Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 13.7s

Alternatives: 9

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: 99.6% 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, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

   2. add-sqr-sqrt 99.3%

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

   3. sqrt-unprod 99.5%

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

   4. sqr-neg 99.5%

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

   5. distribute-frac-neg 99.5%

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

   6. distribute-frac-neg 99.5%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 24.9%

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

   9. *-commutative 24.9%

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

   10. +-commutative 24.9%

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

   11. distribute-rgt1-in 24.9%

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

6. Applied egg-rr 65.9%

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

   1. distribute-frac-neg 65.9%

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

   2. rec-exp 65.9%

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

   3. add-sqr-sqrt 65.9%

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

   4. add-sqr-sqrt 65.9%

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

   5. add-sqr-sqrt 52.8%

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

   6. fabs-sqr 52.8%

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

   7. add-sqr-sqrt 99.5%

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

8. Applied egg-rr 99.5%

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

   1. rec-exp 99.5%

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

   2. distribute-neg-frac 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 74.0% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

   2. add-sqr-sqrt 99.3%

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

   3. sqrt-unprod 99.5%

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

   4. sqr-neg 99.5%

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

   5. distribute-frac-neg 99.5%

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

   6. distribute-frac-neg 99.5%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 24.9%

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

   9. *-commutative 24.9%

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

   10. +-commutative 24.9%

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

   11. distribute-rgt1-in 24.9%

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

6. Applied egg-rr 65.9%

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

   1. distribute-frac-neg 65.9%

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

   2. rec-exp 65.9%

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

   3. add-sqr-sqrt 65.9%

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

   4. add-sqr-sqrt 65.9%

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

   5. add-sqr-sqrt 52.8%

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

   6. fabs-sqr 52.8%

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

   7. add-sqr-sqrt 99.5%

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

8. Applied egg-rr 99.5%

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

   1. rec-exp 99.5%

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

   2. distribute-neg-frac 99.5%

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

10. Simplified 99.5%

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

11. Taylor expanded in x around 0 74.4%

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

    1. mul-1-neg 74.4%

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

    2. unsub-neg 74.4%

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

13. Simplified 74.4%

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

14. Final simplification 74.4%

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

Alternative 3: 60.5% 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 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

   2. add-sqr-sqrt 99.3%

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

   3. sqrt-unprod 99.5%

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

   4. sqr-neg 99.5%

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

   5. distribute-frac-neg 99.5%

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

   6. distribute-frac-neg 99.5%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 24.9%

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

   9. *-commutative 24.9%

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

   10. +-commutative 24.9%

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

   11. distribute-rgt1-in 24.9%

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

6. Applied egg-rr 65.9%

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

7. Taylor expanded in s around inf 61.6%

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

8. Taylor expanded in x around inf 61.6%

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

9. Final simplification 61.6%

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

Alternative 4: 60.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{s}}{1 + e^{\frac{x}{s}}} \end{array} \]

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(Float32(0.5) / 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 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

   2. add-sqr-sqrt 99.3%

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

   3. sqrt-unprod 99.5%

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

   4. sqr-neg 99.5%

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

   5. distribute-frac-neg 99.5%

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

   6. distribute-frac-neg 99.5%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 24.9%

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

   9. *-commutative 24.9%

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

   10. +-commutative 24.9%

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

   11. distribute-rgt1-in 24.9%

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

6. Applied egg-rr 65.9%

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

7. Taylor expanded in s around inf 61.6%

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

8. Taylor expanded in x around inf 61.6%

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

   1. associate-/r* 61.6%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{s}}{1 + e^{\frac{x}{s}}}} \]

10. Simplified 61.6%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{s}}{1 + e^{\frac{x}{s}}}} \]

11. Final simplification 61.6%

    \[\leadsto \frac{\frac{0.5}{s}}{1 + e^{\frac{x}{s}}} \]
12. Add Preprocessing

Alternative 5: 64.4% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

   2. add-sqr-sqrt 99.3%

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

   3. sqrt-unprod 99.5%

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

   4. sqr-neg 99.5%

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

   5. distribute-frac-neg 99.5%

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

   6. distribute-frac-neg 99.5%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 24.9%

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

   9. *-commutative 24.9%

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

   10. +-commutative 24.9%

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

   11. distribute-rgt1-in 24.9%

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

6. Applied egg-rr 65.9%

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

   1. distribute-frac-neg 65.9%

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

   2. rec-exp 65.9%

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

   3. add-sqr-sqrt 65.9%

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

   4. add-sqr-sqrt 65.9%

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

   5. add-sqr-sqrt 52.8%

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

   6. fabs-sqr 52.8%

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

   7. add-sqr-sqrt 99.5%

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

8. Applied egg-rr 99.5%

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

   1. rec-exp 99.5%

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

   2. distribute-neg-frac 99.5%

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

10. Simplified 99.5%

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

11. Taylor expanded in x around 0 61.0%

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

    1. *-commutative 29.3%

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

13. Simplified 61.0%

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

14. Taylor expanded in x around 0 65.8%

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

    1. mul-1-neg 65.8%

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

    2. unsub-neg 65.8%

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

16. Simplified 65.8%

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

17. Final simplification 65.8%

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

Alternative 6: 50.6% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

   2. add-sqr-sqrt 99.3%

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

   3. sqrt-unprod 99.5%

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

   4. sqr-neg 99.5%

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

   5. distribute-frac-neg 99.5%

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

   6. distribute-frac-neg 99.5%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 24.9%

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

   9. *-commutative 24.9%

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

   10. +-commutative 24.9%

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

   11. distribute-rgt1-in 24.9%

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

6. Applied egg-rr 65.9%

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

7. Taylor expanded in s around inf 61.6%

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

8. Taylor expanded in x around 0 54.2%

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

   1. +-commutative 54.2%

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

10. Simplified 54.2%

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

11. Final simplification 54.2%

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

Alternative 7: 29.4% accurate, 68.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

   2. add-sqr-sqrt 99.3%

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

   3. sqrt-unprod 99.5%

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

   4. sqr-neg 99.5%

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

   5. distribute-frac-neg 99.5%

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

   6. distribute-frac-neg 99.5%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 24.9%

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

   9. *-commutative 24.9%

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

   10. +-commutative 24.9%

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

   11. distribute-rgt1-in 24.9%

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

6. Applied egg-rr 65.9%

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

   1. distribute-frac-neg 65.9%

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

   2. rec-exp 65.9%

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

   3. add-sqr-sqrt 65.9%

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

   4. add-sqr-sqrt 65.9%

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

   5. add-sqr-sqrt 52.8%

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

   6. fabs-sqr 52.8%

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

   7. add-sqr-sqrt 99.5%

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

8. Applied egg-rr 99.5%

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

   1. rec-exp 99.5%

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

   2. distribute-neg-frac 99.5%

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

10. Simplified 99.5%

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

11. Taylor expanded in x around 0 61.0%

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

    1. *-commutative 29.3%

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

13. Simplified 61.0%

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

14. Taylor expanded in x around 0 29.3%

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

15. Final simplification 29.3%

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

Alternative 8: 28.9% accurate, 77.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.05000000074505806:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.05000000074505806) (/ 0.25 s) (/ 0.5 x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.05000000074505806f) {
		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 <= 0.05000000074505806e0) 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(0.05000000074505806))
		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(0.05000000074505806))
		tmp = single(0.25) / s;
	else
		tmp = single(0.5) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.0500000007

   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%

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

   5. Taylor expanded in s around inf 37.3%

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

   if 0.0500000007 < 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%

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

      1. fma-undefine 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. sqrt-unprod 99.9%

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

      4. sqr-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sqrt-unprod -0.0%

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

      8. add-sqr-sqrt 4.6%

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

      9. *-commutative 4.6%

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

      10. +-commutative 4.6%

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

      11. distribute-rgt1-in 4.6%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in s around inf 99.1%

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

   8. Taylor expanded in x around 0 11.7%

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

      1. *-commutative 11.7%

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

   10. Simplified 11.7%

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

   11. Taylor expanded in x around inf 11.7%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.05000000074505806:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 27.2% 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 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around inf 27.2%

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

6. Final simplification 27.2%

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

Reproduce

herbie shell --seed 2024039 
(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))))))