Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 8.5s

Alternatives: 14

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

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

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 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-x}{s}}\\ \mathbf{if}\;t\_0 \leq 0.0005000000237487257:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, \mathsf{fma}\left(2, \frac{\frac{x}{s}}{-2}, 1\right), 1\right)}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;0.5 + \frac{0.25}{s} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{x}{s}}{s} \cdot x\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x) s))))
   (if (<= t_0 0.0005000000237487257)
     (/ 1.0 (fma 1.0 (fma 2.0 (/ (/ x s) -2.0) 1.0) 1.0))
     (if (<= t_0 2.0)
       (+ 0.5 (* (/ 0.25 s) x))
       (/ 1.0 (* (* (/ (/ x s) s) x) 0.5))))))

C

float code(float x, float s) {
	float t_0 = expf((-x / s));
	float tmp;
	if (t_0 <= 0.0005000000237487257f) {
		tmp = 1.0f / fmaf(1.0f, fmaf(2.0f, ((x / s) / -2.0f), 1.0f), 1.0f);
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((0.25f / s) * x);
	} else {
		tmp = 1.0f / ((((x / s) / s) * x) * 0.5f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-x) / s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0005000000237487257))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), fma(Float32(2.0), Float32(Float32(x / s) / Float32(-2.0)), Float32(1.0)), Float32(1.0)));
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(Float32(0.25) / s) * x));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x / s) / s) * x) * Float32(0.5)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 5.00000024e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.5

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 99.5%

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

   if 5.00000024e-4 < (exp.f32 (/.f32 (neg.f32 x) s)) < 2

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{1}{4}, \frac{1}{2}\right)} \]

      4. lower-/.f32 87.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{s}}, 0.25, 0.5\right) \]

   5. Applied rewrites 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.3%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot 1\right) \cdot \frac{1}{s}, 0.25, 0.5\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 96.2%

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

   if 2 < (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. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. lower-/.f32 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

         \[\leadsto \frac{1}{2 - \frac{\color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{s}, \frac{-1}{2}, x\right)}}{s}} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{s}, \frac{-1}{2}, x\right)}{s}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\color{blue}{x \cdot \frac{x}{s}}, \frac{-1}{2}, x\right)}{s}} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\color{blue}{x \cdot \frac{x}{s}}, \frac{-1}{2}, x\right)}{s}} \]

      11. lower-/.f32 43.1

          \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(x \cdot \color{blue}{\frac{x}{s}}, -0.5, x\right)}{s}} \]

   7. Applied rewrites 43.1%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{\mathsf{fma}\left(x \cdot \frac{x}{s}, -0.5, x\right)}{s}}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 69.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.0005000000237487257:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, \mathsf{fma}\left(2, \frac{\frac{x}{s}}{-2}, 1\right), 1\right)}\\ \mathbf{elif}\;e^{\frac{-x}{s}} \leq 2:\\ \;\;\;\;0.5 + \frac{0.25}{s} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{x}{s}}{s} \cdot x\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.0005000000237487257:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, \mathsf{fma}\left(2, \frac{\frac{x}{s}}{-2}, 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{\left(\frac{x}{s} \cdot x\right) \cdot -0.5 + x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 0.0005000000237487257)
   (/ 1.0 (fma 1.0 (fma 2.0 (/ (/ x s) -2.0) 1.0) 1.0))
   (/ 1.0 (- 2.0 (/ (+ (* (* (/ x s) x) -0.5) x) s)))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 0.0005000000237487257f) {
		tmp = 1.0f / fmaf(1.0f, fmaf(2.0f, ((x / s) / -2.0f), 1.0f), 1.0f);
	} else {
		tmp = 1.0f / (2.0f - (((((x / s) * x) * -0.5f) + x) / s));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(0.0005000000237487257))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), fma(Float32(2.0), Float32(Float32(x / s) / Float32(-2.0)), Float32(1.0)), Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(Float32(Float32(Float32(Float32(x / s) * x) * Float32(-0.5)) + x) / s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 5.00000024e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.5

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 99.5%

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

   if 5.00000024e-4 < (exp.f32 (/.f32 (neg.f32 x) s))

   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. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. lower-/.f32 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

         \[\leadsto \frac{1}{2 - \frac{\color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{s}, \frac{-1}{2}, x\right)}}{s}} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{s}, \frac{-1}{2}, x\right)}{s}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\color{blue}{x \cdot \frac{x}{s}}, \frac{-1}{2}, x\right)}{s}} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\color{blue}{x \cdot \frac{x}{s}}, \frac{-1}{2}, x\right)}{s}} \]

      11. lower-/.f32 65.0

          \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(x \cdot \color{blue}{\frac{x}{s}}, -0.5, x\right)}{s}} \]

   7. Applied rewrites 65.0%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{\mathsf{fma}\left(x \cdot \frac{x}{s}, -0.5, x\right)}{s}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 78.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.0005000000237487257:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, \mathsf{fma}\left(2, \frac{\frac{x}{s}}{-2}, 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{\left(\frac{x}{s} \cdot x\right) \cdot -0.5 + x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{0.25}{s} \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 0.5)
   (fma 1.0 (* (/ 0.25 s) x) 0.5)
   (/ 1.0 (+ (- 1.0 (/ x s)) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 0.5f) {
		tmp = fmaf(1.0f, ((0.25f / s) * x), 0.5f);
	} 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(0.5))
		tmp = fma(Float32(1.0), Float32(Float32(Float32(0.25) / s) * x), Float32(0.5));
	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)) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{1}{4}, \frac{1}{2}\right)} \]

      4. lower-/.f32 28.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{s}}, 0.25, 0.5\right) \]

   5. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 28.2%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot 1\right) \cdot \frac{1}{s}, 0.25, 0.5\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 27.9%

      \[\leadsto \mathsf{fma}\left(1, \color{blue}{\frac{0.25}{s} \cdot x}, 0.5\right) \]

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 65.2

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

   5. Applied rewrites 65.2%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{0.25}{s} \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{0.25}{s} \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 0.5)
   (fma 1.0 (* (/ 0.25 s) x) 0.5)
   (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 0.5f) {
		tmp = fmaf(1.0f, ((0.25f / s) * x), 0.5f);
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(0.5))
		tmp = fma(Float32(1.0), Float32(Float32(Float32(0.25) / s) * x), Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{1}{4}, \frac{1}{2}\right)} \]

      4. lower-/.f32 28.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{s}}, 0.25, 0.5\right) \]

   5. Applied rewrites 28.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 27.9%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot 1\right) \cdot \frac{1}{s}, 0.25, 0.5\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 27.9%

      \[\leadsto \mathsf{fma}\left(1, \color{blue}{\frac{0.25}{s} \cdot x}, 0.5\right) \]

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 65.2

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

   5. Applied rewrites 65.2%

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

Alternative 6: 90.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, \mathsf{fma}\left(2, \frac{\frac{x}{s}}{-2}, 1\right), 1\right)}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;0.5 + \frac{0.25}{s} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \left(x \cdot x\right) \cdot \left(\frac{\frac{1}{x}}{s} - \frac{0.5}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -5.0)
     (/ 1.0 (fma 1.0 (fma 2.0 (/ (/ x s) -2.0) 1.0) 1.0))
     (if (<= t_0 1.0)
       (+ 0.5 (* (/ 0.25 s) x))
       (/ 1.0 (- 2.0 (* (* x x) (- (/ (/ 1.0 x) s) (/ 0.5 (* s s))))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = 1.0f / fmaf(1.0f, fmaf(2.0f, ((x / s) / -2.0f), 1.0f), 1.0f);
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f + ((0.25f / s) * x);
	} else {
		tmp = 1.0f / (2.0f - ((x * x) * (((1.0f / x) / s) - (0.5f / (s * s)))));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), fma(Float32(2.0), Float32(Float32(x / s) / Float32(-2.0)), Float32(1.0)), Float32(1.0)));
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(Float32(0.25) / s) * x));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(Float32(x * x) * Float32(Float32(Float32(Float32(1.0) / x) / s) - Float32(Float32(0.5) / Float32(s * s))))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.5

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 99.5%

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

   if -5 < (/.f32 (neg.f32 x) s) < 1

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{1}{4}, \frac{1}{2}\right)} \]

      4. lower-/.f32 87.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{s}}, 0.25, 0.5\right) \]

   5. Applied rewrites 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.3%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot 1\right) \cdot \frac{1}{s}, 0.25, 0.5\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 96.2%

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

   if 1 < (/.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. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. lower-/.f32 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

         \[\leadsto \frac{1}{2 - \frac{\color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{s}, \frac{-1}{2}, x\right)}}{s}} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{s}, \frac{-1}{2}, x\right)}{s}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\color{blue}{x \cdot \frac{x}{s}}, \frac{-1}{2}, x\right)}{s}} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\color{blue}{x \cdot \frac{x}{s}}, \frac{-1}{2}, x\right)}{s}} \]

      11. lower-/.f32 43.1

          \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(x \cdot \color{blue}{\frac{x}{s}}, -0.5, x\right)}{s}} \]

   7. Applied rewrites 43.1%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{\mathsf{fma}\left(x \cdot \frac{x}{s}, -0.5, x\right)}{s}}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 72.7%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, \mathsf{fma}\left(2, \frac{\frac{x}{s}}{-2}, 1\right), 1\right)}\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5 + \frac{0.25}{s} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \left(x \cdot x\right) \cdot \left(\frac{\frac{1}{x}}{s} - \frac{0.5}{s \cdot s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, \mathsf{fma}\left(2, \frac{\frac{x}{s}}{-2}, 1\right), 1\right)}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;0.5 + \frac{0.25}{s} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{x}}{s}\right) \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -5.0)
     (/ 1.0 (fma 1.0 (fma 2.0 (/ (/ x s) -2.0) 1.0) 1.0))
     (if (<= t_0 1.0)
       (+ 0.5 (* (/ 0.25 s) x))
       (/ 1.0 (* (- (/ 0.5 (* s s)) (/ (/ 1.0 x) s)) (* x x)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = 1.0f / fmaf(1.0f, fmaf(2.0f, ((x / s) / -2.0f), 1.0f), 1.0f);
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f + ((0.25f / s) * x);
	} else {
		tmp = 1.0f / (((0.5f / (s * s)) - ((1.0f / x) / s)) * (x * x));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), fma(Float32(2.0), Float32(Float32(x / s) / Float32(-2.0)), Float32(1.0)), Float32(1.0)));
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(Float32(0.25) / s) * x));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) - Float32(Float32(Float32(1.0) / x) / s)) * Float32(x * x)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.5

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

   7. Applied rewrites 99.5%

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

   9. Applied rewrites 99.5%

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

   if -5 < (/.f32 (neg.f32 x) s) < 1

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{1}{4}, \frac{1}{2}\right)} \]

      4. lower-/.f32 87.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{s}}, 0.25, 0.5\right) \]

   5. Applied rewrites 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.3%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot 1\right) \cdot \frac{1}{s}, 0.25, 0.5\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 96.2%

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

   if 1 < (/.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. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. lower-/.f32 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

         \[\leadsto \frac{1}{2 - \frac{\color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{s}, \frac{-1}{2}, x\right)}}{s}} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{s}, \frac{-1}{2}, x\right)}{s}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\color{blue}{x \cdot \frac{x}{s}}, \frac{-1}{2}, x\right)}{s}} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\color{blue}{x \cdot \frac{x}{s}}, \frac{-1}{2}, x\right)}{s}} \]

      11. lower-/.f32 43.1

          \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(x \cdot \color{blue}{\frac{x}{s}}, -0.5, x\right)}{s}} \]

   7. Applied rewrites 43.1%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{\mathsf{fma}\left(x \cdot \frac{x}{s}, -0.5, x\right)}{s}}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 72.7%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, \mathsf{fma}\left(2, \frac{\frac{x}{s}}{-2}, 1\right), 1\right)}\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5 + \frac{0.25}{s} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{x}}{s}\right) \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, \mathsf{fma}\left(x, \frac{-1}{s}, 1\right), 1\right)}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;0.5 + \frac{0.25}{s} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{x}{s}}{s} \cdot x\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -5.0)
     (/ 1.0 (fma 1.0 (fma x (/ -1.0 s) 1.0) 1.0))
     (if (<= t_0 1.0)
       (+ 0.5 (* (/ 0.25 s) x))
       (/ 1.0 (* (* (/ (/ x s) s) x) 0.5))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = 1.0f / fmaf(1.0f, fmaf(x, (-1.0f / s), 1.0f), 1.0f);
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f + ((0.25f / s) * x);
	} else {
		tmp = 1.0f / ((((x / s) / s) * x) * 0.5f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), fma(x, Float32(Float32(-1.0) / s), Float32(1.0)), Float32(1.0)));
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(Float32(0.25) / s) * x));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x / s) / s) * x) * Float32(0.5)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.5

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 99.5%

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

   if -5 < (/.f32 (neg.f32 x) s) < 1

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{1}{4}, \frac{1}{2}\right)} \]

      4. lower-/.f32 87.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{s}}, 0.25, 0.5\right) \]

   5. Applied rewrites 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.3%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot 1\right) \cdot \frac{1}{s}, 0.25, 0.5\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 96.2%

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

   if 1 < (/.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. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. lower-/.f32 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

         \[\leadsto \frac{1}{2 - \frac{\color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{s}, \frac{-1}{2}, x\right)}}{s}} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{s}, \frac{-1}{2}, x\right)}{s}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\color{blue}{x \cdot \frac{x}{s}}, \frac{-1}{2}, x\right)}{s}} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\color{blue}{x \cdot \frac{x}{s}}, \frac{-1}{2}, x\right)}{s}} \]

      11. lower-/.f32 43.1

          \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(x \cdot \color{blue}{\frac{x}{s}}, -0.5, x\right)}{s}} \]

   7. Applied rewrites 43.1%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{\mathsf{fma}\left(x \cdot \frac{x}{s}, -0.5, x\right)}{s}}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 69.5%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, \mathsf{fma}\left(x, \frac{-1}{s}, 1\right), 1\right)}\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5 + \frac{0.25}{s} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{x}{s}}{s} \cdot x\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, \mathsf{fma}\left(x, \frac{-1}{s}, 1\right), 1\right)}\\ \mathbf{elif}\;t\_0 \leq 200:\\ \;\;\;\;0.5 + \frac{0.25}{s} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{\mathsf{fma}\left(-0.5, x, s\right) \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -5.0)
     (/ 1.0 (fma 1.0 (fma x (/ -1.0 s) 1.0) 1.0))
     (if (<= t_0 200.0)
       (+ 0.5 (* (/ 0.25 s) x))
       (/ 1.0 (- 2.0 (/ (* (fma -0.5 x s) x) (* s s))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = 1.0f / fmaf(1.0f, fmaf(x, (-1.0f / s), 1.0f), 1.0f);
	} else if (t_0 <= 200.0f) {
		tmp = 0.5f + ((0.25f / s) * x);
	} else {
		tmp = 1.0f / (2.0f - ((fmaf(-0.5f, x, s) * x) / (s * s)));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), fma(x, Float32(Float32(-1.0) / s), Float32(1.0)), Float32(1.0)));
	elseif (t_0 <= Float32(200.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(Float32(0.25) / s) * x));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(Float32(fma(Float32(-0.5), x, s) * x) / Float32(s * s))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.5

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 99.5%

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

   if -5 < (/.f32 (neg.f32 x) s) < 200

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{1}{4}, \frac{1}{2}\right)} \]

      4. lower-/.f32 83.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{s}}, 0.25, 0.5\right) \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.8%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot 1\right) \cdot \frac{1}{s}, 0.25, 0.5\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 91.7%

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

   if 200 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. lower-/.f32 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

         \[\leadsto \frac{1}{2 - \frac{\color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{s}, \frac{-1}{2}, x\right)}}{s}} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{s}, \frac{-1}{2}, x\right)}{s}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\color{blue}{x \cdot \frac{x}{s}}, \frac{-1}{2}, x\right)}{s}} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(\color{blue}{x \cdot \frac{x}{s}}, \frac{-1}{2}, x\right)}{s}} \]

      11. lower-/.f32 44.2

          \[\leadsto \frac{1}{2 - \frac{\mathsf{fma}\left(x \cdot \color{blue}{\frac{x}{s}}, -0.5, x\right)}{s}} \]

   7. Applied rewrites 44.2%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{\mathsf{fma}\left(x \cdot \frac{x}{s}, -0.5, x\right)}{s}}} \]

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 49.8%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, \mathsf{fma}\left(x, \frac{-1}{s}, 1\right), 1\right)}\\ \mathbf{elif}\;\frac{-x}{s} \leq 200:\\ \;\;\;\;0.5 + \frac{0.25}{s} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{\mathsf{fma}\left(-0.5, x, s\right) \cdot x}{s \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -1.0f) {
		tmp = 1.0f / fmaf(1.0f, fmaf(x, (-1.0f / s), 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 (Float32(Float32(-x) / s) <= Float32(-1.0))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), fma(x, Float32(Float32(-1.0) / 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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 5.4

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

   5. Applied rewrites 5.4%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 98.6

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

   7. Applied rewrites 97.5%

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

   9. Applied rewrites 98.6%

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

   if -1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 65.2

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

   5. Applied rewrites 65.2%

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

4. Final simplification 75.7%

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

Alternative 11: 74.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{s}\\ \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, t\_0, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 5.4

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

   5. Applied rewrites 5.4%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 98.6

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

   7. Applied rewrites 97.5%

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

   9. Applied rewrites 97.5%

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

   if -1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 65.2

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

   5. Applied rewrites 65.2%

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

4. Final simplification 75.7%

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

Alternative 12: 74.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, t\_0, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), t_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 (/.f32 (neg.f32 x) s) < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 5.4

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

   5. Applied rewrites 5.4%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 98.6

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 98.6%

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

   if -1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 65.2

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

   5. Applied rewrites 65.2%

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

4. Final simplification 75.7%

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

Alternative 13: 44.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.25}{s} \cdot x\\ \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (* (/ 0.25 s) x)))
   (if (<= (/ (- x) s) -5.0) (fma 1.0 t_0 0.5) (+ 0.5 t_0))))

C

float code(float x, float s) {
	float t_0 = (0.25f / s) * x;
	float tmp;
	if ((-x / s) <= -5.0f) {
		tmp = fmaf(1.0f, t_0, 0.5f);
	} else {
		tmp = 0.5f + t_0;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(Float32(0.25) / s) * x)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-5.0))
		tmp = fma(Float32(1.0), t_0, Float32(0.5));
	else
		tmp = Float32(Float32(0.5) + t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{1}{4}, \frac{1}{2}\right)} \]

      4. lower-/.f32 28.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{s}}, 0.25, 0.5\right) \]

   5. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 27.9%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot 1\right) \cdot \frac{1}{s}, 0.25, 0.5\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 27.9%

      \[\leadsto \mathsf{fma}\left(1, \color{blue}{\frac{0.25}{s} \cdot x}, 0.5\right) \]

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{1}{4}, \frac{1}{2}\right)} \]

      4. lower-/.f32 42.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{s}}, 0.25, 0.5\right) \]

   5. Applied rewrites 42.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{s}, 0.25, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.3%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot 1\right) \cdot \frac{1}{s}, 0.25, 0.5\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 44.7%

      \[\leadsto \frac{0.25}{s} \cdot x + \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{0.25}{s} \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{0.25}{s} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.1% 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 99.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 37.9%

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

Reproduce

herbie shell --seed 2024285 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))