Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 9.2s

Alternatives: 11

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 99.9%

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

   1. distribute-frac-neg 99.9%

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

   2. exp-neg 99.9%

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

4. Applied egg-rr 99.9%

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

   1. add-exp-log 99.9%

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

   2. log-rec 99.9%

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

   3. log1p-expm1-u 99.9%

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

   4. log1p-define 99.9%

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

   5. rec-exp 100.0%

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

   6. expm1-log1p-u 100.0%

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

   7. distribute-neg-frac2 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. distribute-frac-neg 99.9%

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

   2. exp-neg 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \frac{1}{1 + \frac{1}{e^{\frac{x}{s}}}} \]
6. Add Preprocessing

Alternative 3: 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(x / Float32(-s)))))
end

MATLAB

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

Derivation

1. Initial program 99.9%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto \frac{1}{1 + e^{\frac{x}{-s}}} \]
4. Add Preprocessing

Alternative 4: 84.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{s}{x}\\ \mathbf{if}\;x \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;t\_0 + -1\\ \mathbf{elif}\;x \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;\frac{x \cdot 0.25 + s \cdot 0.5}{s}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (/ s x))))
   (if (<= x -1.999999936531045e-21)
     (+ t_0 -1.0)
     (if (<= x 2.00000009162741e-18) (/ (+ (* x 0.25) (* s 0.5)) s) t_0))))

C

float code(float x, float s) {
	float t_0 = 1.0f + (s / x);
	float tmp;
	if (x <= -1.999999936531045e-21f) {
		tmp = t_0 + -1.0f;
	} else if (x <= 2.00000009162741e-18f) {
		tmp = ((x * 0.25f) + (s * 0.5f)) / s;
	} else {
		tmp = 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 = 1.0e0 + (s / x)
    if (x <= (-1.999999936531045e-21)) then
        tmp = t_0 + (-1.0e0)
    else if (x <= 2.00000009162741e-18) then
        tmp = ((x * 0.25e0) + (s * 0.5e0)) / s
    else
        tmp = t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) + Float32(s / x))
	tmp = Float32(0.0)
	if (x <= Float32(-1.999999936531045e-21))
		tmp = Float32(t_0 + Float32(-1.0));
	elseif (x <= Float32(2.00000009162741e-18))
		tmp = Float32(Float32(Float32(x * Float32(0.25)) + Float32(s * Float32(0.5))) / s);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = single(1.0) + (s / x);
	tmp = single(0.0);
	if (x <= single(-1.999999936531045e-21))
		tmp = t_0 + single(-1.0);
	elseif (x <= single(2.00000009162741e-18))
		tmp = ((x * single(0.25)) + (s * single(0.5))) / s;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.9999999e-21

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 44.6%

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

      1. mul-1-neg 44.6%

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

      2. unsub-neg 44.6%

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

   5. Simplified 44.6%

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

   6. Taylor expanded in x around inf 39.9%

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

      1. associate-*r/ 39.9%

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

      2. neg-mul-1 39.9%

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

   8. Simplified 39.9%

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

      1. clear-num 42.1%

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

      2. associate-/r/ 39.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 54.3%

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

      5. sqr-neg 54.3%

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

      6. sqrt-unprod 38.9%

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

      7. add-sqr-sqrt 38.9%

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

   10. Applied egg-rr 38.9%

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

       1. associate-*l/ 38.9%

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

       2. *-un-lft-identity 38.9%

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

       3. expm1-log1p-u 38.8%

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

       4. expm1-undefine 89.3%

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

       5. log1p-define 89.3%

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

       6. add-exp-log 89.4%

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

   12. Applied egg-rr 89.4%

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

   if -1.9999999e-21 < x < 2.00000009e-18

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 77.2%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   4. Step-by-step derivation

      1. associate-*r/ 77.2%

         \[\leadsto 0.5 + \color{blue}{\frac{0.25 \cdot x}{s}} \]

   5. Simplified 77.2%

      \[\leadsto \color{blue}{0.5 + \frac{0.25 \cdot x}{s}} \]

   6. Taylor expanded in s around 0 77.2%

      \[\leadsto \color{blue}{\frac{0.25 \cdot x + 0.5 \cdot s}{s}} \]

   if 2.00000009e-18 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

      1. frac-2neg 98.5%

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

      2. metadata-eval 98.5%

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

      3. div-inv 98.5%

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

      4. distribute-neg-in 98.5%

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

      5. distribute-frac-neg2 98.5%

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

      6. add-sqr-sqrt -0.0%

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

      7. sqrt-unprod 98.5%

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

      8. sqr-neg 98.5%

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

      9. sqrt-unprod 98.5%

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

      10. add-sqr-sqrt 98.5%

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

      11. metadata-eval 98.5%

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

   9. Applied egg-rr 98.5%

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

       1. mul-1-neg 98.5%

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

       2. distribute-neg-frac2 98.5%

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

       3. +-commutative 98.5%

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

       4. distribute-neg-in 98.5%

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

       5. metadata-eval 98.5%

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

       6. sub-neg 98.5%

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

   11. Simplified 98.5%

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

   12. Taylor expanded in x around inf 96.9%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(1 + \frac{s}{x}\right) + -1\\ \mathbf{elif}\;x \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;\frac{x \cdot 0.25 + s \cdot 0.5}{s}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.03999999910593033:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.03999999910593033)
   (/ 1.0 (/ x s))
   (if (<= x 2.00000009162741e-18) (+ 0.5 (/ (* x 0.25) s)) (+ 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.03999999910593033f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 2.00000009162741e-18f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = 1.0f + (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.03999999910593033e0)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 2.00000009162741e-18) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = 1.0e0 + (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.03999999910593033))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(2.00000009162741e-18))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(Float32(1.0) + Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.03999999910593033))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(2.00000009162741e-18))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(1.0) + (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0399999991

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 50.9%

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

      1. mul-1-neg 50.9%

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

      2. unsub-neg 50.9%

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

   5. Simplified 50.9%

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

   6. Taylor expanded in x around inf 48.1%

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

      1. associate-*r/ 48.1%

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

      2. neg-mul-1 48.1%

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

   8. Simplified 48.1%

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

      1. clear-num 50.9%

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

      2. inv-pow 50.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 59.9%

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

      5. sqr-neg 59.9%

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

      6. sqrt-unprod 50.9%

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

      7. add-sqr-sqrt 50.9%

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

   10. Applied egg-rr 50.9%

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

       1. unpow-1 50.9%

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

   12. Simplified 50.9%

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

   if -0.0399999991 < x < 2.00000009e-18

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 62.0%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   4. Step-by-step derivation

      1. associate-*r/ 62.0%

         \[\leadsto 0.5 + \color{blue}{\frac{0.25 \cdot x}{s}} \]

   5. Simplified 62.0%

      \[\leadsto \color{blue}{0.5 + \frac{0.25 \cdot x}{s}} \]

   if 2.00000009e-18 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

      1. frac-2neg 98.5%

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

      2. metadata-eval 98.5%

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

      3. div-inv 98.5%

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

      4. distribute-neg-in 98.5%

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

      5. distribute-frac-neg2 98.5%

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

      6. add-sqr-sqrt -0.0%

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

      7. sqrt-unprod 98.5%

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

      8. sqr-neg 98.5%

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

      9. sqrt-unprod 98.5%

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

      10. add-sqr-sqrt 98.5%

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

      11. metadata-eval 98.5%

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

   9. Applied egg-rr 98.5%

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

       1. mul-1-neg 98.5%

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

       2. distribute-neg-frac2 98.5%

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

       3. +-commutative 98.5%

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

       4. distribute-neg-in 98.5%

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

       5. metadata-eval 98.5%

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

       6. sub-neg 98.5%

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

   11. Simplified 98.5%

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

   12. Taylor expanded in x around inf 96.9%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03999999910593033:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{s}{x}\\ \mathbf{if}\;x \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;t\_0 + -1\\ \mathbf{elif}\;x \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (/ s x))))
   (if (<= x -1.999999936531045e-21)
     (+ t_0 -1.0)
     (if (<= x 2.00000009162741e-18) (+ 0.5 (/ (* x 0.25) s)) t_0))))

C

float code(float x, float s) {
	float t_0 = 1.0f + (s / x);
	float tmp;
	if (x <= -1.999999936531045e-21f) {
		tmp = t_0 + -1.0f;
	} else if (x <= 2.00000009162741e-18f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = 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 = 1.0e0 + (s / x)
    if (x <= (-1.999999936531045e-21)) then
        tmp = t_0 + (-1.0e0)
    else if (x <= 2.00000009162741e-18) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) + Float32(s / x))
	tmp = Float32(0.0)
	if (x <= Float32(-1.999999936531045e-21))
		tmp = Float32(t_0 + Float32(-1.0));
	elseif (x <= Float32(2.00000009162741e-18))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = single(1.0) + (s / x);
	tmp = single(0.0);
	if (x <= single(-1.999999936531045e-21))
		tmp = t_0 + single(-1.0);
	elseif (x <= single(2.00000009162741e-18))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.9999999e-21

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 44.6%

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

      1. mul-1-neg 44.6%

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

      2. unsub-neg 44.6%

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

   5. Simplified 44.6%

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

   6. Taylor expanded in x around inf 39.9%

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

      1. associate-*r/ 39.9%

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

      2. neg-mul-1 39.9%

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

   8. Simplified 39.9%

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

      1. clear-num 42.1%

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

      2. associate-/r/ 39.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 54.3%

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

      5. sqr-neg 54.3%

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

      6. sqrt-unprod 38.9%

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

      7. add-sqr-sqrt 38.9%

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

   10. Applied egg-rr 38.9%

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

       1. associate-*l/ 38.9%

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

       2. *-un-lft-identity 38.9%

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

       3. expm1-log1p-u 38.8%

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

       4. expm1-undefine 89.3%

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

       5. log1p-define 89.3%

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

       6. add-exp-log 89.4%

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

   12. Applied egg-rr 89.4%

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

   if -1.9999999e-21 < x < 2.00000009e-18

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 77.2%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   4. Step-by-step derivation

      1. associate-*r/ 77.2%

         \[\leadsto 0.5 + \color{blue}{\frac{0.25 \cdot x}{s}} \]

   5. Simplified 77.2%

      \[\leadsto \color{blue}{0.5 + \frac{0.25 \cdot x}{s}} \]

   if 2.00000009e-18 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

      1. frac-2neg 98.5%

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

      2. metadata-eval 98.5%

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

      3. div-inv 98.5%

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

      4. distribute-neg-in 98.5%

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

      5. distribute-frac-neg2 98.5%

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

      6. add-sqr-sqrt -0.0%

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

      7. sqrt-unprod 98.5%

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

      8. sqr-neg 98.5%

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

      9. sqrt-unprod 98.5%

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

      10. add-sqr-sqrt 98.5%

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

      11. metadata-eval 98.5%

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

   9. Applied egg-rr 98.5%

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

       1. mul-1-neg 98.5%

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

       2. distribute-neg-frac2 98.5%

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

       3. +-commutative 98.5%

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

       4. distribute-neg-in 98.5%

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

       5. metadata-eval 98.5%

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

       6. sub-neg 98.5%

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

   11. Simplified 98.5%

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

   12. Taylor expanded in x around inf 96.9%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(1 + \frac{s}{x}\right) + -1\\ \mathbf{elif}\;x \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(1 + \frac{s}{x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.999999936531045e-21)
   (+ (+ 1.0 (/ s x)) -1.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.999999936531045e-21f) {
		tmp = (1.0f + (s / x)) + -1.0f;
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (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 <= (-1.999999936531045e-21)) then
        tmp = (1.0e0 + (s / x)) + (-1.0e0)
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.999999936531045e-21))
		tmp = Float32(Float32(Float32(1.0) + Float32(s / x)) + Float32(-1.0));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.999999936531045e-21))
		tmp = (single(1.0) + (s / x)) + single(-1.0);
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999e-21

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 44.6%

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

      1. mul-1-neg 44.6%

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

      2. unsub-neg 44.6%

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

   5. Simplified 44.6%

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

   6. Taylor expanded in x around inf 39.9%

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

      1. associate-*r/ 39.9%

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

      2. neg-mul-1 39.9%

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

   8. Simplified 39.9%

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

      1. clear-num 42.1%

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

      2. associate-/r/ 39.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 54.3%

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

      5. sqr-neg 54.3%

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

      6. sqrt-unprod 38.9%

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

      7. add-sqr-sqrt 38.9%

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

   10. Applied egg-rr 38.9%

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

       1. associate-*l/ 38.9%

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

       2. *-un-lft-identity 38.9%

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

       3. expm1-log1p-u 38.8%

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

       4. expm1-undefine 89.3%

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

       5. log1p-define 89.3%

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

       6. add-exp-log 89.4%

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

   12. Applied egg-rr 89.4%

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

   if -1.9999999e-21 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 92.7%

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

      1. +-commutative 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(1 + \frac{s}{x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.03999999910593033:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{elif}\;x \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.03999999910593033)
   (/ s x)
   (if (<= x 5.000000229068525e-19) 0.5 (+ 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.03999999910593033f) {
		tmp = s / x;
	} else if (x <= 5.000000229068525e-19f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f + (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.03999999910593033e0)) then
        tmp = s / x
    else if (x <= 5.000000229068525e-19) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 + (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.03999999910593033))
		tmp = Float32(s / x);
	elseif (x <= Float32(5.000000229068525e-19))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) + Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.03999999910593033))
		tmp = s / x;
	elseif (x <= single(5.000000229068525e-19))
		tmp = single(0.5);
	else
		tmp = single(1.0) + (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0399999991

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 50.9%

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

      1. mul-1-neg 50.9%

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

      2. unsub-neg 50.9%

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

   5. Simplified 50.9%

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

   6. Taylor expanded in x around inf 48.1%

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

      1. associate-*r/ 48.1%

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

      2. neg-mul-1 48.1%

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

   8. Simplified 48.1%

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

      1. clear-num 50.9%

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

      2. associate-/r/ 48.1%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 58.4%

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

      5. sqr-neg 58.4%

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

      6. sqrt-unprod 48.1%

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

      7. add-sqr-sqrt 48.1%

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

   10. Applied egg-rr 48.1%

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

   11. Taylor expanded in x around 0 48.1%

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

   if -0.0399999991 < x < 5.00000023e-19

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 59.9%

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

   if 5.00000023e-19 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.2%

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

      1. +-commutative 98.2%

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

   7. Simplified 98.2%

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

      1. frac-2neg 98.2%

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

      2. metadata-eval 98.2%

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

      3. div-inv 98.2%

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

      4. distribute-neg-in 98.2%

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

      5. distribute-frac-neg2 98.2%

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

      6. add-sqr-sqrt -0.0%

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

      7. sqrt-unprod 98.0%

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

      8. sqr-neg 98.0%

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

      9. sqrt-unprod 97.9%

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

      10. add-sqr-sqrt 97.9%

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

      11. metadata-eval 97.9%

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

   9. Applied egg-rr 97.9%

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

       1. mul-1-neg 97.9%

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

       2. distribute-neg-frac2 97.9%

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

       3. +-commutative 97.9%

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

       4. distribute-neg-in 97.9%

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

       5. metadata-eval 97.9%

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

       6. sub-neg 97.9%

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

   11. Simplified 97.9%

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

   12. Taylor expanded in x around inf 96.0%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03999999910593033:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{elif}\;x \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.03999999910593033:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.03999999910593033)
   (/ 1.0 (/ x s))
   (if (<= x 5.000000229068525e-19) 0.5 (+ 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.03999999910593033f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 5.000000229068525e-19f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f + (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.03999999910593033e0)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 5.000000229068525e-19) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 + (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.03999999910593033))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(5.000000229068525e-19))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) + Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.03999999910593033))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(5.000000229068525e-19))
		tmp = single(0.5);
	else
		tmp = single(1.0) + (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0399999991

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 50.9%

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

      1. mul-1-neg 50.9%

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

      2. unsub-neg 50.9%

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

   5. Simplified 50.9%

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

   6. Taylor expanded in x around inf 48.1%

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

      1. associate-*r/ 48.1%

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

      2. neg-mul-1 48.1%

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

   8. Simplified 48.1%

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

      1. clear-num 50.9%

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

      2. inv-pow 50.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 59.9%

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

      5. sqr-neg 59.9%

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

      6. sqrt-unprod 50.9%

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

      7. add-sqr-sqrt 50.9%

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

   10. Applied egg-rr 50.9%

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

       1. unpow-1 50.9%

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

   12. Simplified 50.9%

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

   if -0.0399999991 < x < 5.00000023e-19

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 59.9%

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

   if 5.00000023e-19 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.2%

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

      1. +-commutative 98.2%

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

   7. Simplified 98.2%

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

      1. frac-2neg 98.2%

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

      2. metadata-eval 98.2%

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

      3. div-inv 98.2%

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

      4. distribute-neg-in 98.2%

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

      5. distribute-frac-neg2 98.2%

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

      6. add-sqr-sqrt -0.0%

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

      7. sqrt-unprod 98.0%

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

      8. sqr-neg 98.0%

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

      9. sqrt-unprod 97.9%

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

      10. add-sqr-sqrt 97.9%

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

      11. metadata-eval 97.9%

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

   9. Applied egg-rr 97.9%

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

       1. mul-1-neg 97.9%

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

       2. distribute-neg-frac2 97.9%

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

       3. +-commutative 97.9%

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

       4. distribute-neg-in 97.9%

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

       5. metadata-eval 97.9%

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

       6. sub-neg 97.9%

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

   11. Simplified 97.9%

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

   12. Taylor expanded in x around inf 96.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03999999910593033:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.8% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.03999999910593033:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (if (<= x -0.03999999910593033) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.03999999910593033f) {
		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 <= (-0.03999999910593033e0)) 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(-0.03999999910593033))
		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(-0.03999999910593033))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -0.0399999991

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 50.9%

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

      1. mul-1-neg 50.9%

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

      2. unsub-neg 50.9%

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

   5. Simplified 50.9%

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

   6. Taylor expanded in x around inf 48.1%

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

      1. associate-*r/ 48.1%

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

      2. neg-mul-1 48.1%

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

   8. Simplified 48.1%

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

      1. clear-num 50.9%

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

      2. associate-/r/ 48.1%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 58.4%

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

      5. sqr-neg 58.4%

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

      6. sqrt-unprod 48.1%

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

      7. add-sqr-sqrt 48.1%

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

   10. Applied egg-rr 48.1%

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

   11. Taylor expanded in x around 0 48.1%

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

   if -0.0399999991 < x

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 43.0%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03999999910593033:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.6% 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 99.9%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 32.0%

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

4. Final simplification 32.0%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))