Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 8.9s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{\frac{x}{-s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (exp (- (log1p (exp (/ x (- s)))))))

C

float code(float x, float s) {
	return expf(-log1pf(expf((x / -s))));
}

Julia

function code(x, s)
	return exp(Float32(-log1p(exp(Float32(x / Float32(-s))))))
end

Derivation

1. Initial program 99.7%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Step-by-step derivation

   1. div-inv 99.7%

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

   2. exp-prod 85.2%

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

   3. neg-mul-1 85.2%

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

   4. exp-prod 85.2%

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

   5. pow-pow 99.7%

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

   6. div-inv 99.7%

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

3. Applied egg-rr 99.7%

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

   1. add-sqr-sqrt 99.6%

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

   2. add-sqr-sqrt 99.7%

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

   3. pow-exp 99.7%

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

   4. neg-mul-1 99.7%

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

   5. div-inv 99.7%

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

   6. distribute-rgt-neg-in 99.7%

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

   7. mul-1-neg 99.7%

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

   8. div-inv 99.7%

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

   9. pow-exp 85.4%

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

   10. add-exp-log 85.4%

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

   11. log-rec 85.4%

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

   12. log1p-udef 85.4%

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

   13. pow-exp 99.7%

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

   14. frac-2neg 99.7%

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

   15. metadata-eval 99.7%

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

   16. un-div-inv 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Step-by-step derivation

   1. div-inv 99.7%

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

   2. exp-prod 85.2%

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

   3. neg-mul-1 85.2%

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

   4. exp-prod 85.2%

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

   5. pow-pow 99.7%

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

   6. div-inv 99.7%

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

3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(1 + \mathsf{expm1}\left(\frac{x}{-s}\right)\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (+ 1.0 (expm1 (/ x (- s)))))))

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + expm1(Float32(x / Float32(-s))))))
end

Derivation

1. Initial program 99.7%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Step-by-step derivation

   1. div-inv 99.7%

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

   2. exp-prod 85.2%

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

   3. neg-mul-1 85.2%

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

   4. exp-prod 85.2%

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

   5. pow-pow 99.7%

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

   6. div-inv 99.7%

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

3. Applied egg-rr 99.7%

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

   1. pow-exp 99.7%

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

   2. neg-mul-1 99.7%

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

   3. div-inv 99.7%

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

   4. distribute-rgt-neg-in 99.7%

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

   5. mul-1-neg 99.7%

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

   6. div-inv 99.7%

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

   7. expm1-log1p-u 99.7%

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

   8. pow-exp 85.4%

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

   9. expm1-udef 85.4%

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

   10. log1p-udef 85.4%

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

   11. add-exp-log 85.4%

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

   12. pow-exp 99.7%

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

   13. frac-2neg 99.7%

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

   14. metadata-eval 99.7%

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

   15. un-div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate--l+ 99.7%

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

   2. expm1-def 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

   \[\leadsto \frac{1}{1 + \left(1 + \mathsf{expm1}\left(\frac{x}{-s}\right)\right)} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

2. Final simplification 99.7%

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

Alternative 5: 47.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ t_1 := \frac{-x}{s}\\ \mathbf{if}\;t_1 \leq -2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - t_0 \cdot t_0}{2 - t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))) (t_1 (/ (- x) s)))
   (if (<= t_1 -2.0)
     0.5
     (if (<= t_1 INFINITY)
       (/ 1.0 (/ (- 4.0 (* t_0 t_0)) (- 2.0 t_0)))
       (/ 1.0 t_1)))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float t_1 = -x / s;
	float tmp;
	if (t_1 <= -2.0f) {
		tmp = 0.5f;
	} else if (t_1 <= ((float) INFINITY)) {
		tmp = 1.0f / ((4.0f - (t_0 * t_0)) / (2.0f - t_0));
	} else {
		tmp = 1.0f / t_1;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	t_1 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_1 <= Float32(-2.0))
		tmp = Float32(0.5);
	elseif (t_1 <= Float32(Inf))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(t_0 * t_0)) / Float32(Float32(2.0) - t_0)));
	else
		tmp = Float32(Float32(1.0) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	t_1 = -x / s;
	tmp = single(0.0);
	if (t_1 <= single(-2.0))
		tmp = single(0.5);
	elseif (t_1 <= single(Inf))
		tmp = single(1.0) / ((single(4.0) - (t_0 * t_0)) / (single(2.0) - t_0));
	else
		tmp = single(1.0) / t_1;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -2

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.1%

      \[\leadsto \color{blue}{0.5} \]

   if -2 < (/.f32 (neg.f32 x) s) < +inf.0

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 62.2%

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

      1. mul-1-neg 62.2%

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

      2. unsub-neg 62.2%

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   4. Simplified 62.2%

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

      1. sub-neg 62.2%

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

      2. neg-mul-1 62.2%

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

      3. rem-log-exp 96.4%

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

      4. pow-exp 96.4%

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

      5. flip-+ 36.4%

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

   6. Applied egg-rr 57.2%

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

   if +inf.0 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 41.1%

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

      1. mul-1-neg 41.1%

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

      2. unsub-neg 41.1%

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   4. Simplified 41.1%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   5. Taylor expanded in x around inf 20.1%

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

      1. associate-*r/ 20.1%

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

      2. neg-mul-1 20.1%

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

   7. Simplified 20.1%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \frac{x}{-s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 6: 49.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -2.0) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -2.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f - (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 / s) <= (-2.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-2.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-2.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -2

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.1%

      \[\leadsto \color{blue}{0.5} \]

   if -2 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 62.2%

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

      1. mul-1-neg 62.2%

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

      2. unsub-neg 62.2%

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   4. Simplified 62.2%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]

Alternative 7: 47.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s))) (if (<= t_0 0.5) 0.5 (/ 1.0 t_0))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 0.5f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= 0.5e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.5))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(0.5))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 52.1%

      \[\leadsto \color{blue}{0.5} \]

   if 0.5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 42.2%

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

      1. mul-1-neg 42.2%

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

      2. unsub-neg 42.2%

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   4. Simplified 42.2%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   5. Taylor expanded in x around inf 42.2%

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

      1. associate-*r/ 42.2%

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

      2. neg-mul-1 42.2%

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

   7. Simplified 42.2%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 8: 46.0% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.000000058430487e-8) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.000000058430487e-8f) {
		tmp = -s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.000000058430487e-8)) then
        tmp = -s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.000000058430487e-8))
		tmp = Float32(Float32(-s) / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.000000058430487e-8))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000006e-8

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 49.3%

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

      1. mul-1-neg 49.3%

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

      2. unsub-neg 49.3%

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   4. Simplified 49.3%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   5. Taylor expanded in x around inf 44.1%

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

      1. associate-*r/ 44.1%

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

      2. neg-mul-1 44.1%

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

   7. Simplified 44.1%

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

   if -5.00000006e-8 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 47.5%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9: 45.8% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.000000058430487e-8) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.000000058430487e-8f) {
		tmp = s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.000000058430487e-8)) then
        tmp = s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.000000058430487e-8))
		tmp = Float32(s / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.000000058430487e-8))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000006e-8

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 49.3%

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

      1. mul-1-neg 49.3%

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

      2. unsub-neg 49.3%

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   4. Simplified 49.3%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   5. Taylor expanded in x around inf 49.3%

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

      1. associate-*r/ 49.3%

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

      2. neg-mul-1 49.3%

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

   7. Simplified 49.3%

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

      1. expm1-log1p-u 49.3%

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

      2. expm1-udef 95.5%

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

      3. frac-2neg 95.5%

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

      4. metadata-eval 95.5%

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

      5. metadata-eval 95.5%

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

      6. frac-2neg 95.5%

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

      7. clear-num 95.5%

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

      8. add-sqr-sqrt 95.5%

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

      9. sqrt-unprod 95.5%

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

      10. sqr-neg 95.5%

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

      11. sqrt-unprod -0.0%

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

      12. add-sqr-sqrt 93.2%

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

   9. Applied egg-rr 93.2%

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

       1. expm1-def 43.9%

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

       2. expm1-log1p 43.9%

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

   11. Simplified 43.9%

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

   if -5.00000006e-8 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 47.5%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 10: 34.2% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

function code(x, s)
	return Float32(0.5)
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in x around 0 34.4%

   \[\leadsto \color{blue}{0.5} \]

3. Final simplification 34.4%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023319 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))