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(0.004999999888241291))
		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(0.004999999888241291))
		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 < 0.00499999989

   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%

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

   4. Taylor expanded in s around inf 35.6%

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

   if 0.00499999989 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. fma-udef 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. fma-udef 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. add-sqr-sqrt 100.0%

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

      10. fabs-sqr 100.0%

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

      11. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in s around inf 10.3%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot x + 4 \cdot s}} \]

   8. Taylor expanded in x around inf 10.3%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.004999999888241291:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 7: 26.6% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

4. Taylor expanded in s around inf 26.7%

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

5. Final simplification 26.7%

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

Reproduce

herbie shell --seed 2023333 
(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))))))