Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 17.1s

Alternatives: 9

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{1 + e^{\frac{\left|x\right|}{-s}}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  (/ 1.0 (+ 1.0 (exp (/ (fabs x) (- s)))))
  (expm1 (log1p (fma s (exp (/ (fabs x) s)) s)))))

C

float code(float x, float s) {
	return (1.0f / (1.0f + expf((fabsf(x) / -s)))) / expm1f(log1pf(fmaf(s, expf((fabsf(x) / s)), s)));
}

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(abs(x) / Float32(-s))))) / expm1(log1p(fma(s, exp(Float32(abs(x) / s)), s))))
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. expm1-log1p-u 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{\frac{t_0}{s}}{{\left(1 + t_0\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s)))) (/ (/ t_0 s) (pow (+ 1.0 t_0) 2.0))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return (t_0 / s) / powf((1.0f + t_0), 2.0f);
}

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 / s) / ((1.0e0 + t_0) ** 2.0e0)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(Float32(t_0 / s) / (Float32(Float32(1.0) + t_0) ^ Float32(2.0)))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	tmp = (t_0 / s) / ((single(1.0) + t_0) ^ single(2.0));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in s around 0 99.4%

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

   1. associate-/r* 99.4%

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

   2. mul-1-neg 99.4%

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

   3. distribute-frac-neg 99.4%

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

   4. +-commutative 99.4%

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

   5. rec-exp 99.4%

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

   6. mul-1-neg 99.4%

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

   7. unpow2 99.4%

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

   8. +-commutative 99.4%

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

   9. mul-1-neg 99.4%

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

   10. distribute-frac-neg 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 62.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. expm1-log1p-u 99.5%

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

5. Applied egg-rr 99.5%

   \[\leadsto \frac{\frac{1}{1 + e^{\frac{\left|x\right|}{-s}}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)\right)\right)}} \]
6. Step-by-step derivation

   1. fma-def 99.4%

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

   2. expm1-log1p-u 99.4%

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

   3. *-commutative 99.4%

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

   4. fma-def 99.4%

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

   5. add-sqr-sqrt 47.7%

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

   6. fabs-sqr 47.7%

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

   7. add-sqr-sqrt 62.7%

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

7. Applied egg-rr 62.7%

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

8. Final simplification 62.7%

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

Alternative 4: 62.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. fma-udef 99.4%

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

5. Applied egg-rr 99.4%

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

   1. add-exp-log 98.0%

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

   2. *-commutative 98.0%

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

   3. log-prod 97.8%

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

   4. add-log-exp 98.1%

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

   5. add-sqr-sqrt 47.4%

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

   6. fabs-sqr 47.4%

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

   7. add-sqr-sqrt 61.9%

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

7. Applied egg-rr 61.9%

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

   1. *-un-lft-identity 61.9%

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

   2. +-commutative 61.9%

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

   3. exp-sum 61.7%

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

   4. add-exp-log 62.7%

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

9. Applied egg-rr 62.7%

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

    1. *-lft-identity 62.7%

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

    2. associate-/l/ 62.7%

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

    3. *-commutative 62.7%

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

11. Simplified 62.7%

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

12. Final simplification 62.7%

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

Alternative 5: 60.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. fma-udef 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in s around inf 96.0%

   \[\leadsto \frac{\color{blue}{0.5}}{s \cdot e^{\frac{\left|x\right|}{s}} + s} \]
7. Step-by-step derivation

   1. expm1-log1p-u 96.0%

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

   2. expm1-udef 84.6%

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

   3. add-sqr-sqrt 42.7%

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

   4. fabs-sqr 42.7%

      \[\leadsto \frac{0.5}{\left(e^{\mathsf{log1p}\left(s \cdot e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}\right)} - 1\right) + s} \]

   5. add-sqr-sqrt 50.5%

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

8. Applied egg-rr 50.5%

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

   1. expm1-def 61.8%

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

   2. expm1-log1p 61.8%

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

10. Simplified 61.8%

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

11. Final simplification 61.8%

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

Alternative 6: 30.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\left|x\right| + s \cdot 2} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (+ (fabs x) (* s 2.0))))

C

float code(float x, float s) {
	return 0.5f / (fabsf(x) + (s * 2.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / (abs(x) + (s * 2.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(0.5) / Float32(abs(x) + Float32(s * Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / (abs(x) + (s * single(2.0)));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in s around inf 30.2%

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

   1. *-commutative 30.2%

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

6. Simplified 30.2%

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

7. Taylor expanded in s around inf 28.9%

   \[\leadsto \frac{\color{blue}{0.5}}{\left|x\right| + s \cdot 2} \]

8. Final simplification 28.9%

   \[\leadsto \frac{0.5}{\left|x\right| + s \cdot 2} \]

Alternative 7: 30.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s + \left(\left|x\right| + s\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. fma-udef 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in s around inf 96.0%

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

7. Taylor expanded in s around inf 28.9%

   \[\leadsto \frac{0.5}{\color{blue}{\left(s + \left|x\right|\right)} + s} \]

8. Final simplification 28.9%

   \[\leadsto \frac{0.5}{s + \left(\left|x\right| + s\right)} \]

Alternative 8: 31.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 9.999999747378752e-5) (/ 0.25 s) (/ 0.5 x)))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 9.999999747378752e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.5f / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 9.999999747378752e-5) then
        tmp = 0.25e0 / s
    else
        tmp = 0.5e0 / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(9.999999747378752e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(9.999999747378752e-5))
		tmp = single(0.25) / s;
	else
		tmp = single(0.5) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 9.99999975e-5

   1. Initial program 98.6%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in s around inf 53.6%

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

   if 9.99999975e-5 < (fabs.f32 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. associate-/l/ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-/r* 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

   9. Applied egg-rr 53.6%

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

       1. expm1-def 53.6%

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

       2. expm1-log1p 55.7%

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

       3. associate-/r* 55.0%

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

   11. Simplified 55.0%

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

   12. Taylor expanded in s around inf 10.8%

       \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 9: 28.1% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in s around inf 26.8%

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

5. Final simplification 26.8%

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

Reproduce

herbie shell --seed 2023279 
(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))))))