Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 11.6s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 62.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.5%

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

   2. +-commutative 99.5%

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

5. Applied egg-rr 99.5%

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

   1. expm1-log1p-u 97.8%

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

   2. expm1-udef 97.8%

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

7. Applied egg-rr 58.5%

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

   1. expm1-def 58.5%

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

   2. expm1-log1p 59.7%

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

9. Simplified 59.7%

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

10. Final simplification 59.7%

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

Alternative 3: 62.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.5%

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

   2. +-commutative 99.5%

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

5. Applied egg-rr 99.5%

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

   1. expm1-log1p-u 97.8%

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

   2. expm1-udef 97.8%

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

7. Applied egg-rr 58.5%

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

   1. expm1-def 58.5%

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

   2. expm1-log1p 59.7%

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

9. Simplified 59.7%

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

10. Taylor expanded in x around inf 59.7%

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

    1. +-commutative 59.7%

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

    2. fma-def 59.7%

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

    3. *-commutative 59.7%

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

    4. associate-/r* 59.7%

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

    5. associate-*r/ 59.7%

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

    6. mul-1-neg 59.7%

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

12. Simplified 59.7%

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

13. Final simplification 59.7%

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

Alternative 4: 62.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. mul-1-neg 99.5%

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

   3. distribute-frac-neg 99.5%

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

6. Simplified 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. div-inv 99.5%

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

   3. exp-prod 78.0%

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

   4. add-sqr-sqrt 78.0%

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

   5. sqrt-unprod 78.0%

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

   6. sqr-neg 78.0%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 22.6%

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

   9. exp-prod 22.5%

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

   10. div-inv 22.5%

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

   11. add-sqr-sqrt -0.0%

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

   12. sqrt-unprod 95.2%

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

   13. sqr-neg 95.2%

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

   14. sqrt-unprod 99.5%

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

   15. add-sqr-sqrt 99.5%

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

   16. add-sqr-sqrt 45.7%

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

   17. fabs-sqr 45.7%

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

   18. add-sqr-sqrt 59.7%

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

8. Applied egg-rr 59.7%

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

   1. *-lft-identity 59.7%

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

10. Simplified 59.7%

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

11. Final simplification 59.7%

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

Alternative 5: 62.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.5%

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

   2. +-commutative 99.5%

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

5. Applied egg-rr 99.5%

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

   1. expm1-log1p-u 97.8%

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

   2. expm1-udef 97.8%

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

7. Applied egg-rr 58.5%

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

   1. expm1-def 58.5%

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

   2. expm1-log1p 59.7%

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

9. Simplified 59.7%

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

10. Taylor expanded in x around inf 59.7%

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

    1. +-commutative 59.7%

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

    2. fma-def 59.7%

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

    3. *-commutative 59.7%

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

    4. associate-/r* 59.7%

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

    5. associate-*r/ 59.7%

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

    6. mul-1-neg 59.7%

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

12. Simplified 59.7%

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

13. Taylor expanded in s around 0 59.7%

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

14. Final simplification 59.7%

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

Alternative 6: 81.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}} + s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{x \cdot x}{s \cdot s} + \left(4 + \frac{0}{s}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 9.999999998199587e-24)
   (/ 1.0 (+ (/ x (/ s x)) (* s 4.0)))
   (/ 1.0 (* s (+ (/ (* x x) (* s s)) (+ 4.0 (/ 0.0 s)))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 9.999999998199587e-24f) {
		tmp = 1.0f / ((x / (s / x)) + (s * 4.0f));
	} else {
		tmp = 1.0f / (s * (((x * x) / (s * s)) + (4.0f + (0.0f / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 9.999999998199587e-24) then
        tmp = 1.0e0 / ((x / (s / x)) + (s * 4.0e0))
    else
        tmp = 1.0e0 / (s * (((x * x) / (s * s)) + (4.0e0 + (0.0e0 / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(9.999999998199587e-24))
		tmp = Float32(Float32(1.0) / Float32(Float32(x / Float32(s / x)) + Float32(s * Float32(4.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(Float32(x * x) / Float32(s * s)) + Float32(Float32(4.0) + Float32(Float32(0.0) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(9.999999998199587e-24))
		tmp = single(1.0) / ((x / (s / x)) + (s * single(4.0)));
	else
		tmp = single(1.0) / (s * (((x * x) / (s * s)) + (single(4.0) + (single(0.0) / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 1e-23

   1. Initial program 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in s around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. mul-1-neg 97.7%

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

      3. distribute-frac-neg 97.7%

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

   6. Simplified 97.7%

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

   7. Taylor expanded in s around inf 76.9%

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

      1. +-commutative 76.9%

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

      2. +-commutative 76.9%

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

      3. associate-+r+ 76.9%

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

      4. *-commutative 76.9%

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

      5. metadata-eval 76.9%

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

      6. associate-*l* 76.9%

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

      7. *-commutative 76.9%

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

      8. *-commutative 76.9%

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

      9. metadata-eval 76.9%

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

      10. associate-*l* 76.9%

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

      11. *-commutative 76.9%

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

      12. associate-+r+ 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s} \cdot 1 + s \cdot 4}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 76.9%

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

       2. *-rgt-identity 76.9%

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

       3. associate-/l* 80.8%

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

   11. Applied egg-rr 80.8%

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

   if 1e-23 < (fabs.f32 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in s around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. distribute-frac-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in s around -inf 10.2%

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

      1. +-commutative 10.2%

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

      2. +-commutative 10.2%

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

      3. distribute-rgt-out 82.4%

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

      4. metadata-eval 82.4%

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

      5. *-rgt-identity 82.4%

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

      6. associate-+l+ 82.4%

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

      7. unpow2 82.4%

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

      8. sqr-abs 82.4%

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

      9. unpow2 82.4%

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

      10. +-commutative 82.4%

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

   9. Simplified 82.9%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(\frac{x \cdot x}{s \cdot s} + \left(4 + \frac{0}{s}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}} + s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{x \cdot x}{s \cdot s} + \left(4 + \frac{0}{s}\right)\right)}\\ \end{array} \]

Alternative 7: 60.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 95.3%

   \[\leadsto \frac{\color{blue}{0.5}}{\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)} \]
5. Step-by-step derivation

   1. expm1-log1p-u 94.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}\right)\right)} \]

   2. expm1-udef 94.2%

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

6. Applied egg-rr 57.5%

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

   1. expm1-def 57.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}\right)\right)} \]

   2. expm1-log1p 58.6%

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

8. Simplified 58.6%

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

   1. fma-udef 58.6%

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

10. Applied egg-rr 58.6%

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

11. Final simplification 58.6%

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

Alternative 8: 65.8% accurate, 56.4× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / ((s * 4.0f) + (x * (x / 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)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. mul-1-neg 99.5%

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

   3. distribute-frac-neg 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in s around inf 24.4%

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

   1. +-commutative 24.4%

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

   2. +-commutative 24.4%

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

   3. associate-+r+ 24.4%

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

   4. *-commutative 24.4%

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

   5. metadata-eval 24.4%

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

   6. associate-*l* 24.4%

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

   7. *-commutative 24.4%

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

   8. *-commutative 24.4%

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

   9. metadata-eval 24.4%

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

   10. associate-*l* 24.4%

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

   11. *-commutative 24.4%

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

   12. associate-+r+ 24.4%

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

9. Simplified 63.4%

   \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s} \cdot 1 + s \cdot 4}} \]
10. Step-by-step derivation

    1. *-un-lft-identity 63.4%

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

    2. *-rgt-identity 63.4%

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

    3. associate-/l* 64.2%

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

11. Applied egg-rr 64.2%

    \[\leadsto \color{blue}{1 \cdot \frac{1}{\frac{x}{\frac{s}{x}} + s \cdot 4}} \]
12. Step-by-step derivation

    1. *-lft-identity 64.2%

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

    2. associate-/r/ 64.2%

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

13. Simplified 64.2%

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

14. Final simplification 64.2%

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

Alternative 9: 65.8% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. mul-1-neg 99.5%

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

   3. distribute-frac-neg 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in s around inf 24.4%

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

   1. +-commutative 24.4%

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

   2. +-commutative 24.4%

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

   3. associate-+r+ 24.4%

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

   4. *-commutative 24.4%

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

   5. metadata-eval 24.4%

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

   6. associate-*l* 24.4%

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

   7. *-commutative 24.4%

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

   8. *-commutative 24.4%

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

   9. metadata-eval 24.4%

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

   10. associate-*l* 24.4%

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

   11. *-commutative 24.4%

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

   12. associate-+r+ 24.4%

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

9. Simplified 63.4%

   \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s} \cdot 1 + s \cdot 4}} \]
10. Step-by-step derivation

    1. *-un-lft-identity 63.4%

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

    2. *-rgt-identity 63.4%

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

    3. associate-/l* 64.2%

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

11. Applied egg-rr 64.2%

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

12. Final simplification 64.2%

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

Alternative 10: 45.1% accurate, 67.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5000000934815034 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{\frac{1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 3.5000000934815034e-5) (/ 0.25 s) (* s (/ (/ 1.0 x) x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 3.5000000934815034e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = s * ((1.0f / x) / 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 <= 3.5000000934815034e-5) then
        tmp = 0.25e0 / s
    else
        tmp = s * ((1.0e0 / x) / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(3.5000000934815034e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s * Float32(Float32(Float32(1.0) / x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(3.5000000934815034e-5))
		tmp = single(0.25) / s;
	else
		tmp = s * ((single(1.0) / x) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.50000009e-5

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around inf 31.9%

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

   if 3.50000009e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in s around inf 4.0%

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

      1. +-commutative 4.0%

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

      2. +-commutative 4.0%

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

      3. associate-+r+ 4.0%

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

      4. *-commutative 4.0%

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

      5. metadata-eval 4.0%

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

      6. associate-*l* 4.0%

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

      7. *-commutative 4.0%

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

      8. *-commutative 4.0%

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

      9. metadata-eval 4.0%

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

      10. associate-*l* 4.0%

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

      11. *-commutative 4.0%

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

      12. associate-+r+ 4.0%

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

   9. Simplified 73.3%

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

   10. Taylor expanded in x around inf 72.4%

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

       1. unpow2 72.4%

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

   12. Simplified 72.4%

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

       1. associate-/r* 72.4%

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

       2. div-inv 72.4%

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

       3. *-un-lft-identity 72.4%

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

       4. times-frac 72.4%

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

       5. /-rgt-identity 72.4%

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

   14. Applied egg-rr 72.4%

       \[\leadsto \color{blue}{s \cdot \frac{\frac{1}{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5000000934815034 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{\frac{1}{x}}{x}\\ \end{array} \]

Alternative 11: 45.8% accurate, 67.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5000000934815034 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 3.5000000934815034e-5) (/ 0.25 s) (/ 1.0 (/ (* x x) s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 3.5000000934815034e-5f) {
		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 <= 3.5000000934815034e-5) 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(3.5000000934815034e-5))
		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(3.5000000934815034e-5))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / ((x * x) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.50000009e-5

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around inf 31.9%

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

   if 3.50000009e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in s around inf 4.0%

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

      1. +-commutative 4.0%

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

      2. +-commutative 4.0%

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

      3. associate-+r+ 4.0%

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

      4. *-commutative 4.0%

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

      5. metadata-eval 4.0%

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

      6. associate-*l* 4.0%

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

      7. *-commutative 4.0%

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

      8. *-commutative 4.0%

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

      9. metadata-eval 4.0%

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

      10. associate-*l* 4.0%

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

      11. *-commutative 4.0%

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

      12. associate-+r+ 4.0%

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

   9. Simplified 73.3%

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

   10. Taylor expanded in x around inf 72.4%

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

       1. unpow2 72.4%

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

   12. Simplified 72.4%

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

       1. clear-num 73.3%

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

       2. un-div-inv 73.3%

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

       3. *-rgt-identity 73.3%

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

       4. inv-pow 73.3%

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

       5. *-rgt-identity 73.3%

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

       6. associate-*l* 73.3%

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

       7. div-inv 73.3%

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

   14. Applied egg-rr 73.3%

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

       1. unpow-1 73.3%

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

       2. associate-*r/ 73.3%

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

   16. Simplified 73.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x}{s}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5000000934815034 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \]

Alternative 12: 45.2% accurate, 87.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5000000934815034 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 3.5000000934815034e-5) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 3.5000000934815034e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = s / (x * 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 <= 3.5000000934815034e-5) then
        tmp = 0.25e0 / s
    else
        tmp = s / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(3.5000000934815034e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(3.5000000934815034e-5))
		tmp = single(0.25) / s;
	else
		tmp = s / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.50000009e-5

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around inf 31.9%

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

   if 3.50000009e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in s around inf 4.0%

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

      1. +-commutative 4.0%

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

      2. +-commutative 4.0%

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

      3. associate-+r+ 4.0%

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

      4. *-commutative 4.0%

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

      5. metadata-eval 4.0%

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

      6. associate-*l* 4.0%

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

      7. *-commutative 4.0%

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

      8. *-commutative 4.0%

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

      9. metadata-eval 4.0%

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

      10. associate-*l* 4.0%

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

      11. *-commutative 4.0%

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

      12. associate-+r+ 4.0%

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

   9. Simplified 73.3%

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

   10. Taylor expanded in x around inf 72.4%

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

       1. unpow2 72.4%

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

   12. Simplified 72.4%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5000000934815034 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 13: 45.1% accurate, 87.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5000000934815034 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 3.5000000934815034e-5) (/ 0.25 s) (/ (/ s x) x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 3.5000000934815034e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = (s / x) / 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 <= 3.5000000934815034e-5) then
        tmp = 0.25e0 / s
    else
        tmp = (s / x) / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(3.5000000934815034e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(s / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(3.5000000934815034e-5))
		tmp = single(0.25) / s;
	else
		tmp = (s / x) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.50000009e-5

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around inf 31.9%

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

   if 3.50000009e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in s around inf 4.0%

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

      1. +-commutative 4.0%

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

      2. +-commutative 4.0%

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

      3. associate-+r+ 4.0%

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

      4. *-commutative 4.0%

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

      5. metadata-eval 4.0%

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

      6. associate-*l* 4.0%

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

      7. *-commutative 4.0%

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

      8. *-commutative 4.0%

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

      9. metadata-eval 4.0%

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

      10. associate-*l* 4.0%

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

      11. *-commutative 4.0%

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

      12. associate-+r+ 4.0%

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

   9. Simplified 73.3%

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

   10. Taylor expanded in x around inf 72.4%

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

       1. unpow2 72.4%

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

       2. associate-/r* 72.4%

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

   12. Simplified 72.4%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5000000934815034 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \end{array} \]

Alternative 14: 26.9% accurate, 206.7× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 s))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 23.9%

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

5. Final simplification 23.9%

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

Reproduce

herbie shell --seed 2023293 
(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))))))