Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 10.0s

Alternatives: 9

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(fma(s, exp(Float32(Float32(-abs(x)) / s)), s) * Float32(Float32(1.0) + exp(Float32(abs(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 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 62.9% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. distribute-frac-neg 99.8%

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

   6. exp-neg 99.8%

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

   7. associate-*r/ 99.8%

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

   8. *-rgt-identity 99.8%

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

   9. *-lft-identity 99.8%

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

   10. metadata-eval 99.8%

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

   11. times-frac 99.8%

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

   12. neg-mul-1 99.8%

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

   13. neg-mul-1 99.8%

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

   14. fabs-neg 99.8%

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

3. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. clear-num 99.8%

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

   3. associate-/r/ 99.8%

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

   4. rec-exp 99.8%

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

   5. distribute-frac-neg 99.8%

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

   6. distribute-rgt-in 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-/l/ 99.8%

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

   9. associate-/l/ 99.8%

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

5. Applied egg-rr 58.8%

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

   1. clear-num 58.8%

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

   2. frac-times 58.8%

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

   3. *-un-lft-identity 58.8%

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

   4. +-commutative 58.8%

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

   5. +-commutative 58.8%

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

7. Applied egg-rr 58.8%

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

8. Taylor expanded in x around 0 65.0%

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

9. Final simplification 65.0%

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

Alternative 3: 60.0% 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 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 96.9%

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

   1. distribute-rgt-in 96.9%

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

   2. *-un-lft-identity 96.9%

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

   3. add-sqr-sqrt 51.8%

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

   4. fabs-sqr 51.8%

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

   5. add-sqr-sqrt 62.1%

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

   6. add-sqr-sqrt 51.8%

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

   7. add-sqr-sqrt 62.1%

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

6. Applied egg-rr 62.1%

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

   1. distribute-rgt1-in 62.1%

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

   2. +-commutative 62.1%

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

   3. fma-udef 62.1%

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

   4. *-rgt-identity 62.1%

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

8. Simplified 62.1%

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

9. Taylor expanded in x around inf 62.1%

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

10. Final simplification 62.1%

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

Alternative 4: 63.2% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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. distribute-frac-neg 99.8%

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

   6. exp-neg 99.8%

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

   7. associate-*r/ 99.8%

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

   8. *-rgt-identity 99.8%

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

   9. *-lft-identity 99.8%

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

   10. metadata-eval 99.8%

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

   11. times-frac 99.8%

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

   12. neg-mul-1 99.8%

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

   13. neg-mul-1 99.8%

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

   14. fabs-neg 99.8%

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

3. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. clear-num 99.8%

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

   3. associate-/r/ 99.8%

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

   4. rec-exp 99.8%

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

   5. distribute-frac-neg 99.8%

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

   6. distribute-rgt-in 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-/l/ 99.8%

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

   9. associate-/l/ 99.8%

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

5. Applied egg-rr 58.8%

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

   1. clear-num 58.8%

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

   2. frac-times 58.8%

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

   3. *-un-lft-identity 58.8%

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

   4. +-commutative 58.8%

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

   5. +-commutative 58.8%

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

7. Applied egg-rr 58.8%

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

8. Taylor expanded in x around 0 65.0%

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

9. Taylor expanded in x around 0 58.9%

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

10. Final simplification 58.9%

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

Alternative 5: 50.6% 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 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 96.9%

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

   1. distribute-rgt-in 96.9%

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

   2. *-un-lft-identity 96.9%

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

   3. add-sqr-sqrt 51.8%

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

   4. fabs-sqr 51.8%

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

   5. add-sqr-sqrt 62.1%

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

   6. add-sqr-sqrt 51.8%

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

   7. add-sqr-sqrt 62.1%

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

6. Applied egg-rr 62.1%

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

   1. distribute-rgt1-in 62.1%

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

   2. +-commutative 62.1%

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

   3. fma-udef 62.1%

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

   4. *-rgt-identity 62.1%

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

8. Simplified 62.1%

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

9. Taylor expanded in x around inf 62.1%

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

10. Taylor expanded in x around 0 46.2%

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

11. Final simplification 46.2%

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

Alternative 6: 28.6% accurate, 87.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x + -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.00019999999494757503) (/ 0.25 s) (/ 0.5 (+ x -2.0))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.00019999999494757503f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.5f / (x + -2.0f);
	}
	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 + (-2.0e0))
    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) / Float32(x + Float32(-2.0)));
	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 + single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999995e-4

   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. *-commutative 99.7%

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

      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. distribute-frac-neg 99.8%

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

      6. exp-neg 99.7%

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

      7. associate-*r/ 99.7%

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

      8. *-rgt-identity 99.7%

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

      9. *-lft-identity 99.7%

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

      10. metadata-eval 99.7%

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

      11. times-frac 99.7%

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

      12. neg-mul-1 99.7%

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

      13. neg-mul-1 99.7%

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

      14. fabs-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 33.0%

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

   if 1.99999995e-4 < x

   1. Initial program 99.9%

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

   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 99.1%

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

      1. distribute-rgt-in 99.1%

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

      2. *-un-lft-identity 99.1%

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

      3. add-sqr-sqrt 99.1%

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

      4. fabs-sqr 99.1%

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

      5. add-sqr-sqrt 99.1%

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

      6. add-sqr-sqrt 99.1%

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

      7. add-sqr-sqrt 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. distribute-rgt1-in 99.1%

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

      2. +-commutative 99.1%

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

      3. fma-udef 99.1%

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

      4. *-rgt-identity 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in x around inf 99.1%

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

   10. Taylor expanded in s around inf 10.4%

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

   12. Simplified 10.3%

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

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x + -2}\\ \end{array} \]

Alternative 7: 28.9% accurate, 88.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / (x + (s * 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 99.8%

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

4. Taylor expanded in s around inf 96.9%

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

   1. distribute-rgt-in 96.9%

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

   2. *-un-lft-identity 96.9%

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

   3. add-sqr-sqrt 51.8%

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

   4. fabs-sqr 51.8%

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

   5. add-sqr-sqrt 62.1%

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

   6. add-sqr-sqrt 51.8%

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

   7. add-sqr-sqrt 62.1%

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

6. Applied egg-rr 62.1%

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

   1. distribute-rgt1-in 62.1%

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

   2. +-commutative 62.1%

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

   3. fma-udef 62.1%

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

   4. *-rgt-identity 62.1%

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

8. Simplified 62.1%

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

9. Taylor expanded in x around inf 62.1%

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

10. Taylor expanded in s around inf 26.4%

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

11. Final simplification 26.4%

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

Alternative 8: 27.0% 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%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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. distribute-frac-neg 99.8%

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

   6. exp-neg 99.8%

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

   7. associate-*r/ 99.8%

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

   8. *-rgt-identity 99.8%

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

   9. *-lft-identity 99.8%

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

   10. metadata-eval 99.8%

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

   11. times-frac 99.8%

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

   12. neg-mul-1 99.8%

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

   13. neg-mul-1 99.8%

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

   14. fabs-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 24.3%

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

5. Final simplification 24.3%

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

Alternative 9: 8.2% accurate, 620.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.25)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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. distribute-frac-neg 99.8%

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

   6. exp-neg 99.8%

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

   7. associate-*r/ 99.8%

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

   8. *-rgt-identity 99.8%

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

   9. *-lft-identity 99.8%

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

   10. metadata-eval 99.8%

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

   11. times-frac 99.8%

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

   12. neg-mul-1 99.8%

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

   13. neg-mul-1 99.8%

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

   14. fabs-neg 99.8%

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

3. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. clear-num 99.8%

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

   3. associate-/r/ 99.8%

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

   4. rec-exp 99.8%

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

   5. distribute-frac-neg 99.8%

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

   6. distribute-rgt-in 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-/l/ 99.8%

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

   9. associate-/l/ 99.8%

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

5. Applied egg-rr 58.8%

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

   1. clear-num 58.8%

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

   2. frac-times 58.8%

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

   3. *-un-lft-identity 58.8%

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

   4. +-commutative 58.8%

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

   5. +-commutative 58.8%

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

7. Applied egg-rr 58.8%

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

8. Taylor expanded in x around 0 65.0%

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

9. Taylor expanded in x around 0 24.3%

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

11. Simplified 7.7%

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

12. Final simplification 7.7%

    \[\leadsto 0.25 \]

Reproduce

herbie shell --seed 2023332 
(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))))))