Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 13.9s

Alternatives: 11

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. *-rgt-identity 99.3%

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

   4. fabs-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. fma-def 99.3%

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

   7. fabs-neg 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 2: 99.5% accurate, 2.9× speedup

Math

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

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(s * Float32(Float32(Float32(1.0) + exp(Float32(x / s))) * Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))))
end

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

   1. distribute-frac-neg 99.3%

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

   2. rec-exp 99.3%

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

   3. add-sqr-sqrt 51.8%

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

   4. fabs-sqr 51.8%

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

   5. add-sqr-sqrt 96.9%

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

6. Applied egg-rr 96.9%

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

7. Taylor expanded in x around inf 96.9%

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

   1. distribute-lft-in 55.9%

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

   2. div-inv 55.9%

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

   3. exp-prod 54.0%

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

   4. add-sqr-sqrt 8.8%

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

   5. fabs-sqr 8.8%

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

   6. add-sqr-sqrt 54.0%

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

   7. exp-prod 56.0%

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

   8. div-inv 56.0%

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

9. Applied egg-rr 58.3%

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

    1. distribute-lft-out 99.3%

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

11. Simplified 99.3%

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

12. Taylor expanded in s around 0 99.3%

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

    1. rec-exp 99.3%

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

14. Simplified 99.3%

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

15. Final simplification 99.3%

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

Alternative 3: 99.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

   1. distribute-frac-neg 99.3%

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

   2. rec-exp 99.3%

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

   3. add-sqr-sqrt 51.8%

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

   4. fabs-sqr 51.8%

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

   5. add-sqr-sqrt 96.9%

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

6. Applied egg-rr 96.9%

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

7. Taylor expanded in x around inf 96.9%

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

   1. distribute-lft-in 55.9%

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

   2. div-inv 55.9%

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

   3. exp-prod 54.0%

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

   4. add-sqr-sqrt 8.8%

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

   5. fabs-sqr 8.8%

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

   6. add-sqr-sqrt 54.0%

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

   7. exp-prod 56.0%

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

   8. div-inv 56.0%

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

9. Applied egg-rr 58.3%

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

    1. distribute-lft-out 99.3%

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

11. Simplified 99.3%

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

12. Final simplification 99.3%

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

Alternative 4: 61.3% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

   1. distribute-frac-neg 99.3%

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

   2. rec-exp 99.3%

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

   3. add-sqr-sqrt 51.8%

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

   4. fabs-sqr 51.8%

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

   5. add-sqr-sqrt 96.9%

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

6. Applied egg-rr 96.9%

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

7. Taylor expanded in x around inf 96.9%

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

   1. distribute-lft-in 55.9%

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

   2. div-inv 55.9%

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

   3. exp-prod 54.0%

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

   4. add-sqr-sqrt 8.8%

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

   5. fabs-sqr 8.8%

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

   6. add-sqr-sqrt 54.0%

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

   7. exp-prod 56.0%

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

   8. div-inv 56.0%

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

9. Applied egg-rr 58.3%

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

    1. distribute-lft-out 99.3%

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

11. Simplified 99.3%

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

12. Taylor expanded in x around 0 63.0%

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

13. Final simplification 63.0%

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

Alternative 5: 59.9% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. *-rgt-identity 99.3%

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

   4. fabs-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. fma-def 99.3%

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

   7. fabs-neg 99.3%

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

3. Simplified 99.3%

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

6. Applied egg-rr 58.3%

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

   1. associate-*r/ 58.3%

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

   2. *-rgt-identity 58.3%

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

   3. distribute-neg-frac 58.3%

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

   4. +-commutative 58.3%

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

   5. distribute-neg-in 58.3%

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

   6. metadata-eval 58.3%

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

8. Simplified 58.3%

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

9. Taylor expanded in x around 0 61.4%

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

10. Taylor expanded in s around 0 61.4%

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

11. Final simplification 61.4%

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

Alternative 6: 73.5% accurate, 41.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{s} + 2\\ \frac{1}{s \cdot \left(t_0 \cdot t_0\right)} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. *-rgt-identity 99.3%

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

   4. fabs-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. fma-def 99.3%

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

   7. fabs-neg 99.3%

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

3. Simplified 99.3%

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

6. Applied egg-rr 56.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{e^{\frac{x}{s}}}{s}}{{\left(e^{\frac{x}{s}} + 1\right)}^{2}}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 56.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{e^{\frac{x}{s}}}{s}}{{\left(e^{\frac{x}{s}} + 1\right)}^{2}}\right)\right)} \]

   2. expm1-log1p 57.6%

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

   3. associate-/l/ 58.0%

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

   4. +-commutative 58.0%

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

8. Simplified 58.0%

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

9. Taylor expanded in x around 0 55.4%

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

    1. unpow2 55.4%

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

    2. associate-+r+ 55.4%

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

    3. metadata-eval 55.4%

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

    4. associate-+r+ 55.4%

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

    5. metadata-eval 55.4%

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

11. Applied egg-rr 55.4%

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

12. Taylor expanded in x around 0 73.5%

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

13. Final simplification 73.5%

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

Alternative 7: 50.3% accurate, 62.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. *-rgt-identity 99.3%

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

   4. fabs-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. fma-def 99.3%

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

   7. fabs-neg 99.3%

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

3. Simplified 99.3%

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

6. Applied egg-rr 58.3%

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

   1. associate-*r/ 58.3%

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

   2. *-rgt-identity 58.3%

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

   3. distribute-neg-frac 58.3%

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

   4. +-commutative 58.3%

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

   5. distribute-neg-in 58.3%

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

   6. metadata-eval 58.3%

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

8. Simplified 58.3%

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

9. Taylor expanded in x around 0 61.4%

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

10. Taylor expanded in x around 0 49.1%

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

11. Final simplification 49.1%

    \[\leadsto \frac{\frac{-0.5}{s}}{\frac{-x}{s} - 2} \]
12. Add Preprocessing

Alternative 8: 50.3% accurate, 68.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. *-rgt-identity 99.3%

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

   4. fabs-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. fma-def 99.3%

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

   7. fabs-neg 99.3%

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

3. Simplified 99.3%

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

6. Applied egg-rr 58.3%

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

   1. associate-*r/ 58.3%

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

   2. *-rgt-identity 58.3%

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

   3. distribute-neg-frac 58.3%

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

   4. +-commutative 58.3%

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

   5. distribute-neg-in 58.3%

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

   6. metadata-eval 58.3%

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

8. Simplified 58.3%

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

9. Taylor expanded in x around 0 61.4%

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

10. Taylor expanded in x around 0 49.1%

    \[\leadsto \frac{\frac{-0.5}{s}}{\color{blue}{-1 \cdot \frac{x}{s} - 2}} \]
11. Step-by-step derivation

    1. expm1-log1p-u 47.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-0.5}{s}}{-1 \cdot \frac{x}{s} - 2}\right)\right)} \]

    2. expm1-udef 64.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-0.5}{s}}{-1 \cdot \frac{x}{s} - 2}\right)} - 1} \]

    3. sub-neg 64.1%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-0.5}{s}}{\color{blue}{-1 \cdot \frac{x}{s} + \left(-2\right)}}\right)} - 1 \]

    4. add-sqr-sqrt 32.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-0.5}{s}}{\color{blue}{\sqrt{-1 \cdot \frac{x}{s}} \cdot \sqrt{-1 \cdot \frac{x}{s}}} + \left(-2\right)}\right)} - 1 \]

    5. sqrt-unprod 74.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-0.5}{s}}{\color{blue}{\sqrt{\left(-1 \cdot \frac{x}{s}\right) \cdot \left(-1 \cdot \frac{x}{s}\right)}} + \left(-2\right)}\right)} - 1 \]

    6. mul-1-neg 74.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-0.5}{s}}{\sqrt{\color{blue}{\left(-\frac{x}{s}\right)} \cdot \left(-1 \cdot \frac{x}{s}\right)} + \left(-2\right)}\right)} - 1 \]

    7. mul-1-neg 74.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-0.5}{s}}{\sqrt{\left(-\frac{x}{s}\right) \cdot \color{blue}{\left(-\frac{x}{s}\right)}} + \left(-2\right)}\right)} - 1 \]

    8. sqr-neg 74.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-0.5}{s}}{\sqrt{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}}} + \left(-2\right)}\right)} - 1 \]

    9. sqrt-unprod 30.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-0.5}{s}}{\color{blue}{\sqrt{\frac{x}{s}} \cdot \sqrt{\frac{x}{s}}} + \left(-2\right)}\right)} - 1 \]

    10. add-sqr-sqrt 63.3%

        \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-0.5}{s}}{\color{blue}{\frac{x}{s}} + \left(-2\right)}\right)} - 1 \]

    11. metadata-eval 63.3%

        \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-0.5}{s}}{\frac{x}{s} + \color{blue}{-2}}\right)} - 1 \]

12. Applied egg-rr 63.3%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-0.5}{s}}{\frac{x}{s} + -2}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 47.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-0.5}{s}}{\frac{x}{s} + -2}\right)\right)} \]

    2. expm1-log1p 49.3%

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

    3. associate-/l/ 49.3%

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

    4. *-commutative 49.3%

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

14. Simplified 49.3%

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

15. Final simplification 49.3%

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

Alternative 9: 50.3% accurate, 68.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. *-rgt-identity 99.3%

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

   4. fabs-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. fma-def 99.3%

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

   7. fabs-neg 99.3%

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

3. Simplified 99.3%

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

6. Applied egg-rr 58.3%

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

   1. associate-*r/ 58.3%

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

   2. *-rgt-identity 58.3%

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

   3. distribute-neg-frac 58.3%

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

   4. +-commutative 58.3%

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

   5. distribute-neg-in 58.3%

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

   6. metadata-eval 58.3%

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

8. Simplified 58.3%

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

9. Taylor expanded in x around 0 61.4%

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

10. Taylor expanded in x around 0 49.1%

    \[\leadsto \frac{\frac{-0.5}{s}}{\color{blue}{-1 \cdot \frac{x}{s} - 2}} \]
11. Step-by-step derivation

    1. frac-2neg 49.1%

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

    2. div-inv 49.1%

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

    3. sub-neg 49.1%

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

    4. add-sqr-sqrt 24.7%

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

    5. sqrt-unprod 73.1%

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

    6. mul-1-neg 73.1%

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

    7. mul-1-neg 73.1%

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

    8. sqr-neg 73.1%

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

    9. sqrt-unprod 24.3%

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

    10. add-sqr-sqrt 49.3%

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

    11. metadata-eval 49.3%

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

12. Applied egg-rr 49.3%

    \[\leadsto \color{blue}{\left(-\frac{-0.5}{s}\right) \cdot \frac{1}{-\left(\frac{x}{s} + -2\right)}} \]
13. Step-by-step derivation

    1. associate-*r/ 49.3%

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

    2. *-rgt-identity 49.3%

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

    3. distribute-neg-frac 49.3%

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

    4. metadata-eval 49.3%

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

    5. +-commutative 49.3%

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

    6. distribute-neg-in 49.3%

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

    7. metadata-eval 49.3%

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

    8. sub-neg 49.3%

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

14. Simplified 49.3%

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

15. Final simplification 49.3%

    \[\leadsto \frac{\frac{0.5}{s}}{2 - \frac{x}{s}} \]
16. Add Preprocessing

Alternative 10: 28.5% accurate, 121.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.00019999999494757503) (/ 0.25 s) (/ 0.5 x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.00019999999494757503f) {
		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 <= 0.00019999999494757503e0) 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(0.00019999999494757503))
		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(0.00019999999494757503))
		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 < 1.99999995e-4

   1. Initial program 99.0%

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

      1. *-commutative 99.0%

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

      2. distribute-lft-in 99.1%

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

      3. *-rgt-identity 99.1%

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

      4. fabs-neg 99.1%

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

      5. +-commutative 99.1%

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

      6. fma-def 99.1%

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

      7. fabs-neg 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in s around inf 31.9%

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

   if 1.99999995e-4 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-lft-in 99.8%

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

      3. *-rgt-identity 99.8%

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

      4. fabs-neg 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. fabs-neg 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 1.1%

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

      1. associate-*r/ 1.1%

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

      2. *-rgt-identity 1.1%

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

      3. distribute-neg-frac 1.1%

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

      4. +-commutative 1.1%

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

      5. distribute-neg-in 1.1%

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

      6. metadata-eval 1.1%

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

   8. Simplified 1.1%

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

   9. Taylor expanded in x around 0 99.1%

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

   10. Taylor expanded in x around 0 52.2%

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

   11. Taylor expanded in s around 0 10.6%

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

4. Final simplification 25.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.9% 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 99.3%

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

   1. *-commutative 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. *-rgt-identity 99.3%

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

   4. fabs-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. fma-def 99.3%

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

   7. fabs-neg 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around inf 23.6%

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

6. Final simplification 23.6%

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

Reproduce

herbie shell --seed 2024011 
(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))))))