Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 9.6s

Alternatives: 9

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.2× speedup

Math

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

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 99.4%

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

   1. *-lft-identity 99.4%

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

   2. associate-*r/ 99.4%

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

   3. associate-/l* 99.4%

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

   4. distribute-frac-neg 99.4%

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

   5. exp-neg 99.4%

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

   6. associate-/r/ 99.4%

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

   7. /-rgt-identity 99.4%

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

   8. associate-*l* 99.4%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.3% 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 99.4%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 75.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;\left|x\right| \leq 0.10000000149011612:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(t_0\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= (fabs x) 0.10000000149011612)
     (/ (exp (- (/ x s) (* 2.0 (log1p t_0)))) s)
     (/ (/ 1.0 s) (+ t_0 3.0)))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (fabsf(x) <= 0.10000000149011612f) {
		tmp = expf(((x / s) - (2.0f * log1pf(t_0)))) / s;
	} else {
		tmp = (1.0f / s) / (t_0 + 3.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.10000000149011612))
		tmp = Float32(exp(Float32(Float32(x / s) - Float32(Float32(2.0) * log1p(t_0)))) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(t_0 + Float32(3.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.100000001

   1. Initial program 98.9%

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

      1. add-exp-log 95.5%

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

      2. *-commutative 95.5%

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

      3. log-prod 94.8%

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

      4. log1p-udef 94.9%

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

      5. *-commutative 94.9%

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

      6. log-prod 93.4%

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

      7. log1p-udef 93.4%

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

   3. Applied egg-rr 93.4%

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

      1. associate-+r+ 95.2%

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

      2. exp-sum 95.2%

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

      3. count-2 95.2%

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

      4. rem-exp-log 99.1%

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

   5. Simplified 99.1%

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

      1. expm1-log1p-u 95.5%

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

      2. expm1-udef 95.5%

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

   7. Applied egg-rr 95.5%

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

      1. expm1-def 95.4%

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

      2. expm1-log1p 99.0%

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

   9. Simplified 99.0%

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

   if 0.100000001 < (fabs.f32 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. sqr-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. distribute-frac-neg 100.0%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 4.5%

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

      8. *-un-lft-identity 4.5%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. distribute-frac-neg 100.0%

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

      13. sqr-neg 100.0%

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

      14. sqrt-unprod 100.0%

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

      15. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 56.1%

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

      1. *-lft-identity 56.1%

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

   8. Simplified 56.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.10000000149011612:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\\ \end{array} \]

Alternative 4: 61.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	return (1.0f / s) / (expf((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((x / s)) + 3.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 95.9%

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

   1. add-sqr-sqrt 95.9%

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

   2. sqrt-unprod 95.9%

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

   3. sqr-neg 95.9%

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

   4. distribute-frac-neg 95.9%

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

   5. distribute-frac-neg 95.9%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 27.0%

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

   8. *-un-lft-identity 27.0%

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

   9. add-sqr-sqrt -0.0%

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

   10. sqrt-unprod 95.9%

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

   11. distribute-frac-neg 95.9%

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

   12. distribute-frac-neg 95.9%

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

   13. sqr-neg 95.9%

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

   14. sqrt-unprod 95.9%

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

   15. add-sqr-sqrt 95.9%

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

6. Applied egg-rr 63.0%

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

   1. *-lft-identity 63.0%

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

8. Simplified 63.0%

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

9. Final simplification 63.0%

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

Alternative 5: 78.5% 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 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 53.3%

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

   1. associate-+r+ 53.3%

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

   2. distribute-lft1-in 53.3%

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

   3. metadata-eval 53.3%

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

   4. mul0-lft 82.6%

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

   5. metadata-eval 82.6%

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

   6. associate-+r+ 82.6%

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

   7. metadata-eval 82.6%

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

   8. metadata-eval 82.6%

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

   9. +-commutative 82.6%

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

   10. unpow2 82.6%

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

   11. sqr-abs 82.6%

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

   12. unpow2 82.6%

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

6. Simplified 82.6%

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

7. Final simplification 82.6%

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

Alternative 6: 65.8% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. *-lft-identity 99.4%

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

   2. associate-*r/ 99.4%

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

   3. associate-/l* 99.4%

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

   4. distribute-frac-neg 99.4%

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

   5. exp-neg 99.4%

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

   6. associate-/r/ 99.4%

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

   7. /-rgt-identity 99.4%

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

   8. associate-*l* 99.4%

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

3. Simplified 99.6%

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

4. Taylor expanded in s around -inf 41.0%

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

   1. associate-+r+ 41.0%

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

   2. distribute-rgt-out 41.0%

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

   3. metadata-eval 41.0%

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

   4. mul0-rgt 41.0%

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

   5. fma-def 41.0%

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

   6. mul-1-neg 41.0%

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

   7. distribute-rgt1-in 68.7%

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

   8. metadata-eval 68.7%

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

   9. associate-*r/ 68.7%

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

   10. mul-1-neg 68.7%

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

   11. remove-double-neg 68.7%

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

   12. unpow2 68.7%

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

   13. sqr-abs 68.7%

       \[\leadsto \frac{1}{0 + \mathsf{fma}\left(4, s, \frac{\color{blue}{x \cdot x}}{s}\right)} \]

6. Simplified 68.7%

   \[\leadsto \frac{1}{\color{blue}{0 + \mathsf{fma}\left(4, s, \frac{x \cdot x}{s}\right)}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 67.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{0 + \mathsf{fma}\left(4, s, \frac{x \cdot x}{s}\right)}\right)\right)} \]

   2. expm1-udef 81.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{0 + \mathsf{fma}\left(4, s, \frac{x \cdot x}{s}\right)}\right)} - 1} \]

   3. +-lft-identity 81.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\mathsf{fma}\left(4, s, \frac{x \cdot x}{s}\right)}}\right)} - 1 \]

   4. associate-/l* 81.8%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(4, s, \color{blue}{\frac{x}{\frac{s}{x}}}\right)}\right)} - 1 \]

8. Applied egg-rr 81.8%

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

   1. expm1-def 67.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(4, s, \frac{x}{\frac{s}{x}}\right)}\right)\right)} \]

   2. expm1-log1p 68.9%

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

   3. fma-def 68.9%

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

   4. +-commutative 68.9%

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

   5. associate-/r/ 68.9%

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

   6. *-commutative 68.9%

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

   7. fma-def 68.9%

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

   8. *-commutative 68.9%

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

10. Simplified 68.9%

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

    1. fma-udef 68.9%

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

12. Applied egg-rr 68.9%

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

13. Final simplification 68.9%

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

Alternative 7: 45.8% accurate, 67.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.999999747378752e-5) (/ 0.25 s) (/ 1.0 (/ x (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.999999747378752e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (x / (s / 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 <= 9.999999747378752e-5) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x / (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.999999747378752e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x / Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.999999747378752e-5))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x / (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 9.99999975e-5

   1. Initial program 99.2%

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

      1. *-lft-identity 99.2%

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

      2. associate-*r/ 99.2%

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

      3. associate-/l* 99.2%

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

      4. distribute-frac-neg 99.2%

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

      5. exp-neg 99.2%

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

      6. associate-/r/ 99.2%

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

      7. /-rgt-identity 99.2%

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

      8. associate-*l* 99.2%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in s around inf 37.6%

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

   if 9.99999975e-5 < x

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 25.6%

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

      1. associate-+r+ 25.6%

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

      2. distribute-rgt-out 25.6%

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

      3. metadata-eval 25.6%

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

      4. mul0-rgt 25.6%

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

      5. fma-def 25.6%

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

      6. mul-1-neg 25.6%

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

      7. distribute-rgt1-in 84.5%

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

      8. metadata-eval 84.5%

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

      9. associate-*r/ 84.5%

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

      10. mul-1-neg 84.5%

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

      11. remove-double-neg 84.5%

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

      12. unpow2 84.5%

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

      13. sqr-abs 84.5%

          \[\leadsto \frac{1}{0 + \mathsf{fma}\left(4, s, \frac{\color{blue}{x \cdot x}}{s}\right)} \]

   6. Simplified 84.5%

      \[\leadsto \frac{1}{\color{blue}{0 + \mathsf{fma}\left(4, s, \frac{x \cdot x}{s}\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 84.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{0 + \mathsf{fma}\left(4, s, \frac{x \cdot x}{s}\right)}\right)\right)} \]

      2. expm1-udef 99.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{0 + \mathsf{fma}\left(4, s, \frac{x \cdot x}{s}\right)}\right)} - 1} \]

      3. +-lft-identity 99.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\mathsf{fma}\left(4, s, \frac{x \cdot x}{s}\right)}}\right)} - 1 \]

      4. associate-/l* 99.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(4, s, \color{blue}{\frac{x}{\frac{s}{x}}}\right)}\right)} - 1 \]

   8. Applied egg-rr 99.2%

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

      1. expm1-def 84.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(4, s, \frac{x}{\frac{s}{x}}\right)}\right)\right)} \]

      2. expm1-log1p 84.5%

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

      3. fma-def 84.5%

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

      4. +-commutative 84.5%

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

      5. associate-/r/ 84.5%

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

      6. *-commutative 84.5%

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

      7. fma-def 84.5%

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

      8. *-commutative 84.5%

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

   10. Simplified 84.5%

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

   11. Taylor expanded in x around inf 84.3%

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

       1. unpow2 84.3%

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

       2. associate-/l* 84.3%

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

   13. Simplified 84.3%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \end{array} \]

Alternative 8: 45.1% accurate, 87.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.999999747378752e-5) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.999999747378752e-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 <= 9.999999747378752e-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(9.999999747378752e-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(9.999999747378752e-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 < 9.99999975e-5

   1. Initial program 99.2%

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

      1. *-lft-identity 99.2%

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

      2. associate-*r/ 99.2%

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

      3. associate-/l* 99.2%

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

      4. distribute-frac-neg 99.2%

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

      5. exp-neg 99.2%

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

      6. associate-/r/ 99.2%

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

      7. /-rgt-identity 99.2%

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

      8. associate-*l* 99.2%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in s around inf 37.6%

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

   if 9.99999975e-5 < x

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 25.6%

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

      1. associate-+r+ 25.6%

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

      2. distribute-rgt-out 25.6%

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

      3. metadata-eval 25.6%

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

      4. mul0-rgt 25.6%

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

      5. fma-def 25.6%

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

      6. mul-1-neg 25.6%

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

      7. distribute-rgt1-in 84.5%

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

      8. metadata-eval 84.5%

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

      9. associate-*r/ 84.5%

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

      10. mul-1-neg 84.5%

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

      11. remove-double-neg 84.5%

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

      12. unpow2 84.5%

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

      13. sqr-abs 84.5%

          \[\leadsto \frac{1}{0 + \mathsf{fma}\left(4, s, \frac{\color{blue}{x \cdot x}}{s}\right)} \]

   6. Simplified 84.5%

      \[\leadsto \frac{1}{\color{blue}{0 + \mathsf{fma}\left(4, s, \frac{x \cdot x}{s}\right)}} \]

   7. Taylor expanded in s around 0 83.3%

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

      1. unpow2 83.3%

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

   9. Simplified 83.3%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 9: 27.3% 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 99.4%

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

   1. *-lft-identity 99.4%

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

   2. associate-*r/ 99.4%

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

   3. associate-/l* 99.4%

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

   4. distribute-frac-neg 99.4%

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

   5. exp-neg 99.4%

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

   6. associate-/r/ 99.4%

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

   7. /-rgt-identity 99.4%

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

   8. associate-*l* 99.4%

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

3. Simplified 99.6%

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

4. Taylor expanded in s around inf 28.3%

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

5. Final simplification 28.3%

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

Reproduce

herbie shell --seed 2023230 
(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))))))