Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 14.6s

Alternatives: 5

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, 1.5× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/
  1.0
  (*
   (+ 1.0 (exp (- (/ x_m s))))
   (* s (+ 1.0 (pow (sqrt (exp (/ x_m s))) 2.0))))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 99.7%

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

   1. frac-2neg 99.7%

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

   2. frac-2neg 99.7%

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

   3. add-sqr-sqrt 99.7%

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

   4. sqrt-unprod 97.4%

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

   5. sqr-neg 97.4%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 23.7%

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

   8. fma-udef 23.7%

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

   9. *-un-lft-identity 23.7%

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

   10. *-commutative 23.7%

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

   11. distribute-lft-in 23.7%

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

   12. add-exp-log 22.8%

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

   13. +-commutative 22.8%

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

   14. add-exp-log 22.8%

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

   15. log1p-udef 22.8%

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

   16. prod-exp 22.6%

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

6. Applied egg-rr 62.4%

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

   1. +-commutative 62.4%

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

8. Simplified 62.4%

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

   1. distribute-frac-neg 62.4%

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

   2. rec-exp 62.4%

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

   3. frac-2neg 62.4%

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

   4. frac-2neg 62.4%

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

   5. add-sqr-sqrt 62.4%

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

   6. sqrt-unprod 62.2%

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

   7. sqr-neg 62.2%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 96.3%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 62.2%

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

   12. sqr-neg 62.2%

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

   13. sqrt-unprod 62.4%

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

   14. add-sqr-sqrt 62.4%

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

   15. add-sqr-sqrt 52.1%

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

   16. fabs-sqr 52.1%

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

   17. add-sqr-sqrt 98.0%

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

10. Applied egg-rr 98.0%

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

    1. rec-exp 98.0%

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

    2. distribute-neg-frac 98.0%

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

12. Simplified 98.0%

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

    1. exp-sum 98.4%

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

    2. log1p-udef 98.4%

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

    3. +-commutative 98.4%

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

    4. add-exp-log 98.3%

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

    5. add-exp-log 99.8%

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

14. Applied egg-rr 99.8%

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

    1. add-sqr-sqrt 99.8%

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

    2. pow2 99.8%

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

16. Applied egg-rr 99.8%

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

17. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 2.9× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ 1.0 (* (+ 1.0 (exp (- (/ x_m s)))) (* s (+ 1.0 (exp (/ x_m s)))))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 99.7%

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

   1. frac-2neg 99.7%

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

   2. frac-2neg 99.7%

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

   3. add-sqr-sqrt 99.7%

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

   4. sqrt-unprod 97.4%

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

   5. sqr-neg 97.4%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 23.7%

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

   8. fma-udef 23.7%

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

   9. *-un-lft-identity 23.7%

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

   10. *-commutative 23.7%

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

   11. distribute-lft-in 23.7%

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

   12. add-exp-log 22.8%

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

   13. +-commutative 22.8%

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

   14. add-exp-log 22.8%

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

   15. log1p-udef 22.8%

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

   16. prod-exp 22.6%

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

6. Applied egg-rr 62.4%

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

   1. +-commutative 62.4%

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

8. Simplified 62.4%

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

   1. distribute-frac-neg 62.4%

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

   2. rec-exp 62.4%

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

   3. frac-2neg 62.4%

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

   4. frac-2neg 62.4%

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

   5. add-sqr-sqrt 62.4%

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

   6. sqrt-unprod 62.2%

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

   7. sqr-neg 62.2%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 96.3%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 62.2%

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

   12. sqr-neg 62.2%

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

   13. sqrt-unprod 62.4%

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

   14. add-sqr-sqrt 62.4%

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

   15. add-sqr-sqrt 52.1%

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

   16. fabs-sqr 52.1%

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

   17. add-sqr-sqrt 98.0%

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

10. Applied egg-rr 98.0%

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

    1. rec-exp 98.0%

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

    2. distribute-neg-frac 98.0%

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

12. Simplified 98.0%

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

    1. exp-sum 98.4%

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

    2. log1p-udef 98.4%

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

    3. +-commutative 98.4%

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

    4. add-exp-log 98.3%

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

    5. add-exp-log 99.8%

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

14. Applied egg-rr 99.8%

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

15. Final simplification 99.8%

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

Alternative 3: 95.2% accurate, 5.6× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ 1.0 (* 2.0 (+ s (* s (exp (/ x_m s)))))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 99.7%

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

5. Taylor expanded in s around inf 95.9%

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

   1. fma-udef 95.9%

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

7. Applied egg-rr 62.2%

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

8. Final simplification 62.2%

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

Alternative 4: 29.7% accurate, 5.7× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 1.0 (* 2.0 (+ (fabs x_m) (* s 2.0)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 99.7%

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

5. Taylor expanded in s around inf 95.9%

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

6. Taylor expanded in s around inf 27.8%

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

   1. *-commutative 27.8%

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

8. Simplified 27.8%

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

9. Final simplification 27.8%

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

Alternative 5: 27.3% accurate, 206.7× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 0.25 s))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 25.7%

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

6. Final simplification 25.7%

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

Reproduce

herbie shell --seed 2024020 
(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))))))