Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 11.5s

Alternatives: 16

Speedup: 1.2×

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. *-lft-identity 99.7%

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

   2. associate-*r/ 99.7%

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

   3. associate-/l* 99.7%

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

   4. distribute-frac-neg 99.7%

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

   5. exp-neg 99.8%

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

   6. associate-/r/ 99.8%

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

   7. /-rgt-identity 99.8%

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

   8. associate-*l* 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. *-lft-identity 99.7%

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

   2. associate-*r/ 99.7%

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

   3. associate-*l* 99.8%

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

   4. times-frac 98.9%

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

   5. associate-*r/ 99.0%

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

   6. associate-/l* 98.9%

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

   7. distribute-frac-neg 98.9%

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

   8. exp-neg 99.0%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 3: 97.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. *-lft-identity 99.7%

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

   2. associate-*r/ 99.7%

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

   3. associate-*l* 99.8%

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

   4. times-frac 98.9%

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

   5. associate-*r/ 99.0%

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

   6. associate-/l* 98.9%

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

   7. distribute-frac-neg 98.9%

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

   8. exp-neg 99.0%

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

3. Simplified 99.1%

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

   1. expm1-log1p-u 99.1%

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

   2. expm1-udef 99.1%

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

   3. log1p-udef 99.1%

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

   4. +-commutative 99.1%

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

   5. add-exp-log 99.1%

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

   6. add-sqr-sqrt 53.1%

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

   7. fabs-sqr 53.1%

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

   8. add-sqr-sqrt 97.2%

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

   9. add-sqr-sqrt -0.0%

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

   10. sqrt-unprod 94.5%

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

   11. sqr-neg 94.5%

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

   12. sqrt-unprod 97.2%

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

   13. add-sqr-sqrt 97.2%

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

5. Applied egg-rr 97.2%

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

   1. associate--l+ 97.2%

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

   2. metadata-eval 97.2%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(e^{\frac{x}{s}} + \color{blue}{0}\right) + 2\right)} \]

   3. +-rgt-identity 97.2%

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

7. Simplified 97.2%

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

8. Final simplification 97.2%

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

Alternative 4: 95.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.000000023742228 \cdot 10^{-33}:\\ \;\;\;\;{\left(\frac{s \cdot 4}{t_0}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= x -1.000000023742228e-33)
     (pow (/ (* s 4.0) t_0) -1.0)
     (/ 1.0 (* s (+ t_0 3.0))))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (x <= -1.000000023742228e-33f) {
		tmp = powf(((s * 4.0f) / t_0), -1.0f);
	} else {
		tmp = 1.0f / (s * (t_0 + 3.0f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = exp((x / s))
    if (x <= (-1.000000023742228e-33)) then
        tmp = ((s * 4.0e0) / t_0) ** (-1.0e0)
    else
        tmp = 1.0e0 / (s * (t_0 + 3.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (x <= Float32(-1.000000023742228e-33))
		tmp = Float32(Float32(s * Float32(4.0)) / t_0) ^ Float32(-1.0);
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(t_0 + Float32(3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((x / s));
	tmp = single(0.0);
	if (x <= single(-1.000000023742228e-33))
		tmp = ((s * single(4.0)) / t_0) ^ single(-1.0);
	else
		tmp = single(1.0) / (s * (t_0 + single(3.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000002e-33

   1. Initial program 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 95.4%

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

   if -1.00000002e-33 < 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. *-lft-identity 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l* 99.8%

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

      4. times-frac 99.2%

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

      5. associate-*r/ 99.2%

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

      6. associate-/l* 99.1%

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

      7. distribute-frac-neg 99.1%

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

      8. exp-neg 99.2%

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

   3. Simplified 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 99.2%

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

      3. log1p-udef 99.3%

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

      4. +-commutative 99.3%

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

      5. add-exp-log 99.3%

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

      6. add-sqr-sqrt 92.5%

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

      7. fabs-sqr 92.5%

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

      8. add-sqr-sqrt 99.0%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 91.3%

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

      11. sqr-neg 91.3%

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

      12. sqrt-unprod 96.0%

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

      13. add-sqr-sqrt 96.0%

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

   5. Applied egg-rr 96.0%

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

      1. associate--l+ 96.0%

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

      2. metadata-eval 96.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(e^{\frac{x}{s}} + \color{blue}{0}\right) + 2\right)} \]

      3. +-rgt-identity 96.0%

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

   7. Simplified 96.0%

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

   8. Taylor expanded in x around 0 96.7%

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

      1. +-commutative 96.7%

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

   10. Simplified 96.7%

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

       1. distribute-lft-in 96.7%

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

       2. div-inv 96.7%

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

       3. exp-prod 81.5%

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

       4. add-sqr-sqrt 74.8%

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

       5. fabs-sqr 74.8%

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

       6. add-sqr-sqrt 81.5%

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

       7. exp-prod 96.7%

          \[\leadsto \frac{1}{s \cdot \color{blue}{e^{x \cdot \frac{1}{s}}} + s \cdot 3} \]

       8. div-inv 96.7%

          \[\leadsto \frac{1}{s \cdot e^{\color{blue}{\frac{x}{s}}} + s \cdot 3} \]

   12. Applied egg-rr 96.7%

       \[\leadsto \frac{1}{\color{blue}{s \cdot e^{\frac{x}{s}} + s \cdot 3}} \]
   13. Step-by-step derivation

       1. distribute-lft-in 96.7%

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

       2. +-commutative 96.7%

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

   14. Simplified 96.7%

       \[\leadsto \frac{1}{\color{blue}{s \cdot \left(3 + e^{\frac{x}{s}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.000000023742228 \cdot 10^{-33}:\\ \;\;\;\;{\left(\frac{s \cdot 4}{e^{\frac{x}{s}}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)}\\ \end{array} \]

Alternative 5: 89.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9999998795923744 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{x}{s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.9999998795923744e-13)
   (/ 1.0 (log1p (expm1 (* x (/ x s)))))
   (/ 1.0 (* s (+ (exp (/ x s)) 3.0)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.9999998795923744e-13f) {
		tmp = 1.0f / log1pf(expm1f((x * (x / s))));
	} else {
		tmp = 1.0f / (s * (expf((x / s)) + 3.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.9999998795923744e-13))
		tmp = Float32(Float32(1.0) / log1p(expm1(Float32(x * Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(exp(Float32(x / s)) + Float32(3.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -2.99999988e-13

   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. *-lft-identity 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-/l* 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. exp-neg 99.9%

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

      6. associate-/r/ 99.9%

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

      7. /-rgt-identity 99.9%

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

      8. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in s around inf 6.1%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+ 6.1%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative 6.1%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+ 6.1%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified 6.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 59.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   8. Step-by-step derivation

      1. unpow2 59.6%

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

      2. associate-/l* 59.6%

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

   9. Simplified 59.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{s}{x}}}} \]
   10. Step-by-step derivation

       1. associate-/l* 59.6%

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

       2. add-sqr-sqrt 59.6%

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

       3. fabs-sqr 59.6%

          \[\leadsto \frac{1}{\color{blue}{\left|\sqrt{\frac{x \cdot x}{s}} \cdot \sqrt{\frac{x \cdot x}{s}}\right|}} \]

       4. add-sqr-sqrt 59.6%

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

       5. associate-/l* 59.6%

          \[\leadsto \frac{1}{\left|\color{blue}{\frac{x}{\frac{s}{x}}}\right|} \]

       6. log1p-expm1-u 93.9%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left|\frac{x}{\frac{s}{x}}\right|\right)\right)}} \]

       7. associate-/l* 93.9%

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

       8. add-sqr-sqrt 93.9%

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

       9. fabs-sqr 93.9%

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

       10. add-sqr-sqrt 93.9%

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

       11. associate-*r/ 93.9%

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

   11. Applied egg-rr 93.9%

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

   if -2.99999988e-13 < x

   1. Initial program 99.7%

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

      1. *-lft-identity 99.7%

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

      2. associate-*r/ 99.7%

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

      3. associate-*l* 99.7%

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

      4. times-frac 99.1%

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

      5. associate-*r/ 99.2%

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

      6. associate-/l* 99.1%

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

      7. distribute-frac-neg 99.1%

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

      8. exp-neg 99.2%

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

   3. Simplified 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 99.3%

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

      3. log1p-udef 99.3%

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

      4. +-commutative 99.3%

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

      5. add-exp-log 99.3%

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

      6. add-sqr-sqrt 74.7%

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

      7. fabs-sqr 74.7%

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

      8. add-sqr-sqrt 97.3%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 92.8%

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

      11. sqr-neg 92.8%

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

      12. sqrt-unprod 96.7%

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

      13. add-sqr-sqrt 96.7%

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

   5. Applied egg-rr 96.7%

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

      1. associate--l+ 96.7%

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

      2. metadata-eval 96.7%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(e^{\frac{x}{s}} + \color{blue}{0}\right) + 2\right)} \]

      3. +-rgt-identity 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in x around 0 96.4%

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

      1. +-commutative 96.4%

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

   10. Simplified 96.4%

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

       1. distribute-lft-in 96.4%

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

       2. div-inv 96.4%

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

       3. exp-prod 77.8%

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

       4. add-sqr-sqrt 60.4%

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

       5. fabs-sqr 60.4%

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

       6. add-sqr-sqrt 77.8%

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

       7. exp-prod 89.5%

          \[\leadsto \frac{1}{s \cdot \color{blue}{e^{x \cdot \frac{1}{s}}} + s \cdot 3} \]

       8. div-inv 89.5%

          \[\leadsto \frac{1}{s \cdot e^{\color{blue}{\frac{x}{s}}} + s \cdot 3} \]

   12. Applied egg-rr 89.5%

       \[\leadsto \frac{1}{\color{blue}{s \cdot e^{\frac{x}{s}} + s \cdot 3}} \]
   13. Step-by-step derivation

       1. distribute-lft-in 89.5%

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

       2. +-commutative 89.5%

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

   14. Simplified 89.5%

       \[\leadsto \frac{1}{\color{blue}{s \cdot \left(3 + e^{\frac{x}{s}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9999998795923744 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \frac{x}{s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)}\\ \end{array} \]

Alternative 6: 96.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \left(e^{\frac{\left|x\right|}{s}} + 3\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (* s (+ (exp (/ (fabs x) s)) 3.0))))

C

float code(float x, float s) {
	return 1.0f / (s * (expf((fabsf(x) / s)) + 3.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * (exp((abs(x) / s)) + 3.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(exp(Float32(abs(x) / s)) + Float32(3.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * (exp((abs(x) / s)) + single(3.0)));
end

Derivation

1. Initial program 99.7%

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

   1. *-lft-identity 99.7%

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

   2. associate-*r/ 99.7%

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

   3. associate-*l* 99.8%

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

   4. times-frac 98.9%

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

   5. associate-*r/ 99.0%

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

   6. associate-/l* 98.9%

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

   7. distribute-frac-neg 98.9%

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

   8. exp-neg 99.0%

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

3. Simplified 99.1%

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

   1. expm1-log1p-u 99.1%

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

   2. expm1-udef 99.1%

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

   3. log1p-udef 99.1%

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

   4. +-commutative 99.1%

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

   5. add-exp-log 99.1%

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

   6. add-sqr-sqrt 53.1%

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

   7. fabs-sqr 53.1%

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

   8. add-sqr-sqrt 97.2%

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

   9. add-sqr-sqrt -0.0%

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

   10. sqrt-unprod 94.5%

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

   11. sqr-neg 94.5%

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

   12. sqrt-unprod 97.2%

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

   13. add-sqr-sqrt 97.2%

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

5. Applied egg-rr 97.2%

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

   1. associate--l+ 97.2%

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

   2. metadata-eval 97.2%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(e^{\frac{x}{s}} + \color{blue}{0}\right) + 2\right)} \]

   3. +-rgt-identity 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in x around 0 97.2%

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

   1. +-commutative 97.2%

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

10. Simplified 97.2%

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

11. Final simplification 97.2%

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

Alternative 7: 89.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.000000156871975e-23)
   (/ (/ 1.0 s) (+ 4.0 (/ (* x x) (* s s))))
   (/ 1.0 (* s (+ (exp (/ x s)) 3.0)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.000000156871975e-23f) {
		tmp = (1.0f / s) / (4.0f + ((x * x) / (s * s)));
	} else {
		tmp = 1.0f / (s * (expf((x / s)) + 3.0f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.000000156871975e-23)) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x * x) / (s * s)))
    else
        tmp = 1.0e0 / (s * (exp((x / s)) + 3.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.000000156871975e-23))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) / Float32(s * s))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(exp(Float32(x / s)) + Float32(3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.000000156871975e-23))
		tmp = (single(1.0) / s) / (single(4.0) + ((x * x) / (s * s)));
	else
		tmp = single(1.0) / (s * (exp((x / s)) + single(3.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000016e-23

   1. Initial program 99.7%

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

      1. *-lft-identity 99.7%

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

      2. associate-*r/ 99.7%

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

      3. associate-*l* 99.8%

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

      4. times-frac 98.6%

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

      5. associate-*r/ 98.7%

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

      6. associate-/l* 98.6%

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

      7. distribute-frac-neg 98.6%

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

      8. exp-neg 98.8%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in s around inf 51.4%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+ 51.4%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 51.4%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 51.4%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 80.2%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 80.2%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 80.2%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 80.2%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 80.2%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 80.2%

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

   6. Simplified 80.2%

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

   if -5.00000016e-23 < x

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

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

      2. associate-*r/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. times-frac 99.1%

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

      5. associate-*r/ 99.2%

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

      6. associate-/l* 99.1%

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

      7. distribute-frac-neg 99.1%

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

      8. exp-neg 99.2%

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

   3. Simplified 99.2%

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

      1. expm1-log1p-u 99.2%

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

      2. expm1-udef 99.2%

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

      3. log1p-udef 99.2%

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

      4. +-commutative 99.2%

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

      5. add-exp-log 99.2%

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

      6. add-sqr-sqrt 85.5%

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

      7. fabs-sqr 85.5%

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

      8. add-sqr-sqrt 98.1%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 91.8%

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

      11. sqr-neg 91.8%

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

      12. sqrt-unprod 96.2%

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

      13. add-sqr-sqrt 96.2%

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

   5. Applied egg-rr 96.2%

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

      1. associate--l+ 96.2%

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

      2. metadata-eval 96.2%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(e^{\frac{x}{s}} + \color{blue}{0}\right) + 2\right)} \]

      3. +-rgt-identity 96.2%

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

   7. Simplified 96.2%

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

   8. Taylor expanded in x around 0 96.9%

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

      1. +-commutative 96.9%

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

   10. Simplified 96.9%

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

       1. distribute-lft-in 96.9%

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

       2. div-inv 96.9%

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

       3. exp-prod 79.3%

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

       4. add-sqr-sqrt 69.1%

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

       5. fabs-sqr 69.1%

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

       6. add-sqr-sqrt 79.3%

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

       7. exp-prod 93.3%

          \[\leadsto \frac{1}{s \cdot \color{blue}{e^{x \cdot \frac{1}{s}}} + s \cdot 3} \]

       8. div-inv 93.3%

          \[\leadsto \frac{1}{s \cdot e^{\color{blue}{\frac{x}{s}}} + s \cdot 3} \]

   12. Applied egg-rr 93.3%

       \[\leadsto \frac{1}{\color{blue}{s \cdot e^{\frac{x}{s}} + s \cdot 3}} \]
   13. Step-by-step derivation

       1. distribute-lft-in 93.3%

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

       2. +-commutative 93.3%

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

   14. Simplified 93.3%

       \[\leadsto \frac{1}{\color{blue}{s \cdot \left(3 + e^{\frac{x}{s}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)}\\ \end{array} \]

Alternative 8: 95.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.000000023742228 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{t_0}{s}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= x -1.000000023742228e-33)
     (/ (/ t_0 s) 4.0)
     (/ 1.0 (* s (+ t_0 3.0))))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (x <= -1.000000023742228e-33f) {
		tmp = (t_0 / s) / 4.0f;
	} else {
		tmp = 1.0f / (s * (t_0 + 3.0f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = exp((x / s))
    if (x <= (-1.000000023742228e-33)) then
        tmp = (t_0 / s) / 4.0e0
    else
        tmp = 1.0e0 / (s * (t_0 + 3.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (x <= Float32(-1.000000023742228e-33))
		tmp = Float32(Float32(t_0 / s) / Float32(4.0));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(t_0 + Float32(3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((x / s));
	tmp = single(0.0);
	if (x <= single(-1.000000023742228e-33))
		tmp = (t_0 / s) / single(4.0);
	else
		tmp = single(1.0) / (s * (t_0 + single(3.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000002e-33

   1. Initial program 99.6%

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

      1. expm1-log1p-u 98.7%

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

      2. expm1-udef 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. expm1-def 98.7%

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

      2. expm1-log1p 99.7%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 95.3%

      \[\leadsto \frac{\frac{e^{\frac{x}{s}}}{s}}{{\color{blue}{2}}^{2}} \]

   if -1.00000002e-33 < 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. *-lft-identity 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l* 99.8%

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

      4. times-frac 99.2%

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

      5. associate-*r/ 99.2%

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

      6. associate-/l* 99.1%

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

      7. distribute-frac-neg 99.1%

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

      8. exp-neg 99.2%

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

   3. Simplified 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 99.2%

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

      3. log1p-udef 99.3%

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

      4. +-commutative 99.3%

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

      5. add-exp-log 99.3%

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

      6. add-sqr-sqrt 92.5%

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

      7. fabs-sqr 92.5%

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

      8. add-sqr-sqrt 99.0%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 91.3%

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

      11. sqr-neg 91.3%

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

      12. sqrt-unprod 96.0%

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

      13. add-sqr-sqrt 96.0%

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

   5. Applied egg-rr 96.0%

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

      1. associate--l+ 96.0%

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

      2. metadata-eval 96.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(e^{\frac{x}{s}} + \color{blue}{0}\right) + 2\right)} \]

      3. +-rgt-identity 96.0%

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

   7. Simplified 96.0%

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

   8. Taylor expanded in x around 0 96.7%

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

      1. +-commutative 96.7%

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

   10. Simplified 96.7%

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

       1. distribute-lft-in 96.7%

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

       2. div-inv 96.7%

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

       3. exp-prod 81.5%

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

       4. add-sqr-sqrt 74.8%

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

       5. fabs-sqr 74.8%

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

       6. add-sqr-sqrt 81.5%

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

       7. exp-prod 96.7%

          \[\leadsto \frac{1}{s \cdot \color{blue}{e^{x \cdot \frac{1}{s}}} + s \cdot 3} \]

       8. div-inv 96.7%

          \[\leadsto \frac{1}{s \cdot e^{\color{blue}{\frac{x}{s}}} + s \cdot 3} \]

   12. Applied egg-rr 96.7%

       \[\leadsto \frac{1}{\color{blue}{s \cdot e^{\frac{x}{s}} + s \cdot 3}} \]
   13. Step-by-step derivation

       1. distribute-lft-in 96.7%

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

       2. +-commutative 96.7%

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

   14. Simplified 96.7%

       \[\leadsto \frac{1}{\color{blue}{s \cdot \left(3 + e^{\frac{x}{s}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.000000023742228 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{e^{\frac{x}{s}}}{s}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)}\\ \end{array} \]

Alternative 9: 64.7% accurate, 46.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257 \lor \neg \left(x \leq 9.999999974752427 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{1}{\frac{x}{s} \cdot \left(x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -0.0005000000237487257) (not (<= x 9.999999974752427e-7)))
   (/ 1.0 (* (/ x s) (* x 2.0)))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -0.0005000000237487257f) || !(x <= 9.999999974752427e-7f)) {
		tmp = 1.0f / ((x / s) * (x * 2.0f));
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-0.0005000000237487257e0)) .or. (.not. (x <= 9.999999974752427e-7))) then
        tmp = 1.0e0 / ((x / s) * (x * 2.0e0))
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-0.0005000000237487257)) || !(x <= Float32(9.999999974752427e-7)))
		tmp = Float32(Float32(1.0) / Float32(Float32(x / s) * Float32(x * Float32(2.0))));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-0.0005000000237487257)) || ~((x <= single(9.999999974752427e-7))))
		tmp = single(1.0) / ((x / s) * (x * single(2.0)));
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000024e-4 or 9.99999997e-7 < 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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 4.8%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+ 4.8%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative 4.8%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+ 4.8%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified 4.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]
   7. Step-by-step derivation

      1. fma-udef 4.8%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right)} - \frac{x \cdot x}{s}} \]

   8. Applied egg-rr 4.8%

      \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right)} - \frac{x \cdot x}{s}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 4.8%

         \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\sqrt{\frac{x \cdot x}{s}} \cdot \sqrt{\frac{x \cdot x}{s}}}} \]

      2. sqrt-unprod 27.0%

         \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\sqrt{\frac{x \cdot x}{s} \cdot \frac{x \cdot x}{s}}}} \]

      3. pow2 27.0%

         \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \sqrt{\color{blue}{{\left(\frac{x \cdot x}{s}\right)}^{2}}}} \]

   10. Applied egg-rr 27.0%

       \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\sqrt{{\left(\frac{x \cdot x}{s}\right)}^{2}}}} \]
   11. Step-by-step derivation

       1. unpow2 27.0%

          \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \sqrt{\color{blue}{\frac{x \cdot x}{s} \cdot \frac{x \cdot x}{s}}}} \]

       2. rem-sqrt-square 4.8%

          \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\left|\frac{x \cdot x}{s}\right|}} \]

       3. associate-/l* 4.8%

          \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \left|\color{blue}{\frac{x}{\frac{s}{x}}}\right|} \]

   12. Simplified 4.8%

       \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\left|\frac{x}{\frac{s}{x}}\right|}} \]

   13. Taylor expanded in x around inf 69.4%

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

       1. *-commutative 69.4%

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

       2. unpow2 69.4%

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

       3. associate-*l/ 69.4%

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

       4. associate-*r* 69.4%

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

   15. Simplified 69.4%

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

   if -5.00000024e-4 < x < 9.99999997e-7

   1. Initial program 99.4%

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

   2. Taylor expanded in s around inf 59.3%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257 \lor \neg \left(x \leq 9.999999974752427 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{1}{\frac{x}{s} \cdot \left(x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 10: 78.6% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. *-lft-identity 99.7%

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

   2. associate-*r/ 99.7%

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

   3. associate-*l* 99.8%

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

   4. times-frac 98.9%

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

   5. associate-*r/ 99.0%

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

   6. associate-/l* 98.9%

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

   7. distribute-frac-neg 98.9%

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

   8. exp-neg 99.0%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around inf 52.6%

   \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)\right)}} \]
5. Step-by-step derivation

   1. associate-+r+ 52.6%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

   2. distribute-lft1-in 52.6%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

   3. metadata-eval 52.6%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

   4. mul0-lft 76.9%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

   5. associate-+r+ 76.9%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

   6. metadata-eval 76.9%

      \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

   7. unpow2 76.9%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

   8. sqr-abs 76.9%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

   9. unpow2 76.9%

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

6. Simplified 76.9%

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

7. Final simplification 76.9%

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

Alternative 11: 63.5% accurate, 55.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257 \lor \neg \left(x \leq 9.999999974752427 \cdot 10^{-7}\right):\\ \;\;\;\;0.5 \cdot \frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -0.0005000000237487257) (not (<= x 9.999999974752427e-7)))
   (* 0.5 (/ s (* x x)))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -0.0005000000237487257f) || !(x <= 9.999999974752427e-7f)) {
		tmp = 0.5f * (s / (x * x));
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-0.0005000000237487257e0)) .or. (.not. (x <= 9.999999974752427e-7))) then
        tmp = 0.5e0 * (s / (x * x))
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-0.0005000000237487257)) || !(x <= Float32(9.999999974752427e-7)))
		tmp = Float32(Float32(0.5) * Float32(s / Float32(x * x)));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-0.0005000000237487257)) || ~((x <= single(9.999999974752427e-7))))
		tmp = single(0.5) * (s / (x * x));
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000024e-4 or 9.99999997e-7 < 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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 4.8%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+ 4.8%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative 4.8%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+ 4.8%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified 4.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]
   7. Step-by-step derivation

      1. fma-udef 4.8%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right)} - \frac{x \cdot x}{s}} \]

   8. Applied egg-rr 4.8%

      \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right)} - \frac{x \cdot x}{s}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 4.8%

         \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\sqrt{\frac{x \cdot x}{s}} \cdot \sqrt{\frac{x \cdot x}{s}}}} \]

      2. sqrt-unprod 27.0%

         \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\sqrt{\frac{x \cdot x}{s} \cdot \frac{x \cdot x}{s}}}} \]

      3. pow2 27.0%

         \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \sqrt{\color{blue}{{\left(\frac{x \cdot x}{s}\right)}^{2}}}} \]

   10. Applied egg-rr 27.0%

       \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\sqrt{{\left(\frac{x \cdot x}{s}\right)}^{2}}}} \]
   11. Step-by-step derivation

       1. unpow2 27.0%

          \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \sqrt{\color{blue}{\frac{x \cdot x}{s} \cdot \frac{x \cdot x}{s}}}} \]

       2. rem-sqrt-square 4.8%

          \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\left|\frac{x \cdot x}{s}\right|}} \]

       3. associate-/l* 4.8%

          \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \left|\color{blue}{\frac{x}{\frac{s}{x}}}\right|} \]

   12. Simplified 4.8%

       \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\left|\frac{x}{\frac{s}{x}}\right|}} \]

   13. Taylor expanded in x around inf 67.5%

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

       1. unpow2 67.5%

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

   15. Simplified 67.5%

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

   if -5.00000024e-4 < x < 9.99999997e-7

   1. Initial program 99.4%

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

   2. Taylor expanded in s around inf 59.3%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257 \lor \neg \left(x \leq 9.999999974752427 \cdot 10^{-7}\right):\\ \;\;\;\;0.5 \cdot \frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 12: 64.5% accurate, 55.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257 \lor \neg \left(x \leq 9.999999974752427 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -0.0005000000237487257) (not (<= x 9.999999974752427e-7)))
   (/ 1.0 (/ x (/ s x)))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -0.0005000000237487257f) || !(x <= 9.999999974752427e-7f)) {
		tmp = 1.0f / (x / (s / x));
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-0.0005000000237487257e0)) .or. (.not. (x <= 9.999999974752427e-7))) then
        tmp = 1.0e0 / (x / (s / x))
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-0.0005000000237487257)) || !(x <= Float32(9.999999974752427e-7)))
		tmp = Float32(Float32(1.0) / Float32(x / Float32(s / x)));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-0.0005000000237487257)) || ~((x <= single(9.999999974752427e-7))))
		tmp = single(1.0) / (x / (s / x));
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000024e-4 or 9.99999997e-7 < 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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 4.8%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+ 4.8%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative 4.8%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+ 4.8%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified 4.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 69.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   8. Step-by-step derivation

      1. unpow2 69.3%

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

      2. associate-/l* 69.3%

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

   9. Simplified 69.3%

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

   if -5.00000024e-4 < x < 9.99999997e-7

   1. Initial program 99.4%

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

   2. Taylor expanded in s around inf 59.3%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257 \lor \neg \left(x \leq 9.999999974752427 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 13: 63.4% accurate, 55.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257:\\ \;\;\;\;0.5 \cdot \frac{s}{x \cdot x}\\ \mathbf{elif}\;x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0005000000237487257)
   (* 0.5 (/ s (* x x)))
   (if (<= x 9.999999974752427e-7) (/ 0.25 s) (* (/ s x) (/ 0.5 x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0005000000237487257f) {
		tmp = 0.5f * (s / (x * x));
	} else if (x <= 9.999999974752427e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = (s / x) * (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 <= (-0.0005000000237487257e0)) then
        tmp = 0.5e0 * (s / (x * x))
    else if (x <= 9.999999974752427e-7) then
        tmp = 0.25e0 / s
    else
        tmp = (s / x) * (0.5e0 / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0005000000237487257))
		tmp = Float32(Float32(0.5) * Float32(s / Float32(x * x)));
	elseif (x <= Float32(9.999999974752427e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(s / x) * Float32(Float32(0.5) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0005000000237487257))
		tmp = single(0.5) * (s / (x * x));
	elseif (x <= single(9.999999974752427e-7))
		tmp = single(0.25) / s;
	else
		tmp = (s / x) * (single(0.5) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5.00000024e-4

   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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 5.1%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+ 5.1%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative 5.1%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+ 5.1%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified 5.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]
   7. Step-by-step derivation

      1. fma-udef 5.1%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right)} - \frac{x \cdot x}{s}} \]

   8. Applied egg-rr 5.1%

      \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right)} - \frac{x \cdot x}{s}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 5.1%

         \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\sqrt{\frac{x \cdot x}{s}} \cdot \sqrt{\frac{x \cdot x}{s}}}} \]

      2. sqrt-unprod 31.1%

         \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\sqrt{\frac{x \cdot x}{s} \cdot \frac{x \cdot x}{s}}}} \]

      3. pow2 31.1%

         \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \sqrt{\color{blue}{{\left(\frac{x \cdot x}{s}\right)}^{2}}}} \]

   10. Applied egg-rr 31.1%

       \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\sqrt{{\left(\frac{x \cdot x}{s}\right)}^{2}}}} \]
   11. Step-by-step derivation

       1. unpow2 31.1%

          \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \sqrt{\color{blue}{\frac{x \cdot x}{s} \cdot \frac{x \cdot x}{s}}}} \]

       2. rem-sqrt-square 5.1%

          \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\left|\frac{x \cdot x}{s}\right|}} \]

       3. associate-/l* 5.1%

          \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \left|\color{blue}{\frac{x}{\frac{s}{x}}}\right|} \]

   12. Simplified 5.1%

       \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\left|\frac{x}{\frac{s}{x}}\right|}} \]

   13. Taylor expanded in x around inf 63.3%

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

       1. unpow2 63.3%

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

   15. Simplified 63.3%

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

   if -5.00000024e-4 < x < 9.99999997e-7

   1. Initial program 99.4%

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

   2. Taylor expanded in s around inf 59.3%

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

   if 9.99999997e-7 < 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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 4.5%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+ 4.5%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative 4.5%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+ 4.5%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified 4.5%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]
   7. Step-by-step derivation

      1. fma-udef 4.5%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right)} - \frac{x \cdot x}{s}} \]

   8. Applied egg-rr 4.5%

      \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right)} - \frac{x \cdot x}{s}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 4.5%

         \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\sqrt{\frac{x \cdot x}{s}} \cdot \sqrt{\frac{x \cdot x}{s}}}} \]

      2. sqrt-unprod 23.4%

         \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\sqrt{\frac{x \cdot x}{s} \cdot \frac{x \cdot x}{s}}}} \]

      3. pow2 23.4%

         \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \sqrt{\color{blue}{{\left(\frac{x \cdot x}{s}\right)}^{2}}}} \]

   10. Applied egg-rr 23.4%

       \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\sqrt{{\left(\frac{x \cdot x}{s}\right)}^{2}}}} \]
   11. Step-by-step derivation

       1. unpow2 23.4%

          \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \sqrt{\color{blue}{\frac{x \cdot x}{s} \cdot \frac{x \cdot x}{s}}}} \]

       2. rem-sqrt-square 4.5%

          \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\left|\frac{x \cdot x}{s}\right|}} \]

       3. associate-/l* 4.5%

          \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \left|\color{blue}{\frac{x}{\frac{s}{x}}}\right|} \]

   12. Simplified 4.5%

       \[\leadsto \frac{1}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right) - \color{blue}{\left|\frac{x}{\frac{s}{x}}\right|}} \]

   13. Taylor expanded in x around inf 71.4%

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

       1. associate-*r/ 71.4%

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

       2. unpow2 71.4%

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

       3. times-frac 71.4%

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

   15. Simplified 71.4%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257:\\ \;\;\;\;0.5 \cdot \frac{s}{x \cdot x}\\ \mathbf{elif}\;x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{0.5}{x}\\ \end{array} \]

Alternative 14: 64.5% accurate, 55.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \mathbf{elif}\;x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0005000000237487257)
   (/ 1.0 (/ x (/ s x)))
   (if (<= x 9.999999974752427e-7) (/ 0.25 s) (/ 1.0 (/ (* x x) s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0005000000237487257f) {
		tmp = 1.0f / (x / (s / x));
	} else if (x <= 9.999999974752427e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / ((x * x) / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.0005000000237487257e0)) then
        tmp = 1.0e0 / (x / (s / x))
    else if (x <= 9.999999974752427e-7) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / ((x * x) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0005000000237487257))
		tmp = Float32(Float32(1.0) / Float32(x / Float32(s / x)));
	elseif (x <= Float32(9.999999974752427e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * x) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0005000000237487257))
		tmp = single(1.0) / (x / (s / x));
	elseif (x <= single(9.999999974752427e-7))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / ((x * x) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5.00000024e-4

   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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 5.1%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+ 5.1%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative 5.1%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+ 5.1%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified 5.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 65.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   8. Step-by-step derivation

      1. unpow2 65.7%

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

      2. associate-/l* 65.7%

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

   9. Simplified 65.7%

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

   if -5.00000024e-4 < x < 9.99999997e-7

   1. Initial program 99.4%

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

   2. Taylor expanded in s around inf 59.3%

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

   if 9.99999997e-7 < 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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 4.5%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+ 4.5%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative 4.5%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+ 4.5%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified 4.5%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]
   7. Step-by-step derivation

      1. fma-udef 4.5%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right)} - \frac{x \cdot x}{s}} \]

   8. Applied egg-rr 4.5%

      \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right)} - \frac{x \cdot x}{s}} \]

   9. Taylor expanded in x around inf 72.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   10. Step-by-step derivation

       1. unpow2 72.5%

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

   11. Simplified 72.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \mathbf{elif}\;x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \]

Alternative 15: 63.3% accurate, 66.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257 \lor \neg \left(x \leq 9.999999974752427 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -0.0005000000237487257) (not (<= x 9.999999974752427e-7)))
   (/ s (* x x))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -0.0005000000237487257f) || !(x <= 9.999999974752427e-7f)) {
		tmp = s / (x * x);
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-0.0005000000237487257e0)) .or. (.not. (x <= 9.999999974752427e-7))) then
        tmp = s / (x * x)
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-0.0005000000237487257)) || !(x <= Float32(9.999999974752427e-7)))
		tmp = Float32(s / Float32(x * x));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-0.0005000000237487257)) || ~((x <= single(9.999999974752427e-7))))
		tmp = s / (x * x);
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000024e-4 or 9.99999997e-7 < 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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 4.8%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+ 4.8%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative 4.8%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+ 4.8%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified 4.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 67.2%

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 67.2%

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

   9. Simplified 67.2%

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

   if -5.00000024e-4 < x < 9.99999997e-7

   1. Initial program 99.4%

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

   2. Taylor expanded in s around inf 59.3%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257 \lor \neg \left(x \leq 9.999999974752427 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 16: 26.7% 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 99.7%

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

2. Taylor expanded in s around inf 29.7%

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

3. Final simplification 29.7%

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

Reproduce

herbie shell --seed 2023178 
(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))))))