Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 15.3s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = t_0 + 1.0f;
	return t_0 / (t_1 * (s * 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 = t_0 + 1.0e0
    code = t_0 / (t_1 * (s * t_1))
end function

Julia

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

MATLAB

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

Derivation

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. Add Preprocessing

3. Final simplification 99.3%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. fabs-neg 99.3%

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

   3. +-commutative 99.3%

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

   4. fabs-neg 99.3%

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

   5. distribute-lft-in 99.3%

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

   6. *-rgt-identity 99.3%

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

   7. +-commutative 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 3: 70.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{-s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 2.0000000233721948e-7)
   (/ (exp (+ (/ x s) (* -2.0 (log1p (exp (/ x s)))))) s)
   (exp (/ x (- s)))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 2.0000000233721948e-7f) {
		tmp = expf(((x / s) + (-2.0f * log1pf(expf((x / s)))))) / s;
	} else {
		tmp = expf((x / -s));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(2.0000000233721948e-7))
		tmp = Float32(exp(Float32(Float32(x / s) + Float32(Float32(-2.0) * log1p(exp(Float32(x / s)))))) / s);
	else
		tmp = exp(Float32(x / Float32(-s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 2.00000002e-7

   1. Initial program 98.1%

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

      1. fabs-neg 98.1%

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

      2. distribute-frac-neg 98.1%

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

      3. distribute-frac-neg2 98.1%

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

      4. fabs-neg 98.1%

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

      5. *-commutative 98.1%

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

      6. fabs-neg 98.1%

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

      7. +-commutative 98.1%

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

      8. fabs-neg 98.1%

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

   3. Simplified 98.1%

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

      1. associate-/r* 98.0%

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

      2. div-inv 98.0%

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

   6. Applied egg-rr 82.6%

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

      1. associate-*l/ 82.8%

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

      2. add-exp-log 82.7%

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

      3. prod-exp 98.0%

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

      4. pow-flip 98.1%

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

      5. log-pow 97.9%

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

      6. metadata-eval 97.9%

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

      7. log1p-define 98.1%

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

   8. Applied egg-rr 98.1%

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

   if 2.00000002e-7 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in s around inf 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   7. Applied egg-rr 52.8%

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

      1. unpow-1 52.8%

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

      2. associate-*r* 52.8%

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

      3. *-commutative 52.8%

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

      4. associate-/r* 52.8%

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

      5. *-commutative 52.8%

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

      6. associate-/r* 52.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s}}}{e^{\frac{x}{s}}} \]

      7. metadata-eval 52.8%

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

   9. Simplified 52.8%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{s}}{e^{\frac{x}{s}}}} \]
   10. Step-by-step derivation

       1. add-exp-log 52.8%

          \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{0.25}{s}\right)}}}{e^{\frac{x}{s}}} \]

       2. div-exp 52.8%

          \[\leadsto \color{blue}{e^{\log \left(\frac{0.25}{s}\right) - \frac{x}{s}}} \]

   11. Applied egg-rr 52.8%

       \[\leadsto \color{blue}{e^{\log \left(\frac{0.25}{s}\right) - \frac{x}{s}}} \]

   12. Taylor expanded in s around 0 52.8%

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

       1. mul-1-neg 52.8%

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

       2. distribute-frac-neg2 52.8%

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

   14. Simplified 52.8%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{-s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% 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 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around inf 95.6%

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

6. Final simplification 95.6%

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

Alternative 5: 56.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;\frac{0.25 + \left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{-s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999920083944e-12)
   (/ (+ 0.25 (* (* (/ x s) (/ x s)) -0.0625)) s)
   (exp (/ x (- s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999920083944e-12f) {
		tmp = (0.25f + (((x / s) * (x / s)) * -0.0625f)) / s;
	} else {
		tmp = expf((x / -s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.9999999920083944e-12) then
        tmp = (0.25e0 + (((x / s) * (x / s)) * (-0.0625e0))) / s
    else
        tmp = exp((x / -s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999920083944e-12))
		tmp = Float32(Float32(Float32(0.25) + Float32(Float32(Float32(x / s) * Float32(x / s)) * Float32(-0.0625))) / s);
	else
		tmp = exp(Float32(x / Float32(-s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999920083944e-12))
		tmp = (single(0.25) + (((x / s) * (x / s)) * single(-0.0625))) / s;
	else
		tmp = exp((x / -s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999e-12

   1. Initial program 98.9%

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

      1. fabs-neg 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-frac-neg2 98.9%

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

      4. fabs-neg 98.9%

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

      5. *-commutative 98.9%

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

      6. fabs-neg 98.9%

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

      7. +-commutative 98.9%

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

      8. fabs-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in s around inf 32.2%

      \[\leadsto \color{blue}{\frac{\left(0.25 + 0.125 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - 0.0625 \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]
   6. Step-by-step derivation

   7. Simplified 32.9%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\mathsf{fma}\left(0.125, x \cdot x, 0.0625 \cdot \left(-3 \cdot \left(x \cdot x\right)\right)\right)}{{s}^{2}}}{s}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 32.9%

         \[\leadsto \frac{0.25 + \frac{\color{blue}{1 \cdot \mathsf{fma}\left(0.125, x \cdot x, 0.0625 \cdot \left(-3 \cdot \left(x \cdot x\right)\right)\right)}}{{s}^{2}}}{s} \]

      2. unpow2 32.9%

         \[\leadsto \frac{0.25 + \frac{1 \cdot \mathsf{fma}\left(0.125, x \cdot x, 0.0625 \cdot \left(-3 \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{s \cdot s}}}{s} \]

      3. times-frac 37.0%

         \[\leadsto \frac{0.25 + \color{blue}{\frac{1}{s} \cdot \frac{\mathsf{fma}\left(0.125, x \cdot x, 0.0625 \cdot \left(-3 \cdot \left(x \cdot x\right)\right)\right)}{s}}}{s} \]

      4. fma-undefine 37.0%

         \[\leadsto \frac{0.25 + \frac{1}{s} \cdot \frac{\color{blue}{0.125 \cdot \left(x \cdot x\right) + 0.0625 \cdot \left(-3 \cdot \left(x \cdot x\right)\right)}}{s}}{s} \]

      5. associate-*r* 37.0%

         \[\leadsto \frac{0.25 + \frac{1}{s} \cdot \frac{0.125 \cdot \left(x \cdot x\right) + \color{blue}{\left(0.0625 \cdot -3\right) \cdot \left(x \cdot x\right)}}{s}}{s} \]

      6. distribute-rgt-out 37.6%

         \[\leadsto \frac{0.25 + \frac{1}{s} \cdot \frac{\color{blue}{\left(x \cdot x\right) \cdot \left(0.125 + 0.0625 \cdot -3\right)}}{s}}{s} \]

      7. pow2 37.6%

         \[\leadsto \frac{0.25 + \frac{1}{s} \cdot \frac{\color{blue}{{x}^{2}} \cdot \left(0.125 + 0.0625 \cdot -3\right)}{s}}{s} \]

      8. metadata-eval 37.6%

         \[\leadsto \frac{0.25 + \frac{1}{s} \cdot \frac{{x}^{2} \cdot \left(0.125 + \color{blue}{-0.1875}\right)}{s}}{s} \]

      9. metadata-eval 37.6%

         \[\leadsto \frac{0.25 + \frac{1}{s} \cdot \frac{{x}^{2} \cdot \color{blue}{-0.0625}}{s}}{s} \]

   9. Applied egg-rr 37.6%

      \[\leadsto \frac{0.25 + \color{blue}{\frac{1}{s} \cdot \frac{{x}^{2} \cdot -0.0625}{s}}}{s} \]

   10. Taylor expanded in s around 0 33.6%

       \[\leadsto \frac{0.25 + \color{blue}{-0.0625 \cdot \frac{{x}^{2}}{{s}^{2}}}}{s} \]
   11. Step-by-step derivation

       1. *-commutative 33.6%

          \[\leadsto \frac{0.25 + \color{blue}{\frac{{x}^{2}}{{s}^{2}} \cdot -0.0625}}{s} \]

       2. unpow2 33.6%

          \[\leadsto \frac{0.25 + \frac{\color{blue}{x \cdot x}}{{s}^{2}} \cdot -0.0625}{s} \]

       3. unpow2 33.6%

          \[\leadsto \frac{0.25 + \frac{x \cdot x}{\color{blue}{s \cdot s}} \cdot -0.0625}{s} \]

       4. times-frac 38.0%

          \[\leadsto \frac{0.25 + \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} \cdot -0.0625}{s} \]

       5. unpow2 38.0%

          \[\leadsto \frac{0.25 + \color{blue}{{\left(\frac{x}{s}\right)}^{2}} \cdot -0.0625}{s} \]

   12. Simplified 38.0%

       \[\leadsto \frac{0.25 + \color{blue}{{\left(\frac{x}{s}\right)}^{2} \cdot -0.0625}}{s} \]
   13. Step-by-step derivation

       1. unpow2 38.0%

          \[\leadsto \frac{0.25 + \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} \cdot -0.0625}{s} \]

   14. Applied egg-rr 38.0%

       \[\leadsto \frac{0.25 + \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} \cdot -0.0625}{s} \]

   if 1.99999999e-12 < x

   1. Initial program 99.9%

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

      1. fabs-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-frac-neg2 99.9%

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

      4. fabs-neg 99.9%

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

      5. *-commutative 99.9%

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

      6. fabs-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. fabs-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in s around inf 99.0%

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

      1. clear-num 99.0%

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

      2. inv-pow 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. unpow-1 99.0%

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

      2. associate-*r* 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-/r* 99.0%

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

      5. *-commutative 99.0%

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

      6. associate-/r* 99.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s}}}{e^{\frac{x}{s}}} \]

      7. metadata-eval 99.0%

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

   9. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{s}}{e^{\frac{x}{s}}}} \]
   10. Step-by-step derivation

       1. add-exp-log 99.0%

          \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{0.25}{s}\right)}}}{e^{\frac{x}{s}}} \]

       2. div-exp 99.0%

          \[\leadsto \color{blue}{e^{\log \left(\frac{0.25}{s}\right) - \frac{x}{s}}} \]

   11. Applied egg-rr 99.0%

       \[\leadsto \color{blue}{e^{\log \left(\frac{0.25}{s}\right) - \frac{x}{s}}} \]

   12. Taylor expanded in s around 0 97.4%

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

       1. mul-1-neg 97.4%

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

       2. distribute-frac-neg2 97.4%

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

   14. Simplified 97.4%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;\frac{0.25 + \left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{-s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{s}}{{e}^{\left(\frac{x}{s}\right)}} \end{array} \]

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(Float32(0.25) / s) / (Float32(exp(1)) ^ Float32(x / s)))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(0.25) / s) / (single(2.71828182845904523536) ^ (x / s));
end

Derivation

1. Initial program 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around inf 95.6%

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

   1. clear-num 95.6%

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

   2. inv-pow 95.6%

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

7. Applied egg-rr 60.2%

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

   1. unpow-1 60.2%

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

   2. associate-*r* 60.2%

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

   3. *-commutative 60.2%

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

   4. associate-/r* 60.2%

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

   5. *-commutative 60.2%

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

   6. associate-/r* 60.2%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s}}}{e^{\frac{x}{s}}} \]

   7. metadata-eval 60.2%

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

9. Simplified 60.2%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{s}}{e^{\frac{x}{s}}}} \]
10. Step-by-step derivation

    1. *-un-lft-identity 60.2%

       \[\leadsto \frac{\frac{0.25}{s}}{e^{\color{blue}{1 \cdot \frac{x}{s}}}} \]

    2. exp-prod 60.2%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{x}{s}\right)}}} \]

11. Applied egg-rr 60.2%

    \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{x}{s}\right)}}} \]
12. Step-by-step derivation

    1. exp-1-e 60.2%

       \[\leadsto \frac{\frac{0.25}{s}}{{\color{blue}{e}}^{\left(\frac{x}{s}\right)}} \]

13. Simplified 60.2%

    \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{{e}^{\left(\frac{x}{s}\right)}}} \]

14. Final simplification 60.2%

    \[\leadsto \frac{\frac{0.25}{s}}{{e}^{\left(\frac{x}{s}\right)}} \]
15. Add Preprocessing

Alternative 7: 59.7% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around inf 95.6%

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

   1. clear-num 95.6%

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

   2. inv-pow 95.6%

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

7. Applied egg-rr 60.2%

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

   1. unpow-1 60.2%

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

   2. associate-*r* 60.2%

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

   3. *-commutative 60.2%

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

   4. associate-/r* 60.2%

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

   5. *-commutative 60.2%

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

   6. associate-/r* 60.2%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s}}}{e^{\frac{x}{s}}} \]

   7. metadata-eval 60.2%

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

9. Simplified 60.2%

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

10. Final simplification 60.2%

    \[\leadsto \frac{\frac{0.25}{s}}{e^{\frac{x}{s}}} \]
11. Add Preprocessing

Alternative 8: 39.7% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{s}}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.0000000233721948e-7) (/ 0.25 s) (/ (/ 0.25 s) (/ x s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.0000000233721948e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = (0.25f / s) / (x / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 2.0000000233721948e-7) then
        tmp = 0.25e0 / s
    else
        tmp = (0.25e0 / s) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.0000000233721948e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(0.25) / s) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.0000000233721948e-7))
		tmp = single(0.25) / s;
	else
		tmp = (single(0.25) / s) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.00000002e-7

   1. Initial program 99.0%

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

      1. fabs-neg 99.0%

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

      2. distribute-frac-neg 99.0%

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

      3. distribute-frac-neg2 99.0%

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

      4. fabs-neg 99.0%

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

      5. *-commutative 99.0%

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

      6. fabs-neg 99.0%

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

      7. +-commutative 99.0%

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

      8. fabs-neg 99.0%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in s around inf 36.8%

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

   if 2.00000002e-7 < x

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in s around inf 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/r* 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r* 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s}}}{e^{\frac{x}{s}}} \]

      7. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 47.9%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{1 + \frac{x}{s}}} \]
   11. Step-by-step derivation

       1. +-commutative 47.9%

          \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\frac{x}{s} + 1}} \]

   12. Simplified 47.9%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\frac{x}{s} + 1}} \]

   13. Taylor expanded in x around inf 47.9%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{s}}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.6% accurate, 56.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s} \cdot \frac{1}{1 + \frac{x}{s}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (* (/ 0.25 s) (/ 1.0 (+ 1.0 (/ x s)))))

C

float code(float x, float s) {
	return (0.25f / s) * (1.0f / (1.0f + (x / s)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (0.25e0 / s) * (1.0e0 / (1.0e0 + (x / s)))
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(0.25) / s) * Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(0.25) / s) * (single(1.0) / (single(1.0) + (x / s)));
end

Derivation

1. Initial program 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around inf 95.6%

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

   1. clear-num 95.6%

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

   2. inv-pow 95.6%

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

7. Applied egg-rr 60.2%

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

   1. unpow-1 60.2%

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

   2. associate-*r* 60.2%

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

   3. *-commutative 60.2%

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

   4. associate-/r* 60.2%

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

   5. *-commutative 60.2%

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

   6. associate-/r* 60.2%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s}}}{e^{\frac{x}{s}}} \]

   7. metadata-eval 60.2%

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

9. Simplified 60.2%

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

10. Taylor expanded in x around 0 52.8%

    \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{1 + \frac{x}{s}}} \]
11. Step-by-step derivation

    1. +-commutative 52.8%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\frac{x}{s} + 1}} \]

12. Simplified 52.8%

    \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\frac{x}{s} + 1}} \]
13. Step-by-step derivation

    1. clear-num 52.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{x}{s} + 1}{\frac{0.25}{s}}}} \]

    2. associate-/r/ 52.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{x}{s} + 1} \cdot \frac{0.25}{s}} \]

14. Applied egg-rr 52.8%

    \[\leadsto \color{blue}{\frac{1}{\frac{x}{s} + 1} \cdot \frac{0.25}{s}} \]

15. Final simplification 52.8%

    \[\leadsto \frac{0.25}{s} \cdot \frac{1}{1 + \frac{x}{s}} \]
16. Add Preprocessing

Alternative 10: 50.6% accurate, 56.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{s}}{\left(\frac{x}{s} + 2\right) + -1} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ 0.25 s) (+ (+ (/ x s) 2.0) -1.0)))

C

float code(float x, float s) {
	return (0.25f / s) / (((x / s) + 2.0f) + -1.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (0.25e0 / s) / (((x / s) + 2.0e0) + (-1.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(0.25) / s) / Float32(Float32(Float32(x / s) + Float32(2.0)) + Float32(-1.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(0.25) / s) / (((x / s) + single(2.0)) + single(-1.0));
end

Derivation

1. Initial program 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around inf 95.6%

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

   1. clear-num 95.6%

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

   2. inv-pow 95.6%

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

7. Applied egg-rr 60.2%

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

   1. unpow-1 60.2%

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

   2. associate-*r* 60.2%

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

   3. *-commutative 60.2%

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

   4. associate-/r* 60.2%

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

   5. *-commutative 60.2%

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

   6. associate-/r* 60.2%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s}}}{e^{\frac{x}{s}}} \]

   7. metadata-eval 60.2%

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

9. Simplified 60.2%

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

10. Taylor expanded in x around 0 52.8%

    \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{1 + \frac{x}{s}}} \]
11. Step-by-step derivation

    1. +-commutative 52.8%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\frac{x}{s} + 1}} \]

12. Simplified 52.8%

    \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\frac{x}{s} + 1}} \]
13. Step-by-step derivation

    1. expm1-log1p-u 36.9%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{s} + 1\right)\right)}} \]

    2. expm1-undefine 36.9%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{s} + 1\right)} - 1}} \]

14. Applied egg-rr 36.9%

    \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{s} + 1\right)} - 1}} \]
15. Step-by-step derivation

    1. sub-neg 36.9%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{s} + 1\right)} + \left(-1\right)}} \]

    2. log1p-undefine 36.9%

       \[\leadsto \frac{\frac{0.25}{s}}{e^{\color{blue}{\log \left(1 + \left(\frac{x}{s} + 1\right)\right)}} + \left(-1\right)} \]

    3. rem-exp-log 52.9%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\left(1 + \left(\frac{x}{s} + 1\right)\right)} + \left(-1\right)} \]

    4. +-commutative 52.9%

       \[\leadsto \frac{\frac{0.25}{s}}{\left(1 + \color{blue}{\left(1 + \frac{x}{s}\right)}\right) + \left(-1\right)} \]

    5. associate-+r+ 52.8%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\left(\left(1 + 1\right) + \frac{x}{s}\right)} + \left(-1\right)} \]

    6. metadata-eval 52.8%

       \[\leadsto \frac{\frac{0.25}{s}}{\left(\color{blue}{2} + \frac{x}{s}\right) + \left(-1\right)} \]

    7. metadata-eval 52.8%

       \[\leadsto \frac{\frac{0.25}{s}}{\left(2 + \frac{x}{s}\right) + \color{blue}{-1}} \]

16. Simplified 52.8%

    \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\left(2 + \frac{x}{s}\right) + -1}} \]

17. Final simplification 52.8%

    \[\leadsto \frac{\frac{0.25}{s}}{\left(\frac{x}{s} + 2\right) + -1} \]
18. Add Preprocessing

Alternative 11: 50.6% accurate, 68.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s \cdot \left(1 + \frac{x}{s}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 (* s (+ 1.0 (/ x s)))))

C

float code(float x, float s) {
	return 0.25f / (s * (1.0f + (x / s)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / (s * (1.0e0 + (x / s)))
end function

Julia

function code(x, s)
	return Float32(Float32(0.25) / Float32(s * Float32(Float32(1.0) + Float32(x / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.25) / (s * (single(1.0) + (x / s)));
end

Derivation

1. Initial program 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around inf 95.6%

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

   1. clear-num 95.6%

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

   2. inv-pow 95.6%

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

7. Applied egg-rr 60.2%

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

   1. unpow-1 60.2%

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

   2. associate-*r* 60.2%

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

   3. *-commutative 60.2%

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

   4. associate-/r* 60.2%

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

   5. *-commutative 60.2%

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

   6. associate-/r* 60.2%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s}}}{e^{\frac{x}{s}}} \]

   7. metadata-eval 60.2%

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

9. Simplified 60.2%

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

10. Taylor expanded in x around 0 52.8%

    \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{1 + \frac{x}{s}}} \]
11. Step-by-step derivation

    1. +-commutative 52.8%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\frac{x}{s} + 1}} \]

12. Simplified 52.8%

    \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\frac{x}{s} + 1}} \]
13. Step-by-step derivation

    1. div-inv 52.5%

       \[\leadsto \frac{\color{blue}{0.25 \cdot \frac{1}{s}}}{\frac{x}{s} + 1} \]

    2. associate-/l* 52.5%

       \[\leadsto \color{blue}{0.25 \cdot \frac{\frac{1}{s}}{\frac{x}{s} + 1}} \]

14. Applied egg-rr 52.5%

    \[\leadsto \color{blue}{0.25 \cdot \frac{\frac{1}{s}}{\frac{x}{s} + 1}} \]
15. Step-by-step derivation

    1. associate-/r* 52.8%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{1}{s \cdot \left(\frac{x}{s} + 1\right)}} \]

    2. associate-*r/ 52.8%

       \[\leadsto \color{blue}{\frac{0.25 \cdot 1}{s \cdot \left(\frac{x}{s} + 1\right)}} \]

    3. metadata-eval 52.8%

       \[\leadsto \frac{\color{blue}{0.25}}{s \cdot \left(\frac{x}{s} + 1\right)} \]

    4. +-commutative 52.8%

       \[\leadsto \frac{0.25}{s \cdot \color{blue}{\left(1 + \frac{x}{s}\right)}} \]

16. Simplified 52.8%

    \[\leadsto \color{blue}{\frac{0.25}{s \cdot \left(1 + \frac{x}{s}\right)}} \]

17. Final simplification 52.8%

    \[\leadsto \frac{0.25}{s \cdot \left(1 + \frac{x}{s}\right)} \]
18. Add Preprocessing

Alternative 12: 50.6% accurate, 68.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{s}}{1 + \frac{x}{s}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ 0.25 s) (+ 1.0 (/ x s))))

C

float code(float x, float s) {
	return (0.25f / s) / (1.0f + (x / s));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (0.25e0 / s) / (1.0e0 + (x / s))
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(0.25) / s) / Float32(Float32(1.0) + Float32(x / s)))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(0.25) / s) / (single(1.0) + (x / s));
end

Derivation

1. Initial program 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around inf 95.6%

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

   1. clear-num 95.6%

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

   2. inv-pow 95.6%

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

7. Applied egg-rr 60.2%

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

   1. unpow-1 60.2%

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

   2. associate-*r* 60.2%

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

   3. *-commutative 60.2%

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

   4. associate-/r* 60.2%

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

   5. *-commutative 60.2%

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

   6. associate-/r* 60.2%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s}}}{e^{\frac{x}{s}}} \]

   7. metadata-eval 60.2%

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

9. Simplified 60.2%

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

10. Taylor expanded in x around 0 52.8%

    \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{1 + \frac{x}{s}}} \]
11. Step-by-step derivation

    1. +-commutative 52.8%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\frac{x}{s} + 1}} \]

12. Simplified 52.8%

    \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\frac{x}{s} + 1}} \]

13. Final simplification 52.8%

    \[\leadsto \frac{\frac{0.25}{s}}{1 + \frac{x}{s}} \]
14. Add Preprocessing

Alternative 13: 29.2% accurate, 77.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.0000000233721948e-7) (/ 0.25 s) (/ 0.25 x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.0000000233721948e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.25f / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 2.0000000233721948e-7) then
        tmp = 0.25e0 / s
    else
        tmp = 0.25e0 / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.0000000233721948e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.25) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.0000000233721948e-7))
		tmp = single(0.25) / s;
	else
		tmp = single(0.25) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.00000002e-7

   1. Initial program 99.0%

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

      1. fabs-neg 99.0%

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

      2. distribute-frac-neg 99.0%

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

      3. distribute-frac-neg2 99.0%

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

      4. fabs-neg 99.0%

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

      5. *-commutative 99.0%

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

      6. fabs-neg 99.0%

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

      7. +-commutative 99.0%

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

      8. fabs-neg 99.0%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in s around inf 36.8%

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

   if 2.00000002e-7 < x

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in s around inf 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/r* 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r* 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s}}}{e^{\frac{x}{s}}} \]

      7. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 47.9%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{1 + \frac{x}{s}}} \]
   11. Step-by-step derivation

       1. +-commutative 47.9%

          \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\frac{x}{s} + 1}} \]

   12. Simplified 47.9%

       \[\leadsto \frac{\frac{0.25}{s}}{\color{blue}{\frac{x}{s} + 1}} \]

   13. Taylor expanded in s around 0 11.2%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.6% 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 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around inf 26.6%

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

6. Final simplification 26.6%

   \[\leadsto \frac{0.25}{s} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(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))))))