Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 16.5s

Alternatives: 8

Speedup: 1.0×

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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{t\_0}{\left(t\_0 + 1\right) \cdot \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

5. Final simplification 99.5%

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

Alternative 2: 63.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{\frac{-x}{s}}\\ \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(t\_0 \cdot t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (exp (/ (- x) s)))))
   (/
    (/ (exp (* (/ x s) -0.6666666666666666)) (cbrt (exp (/ x s))))
    (* s (* t_0 t_0)))))

C

float code(float x, float s) {
	float t_0 = 1.0f + expf((-x / s));
	return (expf(((x / s) * -0.6666666666666666f)) / cbrtf(expf((x / s)))) / (s * (t_0 * t_0));
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) + exp(Float32(Float32(-x) / s)))
	return Float32(Float32(exp(Float32(Float32(x / s) * Float32(-0.6666666666666666))) / cbrt(exp(Float32(x / s)))) / Float32(s * Float32(t_0 * t_0)))
end

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.5%

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

   1. distribute-frac-neg2 99.5%

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

   2. rec-exp 99.5%

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

   3. add-cube-cbrt 99.3%

      \[\leadsto \frac{\frac{1}{\color{blue}{\left(\sqrt[3]{e^{\frac{\left|x\right|}{s}}} \cdot \sqrt[3]{e^{\frac{\left|x\right|}{s}}}\right) \cdot \sqrt[3]{e^{\frac{\left|x\right|}{s}}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   4. associate-/r* 99.3%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sqrt[3]{e^{\frac{\left|x\right|}{s}}} \cdot \sqrt[3]{e^{\frac{\left|x\right|}{s}}}}}{\sqrt[3]{e^{\frac{\left|x\right|}{s}}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   5. div-inv 99.4%

      \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{e^{\frac{\left|x\right|}{s}}} \cdot \sqrt[3]{e^{\frac{\left|x\right|}{s}}}}}{\sqrt[3]{e^{\color{blue}{\left|x\right| \cdot \frac{1}{s}}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   6. exp-prod 93.9%

      \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{e^{\frac{\left|x\right|}{s}}} \cdot \sqrt[3]{e^{\frac{\left|x\right|}{s}}}}}{\sqrt[3]{\color{blue}{{\left(e^{\left|x\right|}\right)}^{\left(\frac{1}{s}\right)}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   7. add-sqr-sqrt 93.9%

      \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{e^{\frac{\left|x\right|}{s}}} \cdot \sqrt[3]{e^{\frac{\left|x\right|}{s}}}}}{\sqrt[3]{{\left(e^{\left|x\right|}\right)}^{\left(\frac{1}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}\right)}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   8. sqrt-unprod 93.9%

      \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{e^{\frac{\left|x\right|}{s}}} \cdot \sqrt[3]{e^{\frac{\left|x\right|}{s}}}}}{\sqrt[3]{{\left(e^{\left|x\right|}\right)}^{\left(\frac{1}{\color{blue}{\sqrt{s \cdot s}}}\right)}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   9. sqr-neg 93.9%

      \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{e^{\frac{\left|x\right|}{s}}} \cdot \sqrt[3]{e^{\frac{\left|x\right|}{s}}}}}{\sqrt[3]{{\left(e^{\left|x\right|}\right)}^{\left(\frac{1}{\sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}}\right)}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   10. sqrt-unprod 33.5%

       \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{e^{\frac{\left|x\right|}{s}}} \cdot \sqrt[3]{e^{\frac{\left|x\right|}{s}}}}}{\sqrt[3]{{\left(e^{\left|x\right|}\right)}^{\left(\frac{1}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}\right)}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   11. add-sqr-sqrt 33.7%

       \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{e^{\frac{\left|x\right|}{s}}} \cdot \sqrt[3]{e^{\frac{\left|x\right|}{s}}}}}{\sqrt[3]{{\left(e^{\left|x\right|}\right)}^{\left(\frac{1}{\color{blue}{-s}}\right)}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   12. exp-prod 22.0%

       \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{e^{\frac{\left|x\right|}{s}}} \cdot \sqrt[3]{e^{\frac{\left|x\right|}{s}}}}}{\sqrt[3]{\color{blue}{e^{\left|x\right| \cdot \frac{1}{-s}}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   13. div-inv 22.0%

       \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{e^{\frac{\left|x\right|}{s}}} \cdot \sqrt[3]{e^{\frac{\left|x\right|}{s}}}}}{\sqrt[3]{e^{\color{blue}{\frac{\left|x\right|}{-s}}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

6. Applied egg-rr 65.2%

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{{\left(\sqrt[3]{e^{\frac{x}{s}}}\right)}^{2}}}{\sqrt[3]{e^{\frac{x}{s}}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]
7. Step-by-step derivation

   1. *-un-lft-identity 65.2%

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{1}{{\left(\sqrt[3]{e^{\frac{x}{s}}}\right)}^{2}}}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   2. pow-flip 65.3%

      \[\leadsto \frac{\frac{1 \cdot \color{blue}{{\left(\sqrt[3]{e^{\frac{x}{s}}}\right)}^{\left(-2\right)}}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   3. metadata-eval 65.3%

      \[\leadsto \frac{\frac{1 \cdot {\left(\sqrt[3]{e^{\frac{x}{s}}}\right)}^{\color{blue}{-2}}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

8. Applied egg-rr 65.3%

   \[\leadsto \frac{\frac{\color{blue}{1 \cdot {\left(\sqrt[3]{e^{\frac{x}{s}}}\right)}^{-2}}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]
9. Step-by-step derivation

   1. *-lft-identity 65.3%

      \[\leadsto \frac{\frac{\color{blue}{{\left(\sqrt[3]{e^{\frac{x}{s}}}\right)}^{-2}}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   2. unpow1/3 65.3%

      \[\leadsto \frac{\frac{{\color{blue}{\left({\left(e^{\frac{x}{s}}\right)}^{0.3333333333333333}\right)}}^{-2}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   3. exp-prod 65.2%

      \[\leadsto \frac{\frac{{\color{blue}{\left(e^{\frac{x}{s} \cdot 0.3333333333333333}\right)}}^{-2}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   4. *-commutative 65.2%

      \[\leadsto \frac{\frac{{\left(e^{\color{blue}{0.3333333333333333 \cdot \frac{x}{s}}}\right)}^{-2}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   5. exp-prod 65.2%

      \[\leadsto \frac{\frac{\color{blue}{e^{\left(0.3333333333333333 \cdot \frac{x}{s}\right) \cdot -2}}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   6. *-commutative 65.2%

      \[\leadsto \frac{\frac{e^{\color{blue}{-2 \cdot \left(0.3333333333333333 \cdot \frac{x}{s}\right)}}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   7. associate-*r* 65.2%

      \[\leadsto \frac{\frac{e^{\color{blue}{\left(-2 \cdot 0.3333333333333333\right) \cdot \frac{x}{s}}}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   8. metadata-eval 65.2%

      \[\leadsto \frac{\frac{e^{\color{blue}{-0.6666666666666666} \cdot \frac{x}{s}}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

   9. *-commutative 65.2%

      \[\leadsto \frac{\frac{e^{\color{blue}{\frac{x}{s} \cdot -0.6666666666666666}}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]

10. Simplified 65.2%

    \[\leadsto \frac{\frac{\color{blue}{e^{\frac{x}{s} \cdot -0.6666666666666666}}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{-s}}\right)\right)} \]
11. Step-by-step derivation

    1. distribute-frac-neg2 65.2%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\color{blue}{-\frac{\left|x\right|}{s}}}\right)\right)} \]

    2. rec-exp 65.2%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}\right)\right)} \]

    3. div-inv 65.2%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\color{blue}{\left|x\right| \cdot \frac{1}{s}}}}\right)\right)} \]

    4. add-sqr-sqrt 65.2%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\color{blue}{\left(\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}\right)} \cdot \frac{1}{s}}}\right)\right)} \]

    5. sqrt-unprod 63.5%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{s}}}\right)\right)} \]

    6. sqr-neg 63.5%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\sqrt{\color{blue}{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}} \cdot \frac{1}{s}}}\right)\right)} \]

    7. sqrt-unprod -0.0%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\color{blue}{\left(\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}\right)} \cdot \frac{1}{s}}}\right)\right)} \]

    8. add-sqr-sqrt 59.3%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\color{blue}{\left(-\left|x\right|\right)} \cdot \frac{1}{s}}}\right)\right)} \]

    9. div-inv 59.3%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\color{blue}{\frac{-\left|x\right|}{s}}}}\right)\right)} \]

    10. add-sqr-sqrt -0.0%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}}}{s}}}\right)\right)} \]

    11. sqrt-unprod 63.5%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}}{s}}}\right)\right)} \]

    12. sqr-neg 63.5%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\sqrt{\color{blue}{\left|x\right| \cdot \left|x\right|}}}{s}}}\right)\right)} \]

    13. sqrt-unprod 65.2%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}}}{s}}}\right)\right)} \]

    14. add-sqr-sqrt 65.2%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\color{blue}{\left|x\right|}}{s}}}\right)\right)} \]

    15. add-sqr-sqrt 52.7%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}}\right)\right)} \]

    16. fabs-sqr 52.7%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}\right)\right)} \]

    17. add-sqr-sqrt 64.3%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\color{blue}{x}}{s}}}\right)\right)} \]

12. Applied egg-rr 64.3%

    \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right)\right)} \]
13. Step-by-step derivation

    1. rec-exp 64.3%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \color{blue}{e^{-\frac{x}{s}}}\right)\right)} \]

    2. distribute-neg-frac2 64.3%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\color{blue}{\frac{x}{-s}}}\right)\right)} \]

14. Simplified 64.3%

    \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \color{blue}{e^{\frac{x}{-s}}}\right)\right)} \]
15. Step-by-step derivation

    1. distribute-frac-neg2 65.2%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\color{blue}{-\frac{\left|x\right|}{s}}}\right)\right)} \]

    2. rec-exp 65.2%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}\right)\right)} \]

    3. div-inv 65.2%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\color{blue}{\left|x\right| \cdot \frac{1}{s}}}}\right)\right)} \]

    4. add-sqr-sqrt 65.2%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\color{blue}{\left(\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}\right)} \cdot \frac{1}{s}}}\right)\right)} \]

    5. sqrt-unprod 63.5%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{s}}}\right)\right)} \]

    6. sqr-neg 63.5%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\sqrt{\color{blue}{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}} \cdot \frac{1}{s}}}\right)\right)} \]

    7. sqrt-unprod -0.0%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\color{blue}{\left(\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}\right)} \cdot \frac{1}{s}}}\right)\right)} \]

    8. add-sqr-sqrt 59.3%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\color{blue}{\left(-\left|x\right|\right)} \cdot \frac{1}{s}}}\right)\right)} \]

    9. div-inv 59.3%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\color{blue}{\frac{-\left|x\right|}{s}}}}\right)\right)} \]

    10. add-sqr-sqrt -0.0%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}}}{s}}}\right)\right)} \]

    11. sqrt-unprod 63.5%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}}{s}}}\right)\right)} \]

    12. sqr-neg 63.5%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\sqrt{\color{blue}{\left|x\right| \cdot \left|x\right|}}}{s}}}\right)\right)} \]

    13. sqrt-unprod 65.2%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}}}{s}}}\right)\right)} \]

    14. add-sqr-sqrt 65.2%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\color{blue}{\left|x\right|}}{s}}}\right)\right)} \]

    15. add-sqr-sqrt 52.7%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}}\right)\right)} \]

    16. fabs-sqr 52.7%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}\right)\right)} \]

    17. add-sqr-sqrt 64.3%

        \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \frac{1}{e^{\frac{\color{blue}{x}}{s}}}\right)\right)} \]

16. Applied egg-rr 66.1%

    \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + \color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot \left(1 + e^{\frac{x}{-s}}\right)\right)} \]
17. Step-by-step derivation

    1. rec-exp 64.3%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + \color{blue}{e^{-\frac{x}{s}}}\right)\right)} \]

    2. distribute-neg-frac2 64.3%

       \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \left(1 + e^{\color{blue}{\frac{x}{-s}}}\right)\right)} \]

18. Simplified 66.1%

    \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + \color{blue}{e^{\frac{x}{-s}}}\right) \cdot \left(1 + e^{\frac{x}{-s}}\right)\right)} \]

19. Final simplification 66.1%

    \[\leadsto \frac{\frac{e^{\frac{x}{s} \cdot -0.6666666666666666}}{\sqrt[3]{e^{\frac{x}{s}}}}}{s \cdot \left(\left(1 + e^{\frac{-x}{s}}\right) \cdot \left(1 + e^{\frac{-x}{s}}\right)\right)} \]
20. Add Preprocessing

Alternative 3: 74.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;\left|x\right| \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{\frac{s}{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(t\_0\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(s, t\_0, s\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= (fabs x) 0.10000000149011612)
     (/ 1.0 (/ s (exp (+ (/ x s) (* -2.0 (log1p t_0))))))
     (/ 0.5 (fma s t_0 s)))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (fabsf(x) <= 0.10000000149011612f) {
		tmp = 1.0f / (s / expf(((x / s) + (-2.0f * log1pf(t_0)))));
	} else {
		tmp = 0.5f / fmaf(s, t_0, s);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) / Float32(s / exp(Float32(Float32(x / s) + Float32(Float32(-2.0) * log1p(t_0))))));
	else
		tmp = Float32(Float32(0.5) / fma(s, t_0, s));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.100000001

   1. Initial program 98.9%

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

      1. fabs-neg 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-frac-neg2 98.9%

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

      4. fabs-neg 98.9%

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

      5. *-commutative 98.9%

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

      6. fabs-neg 98.9%

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

      7. +-commutative 98.9%

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

      8. fabs-neg 98.9%

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

   3. Simplified 98.9%

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

      1. div-inv 98.9%

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

      2. distribute-frac-neg2 98.9%

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

      3. distribute-frac-neg 98.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 46.5%

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

      6. sqr-neg 46.5%

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

      7. sqrt-unprod 46.1%

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

      8. add-sqr-sqrt 46.1%

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

      9. add-sqr-sqrt 21.2%

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

      10. fabs-sqr 21.2%

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

      11. add-sqr-sqrt 73.4%

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

      12. add-sqr-sqrt 72.9%

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

   6. Applied egg-rr 82.1%

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

      1. *-commutative 82.1%

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

      2. exp-to-pow 82.1%

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

      3. +-commutative 82.1%

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

      4. log1p-undefine 82.2%

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

      5. *-commutative 82.2%

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

      6. rem-exp-log 78.1%

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

      7. prod-exp 79.7%

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

      8. associate-/l* 79.8%

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

      9. *-rgt-identity 79.8%

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

      10. exp-diff 94.7%

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

      11. associate--r+ 94.8%

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

      12. exp-diff 95.0%

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

   8. Simplified 98.9%

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

      1. clear-num 98.9%

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

      2. inv-pow 98.9%

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

      3. sub-neg 98.9%

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

      4. *-commutative 98.9%

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

      5. distribute-rgt-neg-in 98.9%

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

      6. metadata-eval 98.9%

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

   10. Applied egg-rr 98.9%

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

       1. unpow-1 98.9%

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

       2. +-commutative 98.9%

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

       3. +-commutative 98.9%

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

       4. *-commutative 98.9%

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

   12. Simplified 98.9%

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

   if 0.100000001 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. associate-*r* 100.0%

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

      3. times-frac 100.0%

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

   6. Applied egg-rr 41.8%

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

      1. associate-*l/ 41.8%

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

      2. *-lft-identity 41.8%

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

      3. +-commutative 41.8%

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

   8. Simplified 41.8%

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

   9. Taylor expanded in x around 0 60.1%

      \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{\frac{s}{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;\left|x\right| \leq 0.10000000149011612:\\ \;\;\;\;\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(t\_0\right) \cdot 2}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(s, t\_0, s\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= (fabs x) 0.10000000149011612)
     (/ (exp (- (/ x s) (* (log1p t_0) 2.0))) s)
     (/ 0.5 (fma s t_0 s)))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (fabsf(x) <= 0.10000000149011612f) {
		tmp = expf(((x / s) - (log1pf(t_0) * 2.0f))) / s;
	} else {
		tmp = 0.5f / fmaf(s, t_0, s);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.10000000149011612))
		tmp = Float32(exp(Float32(Float32(x / s) - Float32(log1p(t_0) * Float32(2.0)))) / s);
	else
		tmp = Float32(Float32(0.5) / fma(s, t_0, s));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.100000001

   1. Initial program 98.9%

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

      1. fabs-neg 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-frac-neg2 98.9%

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

      4. fabs-neg 98.9%

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

      5. *-commutative 98.9%

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

      6. fabs-neg 98.9%

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

      7. +-commutative 98.9%

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

      8. fabs-neg 98.9%

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

   3. Simplified 98.9%

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

      1. div-inv 98.9%

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

      2. distribute-frac-neg2 98.9%

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

      3. distribute-frac-neg 98.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 46.5%

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

      6. sqr-neg 46.5%

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

      7. sqrt-unprod 46.1%

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

      8. add-sqr-sqrt 46.1%

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

      9. add-sqr-sqrt 21.2%

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

      10. fabs-sqr 21.2%

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

      11. add-sqr-sqrt 73.4%

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

      12. add-sqr-sqrt 72.9%

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

   6. Applied egg-rr 82.1%

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

      1. *-commutative 82.1%

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

      2. exp-to-pow 82.1%

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

      3. +-commutative 82.1%

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

      4. log1p-undefine 82.2%

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

      5. *-commutative 82.2%

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

      6. rem-exp-log 78.1%

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

      7. prod-exp 79.7%

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

      8. associate-/l* 79.8%

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

      9. *-rgt-identity 79.8%

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

      10. exp-diff 94.7%

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

      11. associate--r+ 94.8%

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

      12. exp-diff 95.0%

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

   8. Simplified 98.9%

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

   if 0.100000001 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. associate-*r* 100.0%

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

      3. times-frac 100.0%

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

   6. Applied egg-rr 41.8%

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

      1. associate-*l/ 41.8%

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

      2. *-lft-identity 41.8%

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

      3. +-commutative 41.8%

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

   8. Simplified 41.8%

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

   9. Taylor expanded in x around 0 60.1%

      \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.10000000149011612:\\ \;\;\;\;\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot 2}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (fma s (exp (/ x s)) s)))

C

float code(float x, float s) {
	return 0.5f / fmaf(s, expf((x / s)), s);
}

Julia

function code(x, s)
	return Float32(Float32(0.5) / fma(s, exp(Float32(x / s)), s))
end

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. associate-*r* 99.5%

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

   3. times-frac 99.5%

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

6. Applied egg-rr 62.0%

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

   1. associate-*l/ 62.0%

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

   2. *-lft-identity 62.0%

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

   3. +-commutative 62.0%

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

8. Simplified 62.0%

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

9. Taylor expanded in x around 0 61.6%

   \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]

10. Final simplification 61.6%

    \[\leadsto \frac{0.5}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \]
11. Add Preprocessing

Alternative 6: 50.9% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.5%

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

   2. distribute-frac-neg2 99.5%

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

   3. distribute-frac-neg 99.5%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 23.8%

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

   6. sqr-neg 23.8%

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

   7. sqrt-unprod 23.6%

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

   8. add-sqr-sqrt 23.6%

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

   9. add-sqr-sqrt 11.0%

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

   10. fabs-sqr 11.0%

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

   11. add-sqr-sqrt 57.8%

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

   12. add-sqr-sqrt 57.6%

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

6. Applied egg-rr 61.0%

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

   1. associate-*r/ 61.0%

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

   2. *-rgt-identity 61.0%

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

   3. associate-/r* 60.3%

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

   4. +-commutative 60.3%

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

8. Simplified 60.3%

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

9. Taylor expanded in x around 0 57.4%

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

    1. *-commutative 57.4%

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

11. Simplified 57.4%

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

12. Taylor expanded in x around 0 53.9%

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

13. Final simplification 53.9%

    \[\leadsto \frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot 4} \]
14. Add Preprocessing

Alternative 7: 28.9% accurate, 77.2× speedup

Math

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

FPCore

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

C

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

Derivation

1. Split input into 2 regimes

2. if x < 0.0500000007

   1. Initial program 99.3%

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

      1. fabs-neg 99.3%

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

      2. distribute-frac-neg 99.3%

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

      3. distribute-frac-neg2 99.3%

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

      4. fabs-neg 99.3%

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

      5. *-commutative 99.3%

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

      6. fabs-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. fabs-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in s around inf 37.3%

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

   if 0.0500000007 < x

   1. Initial program 99.9%

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

      1. fabs-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-frac-neg2 99.9%

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

      4. fabs-neg 99.9%

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

      5. *-commutative 99.9%

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

      6. fabs-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. fabs-neg 99.9%

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

   3. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. distribute-frac-neg2 99.9%

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

      3. distribute-frac-neg 99.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 3.3%

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

      6. sqr-neg 3.3%

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

      7. sqrt-unprod 3.3%

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

      8. add-sqr-sqrt 3.3%

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

      9. add-sqr-sqrt 3.3%

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

      10. fabs-sqr 3.3%

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

      11. add-sqr-sqrt 3.3%

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

      12. add-sqr-sqrt 3.3%

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

   6. Applied egg-rr 1.3%

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

      1. associate-*r/ 1.2%

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

      2. *-rgt-identity 1.2%

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

      3. associate-/r* 1.3%

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

      4. +-commutative 1.3%

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

   8. Simplified 1.3%

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

   9. Taylor expanded in x around 0 1.5%

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

       1. *-commutative 1.5%

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

   11. Simplified 1.5%

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

   12. Taylor expanded in x around 0 61.0%

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

   13. Taylor expanded in s around 0 12.0%

       \[\leadsto \color{blue}{\frac{0.25}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.5%

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

Alternative 8: 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%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around inf 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))))))