Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 11.0s

Alternatives: 14

Speedup: 1.5×

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, 1.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (s * (2.0f + (expf((fabsf(x) / s)) + 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 * (2.0e0 + (exp((abs(x) / s)) + exp((abs(x) / -s)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.7%

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

   1. expm1-log1p-u 97.9%

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

   2. expm1-udef 96.9%

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

   3. associate-+r+ 96.9%

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

5. Applied egg-rr 96.9%

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

   1. expm1-def 97.9%

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

   2. expm1-log1p 99.7%

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

   3. associate-/l/ 99.7%

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

   4. *-commutative 99.7%

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

   5. +-commutative 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{e^{\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 1.0000000116860974e-7)
   (/ (exp (+ (/ x s) (* (log1p (exp (/ x s))) -2.0))) s)
   (exp (log (/ (/ 1.0 s) (+ (exp (/ (fabs x) s)) 3.0))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 1.0000000116860974e-7f) {
		tmp = expf(((x / s) + (log1pf(expf((x / s))) * -2.0f))) / s;
	} else {
		tmp = expf(logf(((1.0f / s) / (expf((fabsf(x) / s)) + 3.0f))));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(1.0000000116860974e-7))
		tmp = Float32(exp(Float32(Float32(x / s) + Float32(log1p(exp(Float32(x / s))) * Float32(-2.0)))) / s);
	else
		tmp = exp(log(Float32(Float32(Float32(1.0) / s) / Float32(exp(Float32(abs(x) / s)) + Float32(3.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 1.00000001e-7

   1. Initial program 99.1%

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

      1. associate-*l* 99.1%

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

      2. +-commutative 99.1%

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

      3. +-commutative 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 99.1%

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

      1. associate-/r* 99.3%

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

      2. mul-1-neg 99.3%

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

      3. distribute-frac-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. mul-1-neg 99.3%

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

      6. distribute-frac-neg 99.3%

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

   6. Simplified 99.3%

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

      1. associate-/l/ 99.1%

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

      2. *-un-lft-identity 99.1%

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

      3. times-frac 99.3%

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

   8. Applied egg-rr 76.3%

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

   9. Taylor expanded in x around inf 76.4%

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

       1. +-commutative 76.4%

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

       2. exp-to-pow 76.3%

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

       3. log1p-def 76.4%

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

       4. *-commutative 76.4%

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

       5. rem-exp-log 71.7%

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

       6. exp-sum 71.5%

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

       7. exp-diff 94.2%

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

       8. associate--r+ 94.2%

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

       9. exp-diff 94.5%

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

       10. cancel-sign-sub-inv 94.5%

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

       11. metadata-eval 94.5%

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

       12. *-commutative 94.5%

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

       13. rem-exp-log 99.2%

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

   11. Simplified 99.2%

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

   if 1.00000001e-7 < (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%

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

      1. expm1-log1p-u 99.9%

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

      2. expm1-udef 99.1%

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

      3. associate-+r+ 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. expm1-def 99.9%

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

      2. expm1-log1p 99.9%

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

      3. associate-/l/ 99.9%

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

      4. *-commutative 99.9%

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

      5. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in s around 0 99.9%

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

      1. associate-/r* 99.9%

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

      2. associate-*r/ 99.9%

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

      3. mul-1-neg 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in s around inf 99.9%

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

       1. add-exp-log 99.9%

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

       2. associate-+r+ 99.9%

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

       3. metadata-eval 99.9%

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

   13. Applied egg-rr 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{e^{\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3}\right)}\\ \end{array} \]

Alternative 3: 95.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 0.5f / (s + (s * expf((fabsf(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((abs(x) / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*r/ 99.6%

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

   3. associate-/l* 99.6%

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

   4. distribute-frac-neg 99.6%

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

   5. exp-neg 99.5%

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

   6. associate-/r/ 99.5%

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

   7. /-rgt-identity 99.5%

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

   8. associate-*l* 99.5%

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

3. Simplified 99.7%

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

4. Taylor expanded in s around inf 94.9%

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

5. Taylor expanded in x around 0 95.0%

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

6. Final simplification 95.0%

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

Alternative 4: 96.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.7%

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

   1. expm1-log1p-u 97.9%

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

   2. expm1-udef 96.9%

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

   3. associate-+r+ 96.9%

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

5. Applied egg-rr 96.9%

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

   1. expm1-def 97.9%

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

   2. expm1-log1p 99.7%

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

   3. associate-/l/ 99.7%

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

   4. *-commutative 99.7%

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

   5. +-commutative 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in s around 0 99.6%

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

   1. associate-/r* 99.6%

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

   2. associate-*r/ 99.6%

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

   3. mul-1-neg 99.6%

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

10. Simplified 99.6%

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

11. Taylor expanded in s around inf 96.5%

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

12. Taylor expanded in s around 0 96.5%

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

    1. associate-/r* 96.5%

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

14. Simplified 96.5%

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

15. Final simplification 96.5%

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

Alternative 5: 94.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (exp (/ (- (fabs x)) s)) (* s 4.0)))

C

float code(float x, float s) {
	return expf((-fabsf(x) / s)) / (s * 4.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = exp((-abs(x) / s)) / (s * 4.0e0)
end function

Julia

function code(x, s)
	return Float32(exp(Float32(Float32(-abs(x)) / s)) / Float32(s * Float32(4.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = exp((-abs(x) / s)) / (s * single(4.0));
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf 94.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{4 \cdot s}} \]
3. Step-by-step derivation

   1. *-commutative 94.6%

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

4. Simplified 94.6%

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

5. Final simplification 94.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4} \]

Alternative 6: 94.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -5.000000015855384 \cdot 10^{-31}:\\ \;\;\;\;0.25 \cdot \frac{t_0}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= x -5.000000015855384e-31)
     (* 0.25 (/ t_0 s))
     (/ 1.0 (* s (+ 2.0 (* 2.0 t_0)))))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (x <= -5.000000015855384e-31f) {
		tmp = 0.25f * (t_0 / s);
	} else {
		tmp = 1.0f / (s * (2.0f + (2.0f * t_0)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = exp((x / s))
    if (x <= (-5.000000015855384e-31)) then
        tmp = 0.25e0 * (t_0 / s)
    else
        tmp = 1.0e0 / (s * (2.0e0 + (2.0e0 * t_0)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (x <= Float32(-5.000000015855384e-31))
		tmp = Float32(Float32(0.25) * Float32(t_0 / s));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(2.0) + Float32(Float32(2.0) * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((x / s));
	tmp = single(0.0);
	if (x <= single(-5.000000015855384e-31))
		tmp = single(0.25) * (t_0 / s);
	else
		tmp = single(1.0) / (s * (single(2.0) + (single(2.0) * t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5e-31

   1. Initial program 99.6%

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

      1. associate-*l* 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. associate-/r* 99.7%

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

      2. mul-1-neg 99.7%

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

      3. distribute-frac-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. mul-1-neg 99.7%

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

      6. distribute-frac-neg 99.7%

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

   6. Simplified 99.7%

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

      1. associate-/l/ 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. times-frac 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 95.8%

      \[\leadsto \color{blue}{0.25} \cdot \frac{e^{\frac{x}{s}}}{s} \]

   if -5e-31 < x

   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 99.6%

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

      1. expm1-log1p-u 97.5%

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

      2. expm1-udef 96.1%

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

      3. associate-+r+ 96.1%

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

   5. Applied egg-rr 96.1%

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

      1. expm1-def 97.5%

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

      2. expm1-log1p 99.6%

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

      3. associate-/l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. distribute-lft-in 99.7%

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

   9. Applied egg-rr 93.9%

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

       1. distribute-lft-in 93.9%

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

       2. count-2 93.9%

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

   11. Simplified 93.9%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000015855384 \cdot 10^{-31}:\\ \;\;\;\;0.25 \cdot \frac{e^{\frac{x}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot e^{\frac{x}{s}}\right)}\\ \end{array} \]

Alternative 7: 88.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.0000000781659255 \cdot 10^{-24}:\\ \;\;\;\;0.25 \cdot \frac{e^{\frac{x}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(x \cdot x\right) \cdot \frac{1}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.0000000781659255e-24)
   (* 0.25 (/ (exp (/ x s)) s))
   (/ (/ 1.0 s) (+ 4.0 (* (* x x) (/ 1.0 (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.0000000781659255e-24f) {
		tmp = 0.25f * (expf((x / s)) / s);
	} else {
		tmp = (1.0f / s) / (4.0f + ((x * x) * (1.0f / (s * 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 <= 4.0000000781659255e-24) then
        tmp = 0.25e0 * (exp((x / s)) / s)
    else
        tmp = (1.0e0 / s) / (4.0e0 + ((x * x) * (1.0e0 / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.0000000781659255e-24))
		tmp = Float32(Float32(0.25) * Float32(exp(Float32(x / s)) / s));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) * Float32(Float32(1.0) / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.0000000781659255e-24))
		tmp = single(0.25) * (exp((x / s)) / s);
	else
		tmp = (single(1.0) / s) / (single(4.0) + ((x * x) * (single(1.0) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000008e-24

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. associate-/r* 99.7%

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

      2. mul-1-neg 99.7%

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

      3. distribute-frac-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. mul-1-neg 99.7%

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

      6. distribute-frac-neg 99.7%

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

   6. Simplified 99.7%

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

      1. associate-/l/ 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. times-frac 99.7%

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

   8. Applied egg-rr 94.5%

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

   9. Taylor expanded in x around 0 89.4%

      \[\leadsto \color{blue}{0.25} \cdot \frac{e^{\frac{x}{s}}}{s} \]

   if 4.00000008e-24 < x

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in s around inf 48.7%

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

      1. associate-+r+ 48.7%

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

      2. distribute-lft1-in 48.7%

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

      3. metadata-eval 48.7%

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

      4. mul0-lft 82.0%

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

      5. associate-+r+ 82.0%

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

      6. unpow2 82.0%

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

      7. sqr-abs 82.0%

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

      8. unpow2 82.0%

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

      9. metadata-eval 82.0%

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

   6. Simplified 82.0%

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

      1. div-inv 84.9%

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

   8. Applied egg-rr 84.9%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.0000000781659255 \cdot 10^{-24}:\\ \;\;\;\;0.25 \cdot \frac{e^{\frac{x}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(x \cdot x\right) \cdot \frac{1}{s \cdot s}}\\ \end{array} \]

Alternative 8: 85.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;e^{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.0000000781659255 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{1}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(x \cdot x\right) \cdot \frac{1}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.0000000843119176e-17)
   (exp (/ x s))
   (if (<= x 4.0000000781659255e-24)
     (* (/ 1.0 s) (/ 1.0 (+ 4.0 (* (/ x s) (/ x s)))))
     (/ (/ 1.0 s) (+ 4.0 (* (* x x) (/ 1.0 (* s s))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.0000000843119176e-17f) {
		tmp = expf((x / s));
	} else if (x <= 4.0000000781659255e-24f) {
		tmp = (1.0f / s) * (1.0f / (4.0f + ((x / s) * (x / s))));
	} else {
		tmp = (1.0f / s) / (4.0f + ((x * x) * (1.0f / (s * 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 <= (-5.0000000843119176e-17)) then
        tmp = exp((x / s))
    else if (x <= 4.0000000781659255e-24) then
        tmp = (1.0e0 / s) * (1.0e0 / (4.0e0 + ((x / s) * (x / s))))
    else
        tmp = (1.0e0 / s) / (4.0e0 + ((x * x) * (1.0e0 / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0000000843119176e-17))
		tmp = exp(Float32(x / s));
	elseif (x <= Float32(4.0000000781659255e-24))
		tmp = Float32(Float32(Float32(1.0) / s) * Float32(Float32(1.0) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s)))));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) * Float32(Float32(1.0) / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.0000000843119176e-17))
		tmp = exp((x / s));
	elseif (x <= single(4.0000000781659255e-24))
		tmp = (single(1.0) / s) * (single(1.0) / (single(4.0) + ((x / s) * (x / s))));
	else
		tmp = (single(1.0) / s) / (single(4.0) + ((x * x) * (single(1.0) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5.00000008e-17

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. associate-/r* 99.8%

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

      2. mul-1-neg 99.8%

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

      3. distribute-frac-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. mul-1-neg 99.8%

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

      6. distribute-frac-neg 99.8%

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

   6. Simplified 99.8%

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

      1. add-exp-log 99.6%

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

      2. log-div 99.6%

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

      3. log-div 99.5%

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

      4. add-log-exp 99.5%

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

      5. add-sqr-sqrt -0.0%

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

      6. sqrt-unprod 4.6%

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

      7. sqr-neg 4.6%

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

      8. sqrt-unprod 4.6%

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

      9. add-sqr-sqrt 4.6%

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

      10. add-sqr-sqrt -0.0%

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

      11. fabs-sqr -0.0%

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

      12. add-sqr-sqrt 99.5%

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

      13. log-pow 99.5%

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

      14. log1p-udef 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in x around inf 96.2%

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

   if -5.00000008e-17 < x < 4.00000008e-24

   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 99.5%

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

   4. Taylor expanded in s around inf 54.7%

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

      1. associate-+r+ 54.7%

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

      2. distribute-lft1-in 54.7%

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

      3. metadata-eval 54.7%

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

      4. mul0-lft 54.7%

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

      5. associate-+r+ 54.7%

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

      6. unpow2 54.7%

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

      7. sqr-abs 54.7%

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

      8. unpow2 54.7%

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

      9. metadata-eval 54.7%

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

   6. Simplified 54.7%

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

      1. div-inv 54.7%

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

      2. times-frac 74.3%

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

   8. Applied egg-rr 74.3%

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

   if 4.00000008e-24 < x

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in s around inf 48.7%

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

      1. associate-+r+ 48.7%

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

      2. distribute-lft1-in 48.7%

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

      3. metadata-eval 48.7%

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

      4. mul0-lft 82.0%

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

      5. associate-+r+ 82.0%

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

      6. unpow2 82.0%

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

      7. sqr-abs 82.0%

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

      8. unpow2 82.0%

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

      9. metadata-eval 82.0%

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

   6. Simplified 82.0%

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

      1. div-inv 84.9%

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

   8. Applied egg-rr 84.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;e^{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.0000000781659255 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{1}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(x \cdot x\right) \cdot \frac{1}{s \cdot s}}\\ \end{array} \]

Alternative 9: 78.8% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.0000000781659255 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{1}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.0000000781659255e-24)
   (* (/ 1.0 s) (/ 1.0 (+ 4.0 (* (/ x s) (/ x s)))))
   (/ (/ 1.0 s) (+ 4.0 (/ (* x x) (* s s))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.0000000781659255e-24f) {
		tmp = (1.0f / s) * (1.0f / (4.0f + ((x / s) * (x / s))));
	} else {
		tmp = (1.0f / s) / (4.0f + ((x * x) / (s * 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 <= 4.0000000781659255e-24) then
        tmp = (1.0e0 / s) * (1.0e0 / (4.0e0 + ((x / s) * (x / s))))
    else
        tmp = (1.0e0 / s) / (4.0e0 + ((x * x) / (s * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.0000000781659255e-24))
		tmp = Float32(Float32(Float32(1.0) / s) * Float32(Float32(1.0) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s)))));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) / Float32(s * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.0000000781659255e-24))
		tmp = (single(1.0) / s) * (single(1.0) / (single(4.0) + ((x / s) * (x / s))));
	else
		tmp = (single(1.0) / s) / (single(4.0) + ((x * x) / (s * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000008e-24

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 51.2%

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

      1. associate-+r+ 51.2%

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

      2. distribute-lft1-in 51.2%

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

      3. metadata-eval 51.2%

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

      4. mul0-lft 69.7%

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

      5. associate-+r+ 69.7%

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

      6. unpow2 69.7%

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

      7. sqr-abs 69.7%

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

      8. unpow2 69.7%

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

      9. metadata-eval 69.7%

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

   6. Simplified 69.7%

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

      1. div-inv 69.7%

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

      2. times-frac 74.8%

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

   8. Applied egg-rr 74.8%

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

   if 4.00000008e-24 < x

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in s around inf 48.7%

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

      1. associate-+r+ 48.7%

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

      2. distribute-lft1-in 48.7%

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

      3. metadata-eval 48.7%

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

      4. mul0-lft 82.0%

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

      5. associate-+r+ 82.0%

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

      6. unpow2 82.0%

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

      7. sqr-abs 82.0%

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

      8. unpow2 82.0%

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

      9. metadata-eval 82.0%

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

   6. Simplified 82.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.0000000781659255 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{1}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 10: 79.5% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.0000000781659255 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{1}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(x \cdot x\right) \cdot \frac{1}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.0000000781659255e-24)
   (* (/ 1.0 s) (/ 1.0 (+ 4.0 (* (/ x s) (/ x s)))))
   (/ (/ 1.0 s) (+ 4.0 (* (* x x) (/ 1.0 (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.0000000781659255e-24f) {
		tmp = (1.0f / s) * (1.0f / (4.0f + ((x / s) * (x / s))));
	} else {
		tmp = (1.0f / s) / (4.0f + ((x * x) * (1.0f / (s * 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 <= 4.0000000781659255e-24) then
        tmp = (1.0e0 / s) * (1.0e0 / (4.0e0 + ((x / s) * (x / s))))
    else
        tmp = (1.0e0 / s) / (4.0e0 + ((x * x) * (1.0e0 / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.0000000781659255e-24))
		tmp = Float32(Float32(Float32(1.0) / s) * Float32(Float32(1.0) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s)))));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) * Float32(Float32(1.0) / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.0000000781659255e-24))
		tmp = (single(1.0) / s) * (single(1.0) / (single(4.0) + ((x / s) * (x / s))));
	else
		tmp = (single(1.0) / s) / (single(4.0) + ((x * x) * (single(1.0) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000008e-24

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 51.2%

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

      1. associate-+r+ 51.2%

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

      2. distribute-lft1-in 51.2%

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

      3. metadata-eval 51.2%

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

      4. mul0-lft 69.7%

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

      5. associate-+r+ 69.7%

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

      6. unpow2 69.7%

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

      7. sqr-abs 69.7%

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

      8. unpow2 69.7%

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

      9. metadata-eval 69.7%

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

   6. Simplified 69.7%

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

      1. div-inv 69.7%

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

      2. times-frac 74.8%

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

   8. Applied egg-rr 74.8%

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

   if 4.00000008e-24 < x

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in s around inf 48.7%

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

      1. associate-+r+ 48.7%

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

      2. distribute-lft1-in 48.7%

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

      3. metadata-eval 48.7%

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

      4. mul0-lft 82.0%

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

      5. associate-+r+ 82.0%

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

      6. unpow2 82.0%

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

      7. sqr-abs 82.0%

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

      8. unpow2 82.0%

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

      9. metadata-eval 82.0%

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

   6. Simplified 82.0%

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

      1. div-inv 84.9%

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

   8. Applied egg-rr 84.9%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.0000000781659255 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{1}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(x \cdot x\right) \cdot \frac{1}{s \cdot s}}\\ \end{array} \]

Alternative 11: 78.8% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.0000000781659255 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.0000000781659255e-24)
   (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s))))
   (/ (/ 1.0 s) (+ 4.0 (/ (* x x) (* s s))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.0000000781659255e-24f) {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (x / s)));
	} else {
		tmp = (1.0f / s) / (4.0f + ((x * x) / (s * 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 <= 4.0000000781659255e-24) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
    else
        tmp = (1.0e0 / s) / (4.0e0 + ((x * x) / (s * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.0000000781659255e-24))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s))));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) / Float32(s * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.0000000781659255e-24))
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (x / s)));
	else
		tmp = (single(1.0) / s) / (single(4.0) + ((x * x) / (s * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000008e-24

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 51.2%

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

      1. associate-+r+ 51.2%

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

      2. distribute-lft1-in 51.2%

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

      3. metadata-eval 51.2%

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

      4. mul0-lft 69.7%

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

      5. associate-+r+ 69.7%

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

      6. unpow2 69.7%

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

      7. sqr-abs 69.7%

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

      8. unpow2 69.7%

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

      9. metadata-eval 69.7%

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

   6. Simplified 69.7%

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

   7. Taylor expanded in x around 0 69.7%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{x}^{2}}{{s}^{2}}} + 4} \]
   8. Step-by-step derivation

      1. unpow2 69.7%

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

      2. unpow2 69.7%

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

      3. times-frac 74.8%

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

   9. Simplified 74.8%

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

   if 4.00000008e-24 < x

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in s around inf 48.7%

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

      1. associate-+r+ 48.7%

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

      2. distribute-lft1-in 48.7%

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

      3. metadata-eval 48.7%

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

      4. mul0-lft 82.0%

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

      5. associate-+r+ 82.0%

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

      6. unpow2 82.0%

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

      7. sqr-abs 82.0%

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

      8. unpow2 82.0%

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

      9. metadata-eval 82.0%

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

   6. Simplified 82.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.0000000781659255 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 12: 76.2% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.7%

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

4. Taylor expanded in s around inf 50.2%

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

   1. associate-+r+ 50.2%

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

   2. distribute-lft1-in 50.2%

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

   3. metadata-eval 50.2%

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

   4. mul0-lft 74.5%

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

   5. associate-+r+ 74.5%

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

   6. unpow2 74.5%

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

   7. sqr-abs 74.5%

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

   8. unpow2 74.5%

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

   9. metadata-eval 74.5%

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

6. Simplified 74.5%

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

7. Taylor expanded in x around 0 74.5%

   \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{x}^{2}}{{s}^{2}}} + 4} \]
8. Step-by-step derivation

   1. unpow2 74.5%

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

   2. unpow2 74.5%

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

   3. times-frac 74.0%

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

9. Simplified 74.0%

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

10. Final simplification 74.0%

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

Alternative 13: 61.5% accurate, 66.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-14} \lor \neg \left(x \leq 2.2000000043931323 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -3.99999992980668e-14) (not (<= x 2.2000000043931323e-8)))
   (/ s (* x x))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -3.99999992980668e-14f) || !(x <= 2.2000000043931323e-8f)) {
		tmp = s / (x * x);
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-3.99999992980668e-14)) .or. (.not. (x <= 2.2000000043931323e-8))) then
        tmp = s / (x * x)
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-3.99999992980668e-14)) || !(x <= Float32(2.2000000043931323e-8)))
		tmp = Float32(s / Float32(x * x));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-3.99999992980668e-14)) || ~((x <= single(2.2000000043931323e-8))))
		tmp = s / (x * x);
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -3.99999993e-14 or 2.2e-8 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in s around inf 42.9%

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

      1. associate-+r+ 42.9%

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

      2. distribute-lft1-in 42.9%

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

      3. metadata-eval 42.9%

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

      4. mul0-lft 81.6%

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

      5. associate-+r+ 81.6%

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

      6. unpow2 81.6%

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

      7. sqr-abs 81.6%

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

      8. unpow2 81.6%

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

      9. metadata-eval 81.6%

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

   6. Simplified 81.6%

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

   7. Taylor expanded in s around 0 63.4%

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

      1. unpow2 63.4%

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

   9. Simplified 63.4%

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

   if -3.99999993e-14 < x < 2.2e-8

   1. Initial program 99.3%

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

      1. /-rgt-identity 99.3%

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

      2. associate-/l/ 99.3%

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

      3. *-lft-identity 99.3%

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

      4. +-commutative 99.3%

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

      5. distribute-rgt-in 99.4%

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

      6. *-lft-identity 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-rgt-in 99.4%

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

      9. *-lft-identity 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in s around inf 59.2%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-14} \lor \neg \left(x \leq 2.2000000043931323 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 14: 25.9% 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 99.6%

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

   1. /-rgt-identity 99.6%

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

   2. associate-/l/ 99.6%

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

   3. *-lft-identity 99.6%

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

   4. +-commutative 99.6%

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

   5. distribute-rgt-in 99.6%

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

   6. *-lft-identity 99.6%

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

   7. +-commutative 99.6%

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

   8. distribute-rgt-in 99.6%

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

   9. *-lft-identity 99.6%

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

3. Simplified 99.7%

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

4. Taylor expanded in s around inf 25.3%

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

5. Final simplification 25.3%

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

Reproduce

herbie shell --seed 2023182 
(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))))))