Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 9.2s

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. exp-1-e N/A

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

   6. lower-E.f32 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.6000000238418579:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ (exp (/ (- x) s)) 1.0)) 0.6000000238418579)
   (/ 1.0 (+ (- 1.0 (/ x s)) 1.0))
   (/ 1.0 (+ (/ 1.0 (/ x s)) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (expf((-x / s)) + 1.0f)) <= 0.6000000238418579f) {
		tmp = 1.0f / ((1.0f - (x / s)) + 1.0f);
	} else {
		tmp = 1.0f / ((1.0f / (x / s)) + 1.0f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((1.0e0 / (exp((-x / s)) + 1.0e0)) <= 0.6000000238418579e0) then
        tmp = 1.0e0 / ((1.0e0 - (x / s)) + 1.0e0)
    else
        tmp = 1.0e0 / ((1.0e0 / (x / s)) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(exp(Float32(Float32(-x) / s)) + Float32(1.0))) <= Float32(0.6000000238418579))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) - Float32(x / s)) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(x / s)) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) / (exp((-x / s)) + single(1.0))) <= single(0.6000000238418579))
		tmp = single(1.0) / ((single(1.0) - (x / s)) + single(1.0));
	else
		tmp = single(1.0) / ((single(1.0) / (x / s)) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.600000024

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 64.6

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

   5. Applied rewrites 64.6%

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

   if 0.600000024 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 99.9%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 95.6

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

   7. Applied rewrites 95.6%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 95.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.6000000238418579:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.9975000023841858:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 0.9975000023841858)
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
   (/ 1.0 (+ (- 1.0 (/ x s)) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 0.9975000023841858f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else {
		tmp = 1.0f / ((1.0f - (x / s)) + 1.0f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (exp((-x / s)) <= 0.9975000023841858e0) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else
        tmp = 1.0e0 / ((1.0e0 - (x / s)) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(0.9975000023841858))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0))) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) - Float32(x / s)) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp((-x / s)) <= single(0.9975000023841858))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	else
		tmp = single(1.0) / ((single(1.0) - (x / s)) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 0.997500002

   1. Initial program 99.9%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 93.3

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

   7. Applied rewrites 93.3%

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

   if 0.997500002 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 64.8

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

   5. Applied rewrites 64.8%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.9975000023841858:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.05000000074505806:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{x}{s}, -1, 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 0.05000000074505806)
   (/ 1.0 (+ (fma (/ x s) -1.0 1.0) 1.0))
   (/ 1.0 (+ (- 1.0 (/ x s)) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 0.05000000074505806f) {
		tmp = 1.0f / (fmaf((x / s), -1.0f, 1.0f) + 1.0f);
	} else {
		tmp = 1.0f / ((1.0f - (x / s)) + 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(0.05000000074505806))
		tmp = Float32(Float32(1.0) / Float32(fma(Float32(x / s), Float32(-1.0), Float32(1.0)) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) - Float32(x / s)) + Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 0.0500000007

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 28.1%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 1\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 28.1%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(0.5, \color{blue}{\frac{x}{s}}, -1\right), 1\right)} \]

   8. Taylor expanded in s around inf

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{x}{s}, -1, 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 10.1%

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

   if 0.0500000007 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 64.6

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

   5. Applied rewrites 64.6%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.05000000074505806:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{x}{s}, -1, 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 6: 90.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 5:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\frac{0.5}{s \cdot s} - \frac{\frac{-1}{x} - \frac{-1}{s}}{x}\right) \cdot x\right) \cdot x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 5.0)
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
   (/
    1.0
    (+ (* (* (- (/ 0.5 (* s s)) (/ (- (/ -1.0 x) (/ -1.0 s)) x)) x) x) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 5.0f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else {
		tmp = 1.0f / (((((0.5f / (s * s)) - (((-1.0f / x) - (-1.0f / s)) / x)) * x) * x) + 1.0f);
	}
	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) <= 5.0e0) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else
        tmp = 1.0e0 / (((((0.5e0 / (s * s)) - ((((-1.0e0) / x) - ((-1.0e0) / s)) / x)) * x) * x) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(5.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0))) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) - Float32(Float32(Float32(Float32(-1.0) / x) - Float32(Float32(-1.0) / s)) / x)) * x) * x) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(5.0))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	else
		tmp = single(1.0) / (((((single(0.5) / (s * s)) - (((single(-1.0) / x) - (single(-1.0) / s)) / x)) * x) * x) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5

   1. Initial program 99.7%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 92.8

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

   7. Applied rewrites 92.8%

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

   if 5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.4%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 1\right)}} \]

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 81.2%

      \[\leadsto \frac{1}{1 + \left(\left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s} - \frac{1}{x}}{x}\right) \cdot x\right) \cdot \color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.8%

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

Alternative 7: 90.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 5:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 5.0)
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
   (/ 1.0 (+ (* (* (/ 0.5 (* s s)) x) x) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 5.0f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else {
		tmp = 1.0f / ((((0.5f / (s * s)) * x) * x) + 1.0f);
	}
	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) <= 5.0e0) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else
        tmp = 1.0e0 / ((((0.5e0 / (s * s)) * x) * x) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(5.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0))) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) * x) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(5.0))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	else
		tmp = single(1.0) / ((((single(0.5) / (s * s)) * x) * x) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5

   1. Initial program 99.7%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 92.8

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

   7. Applied rewrites 92.8%

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

   if 5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.4%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 1\right)}} \]

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 81.2%

      \[\leadsto \frac{1}{1 + \left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot \color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.8%

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

Alternative 8: 49.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -2.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((1.0f - (x / s)) + 1.0f);
	}
	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 / ((1.0e0 - (x / s)) + 1.0e0)
    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(Float32(1.0) - Float32(x / s)) + Float32(1.0)));
	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(1.0) - (x / s)) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 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. Add Preprocessing

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 64.6

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

   5. Applied rewrites 64.6%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.6% accurate, 2.8× 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 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 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. Add Preprocessing

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 64.6

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

   5. Applied rewrites 64.6%

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

Alternative 10: 35.2% accurate, 128.0× 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. Add Preprocessing

3. Taylor expanded in s around inf

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

5. Applied rewrites 37.2%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024277 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))