Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 10.1s

Alternatives: 13

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 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
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: 91.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{e^{\frac{-x}{s}} + 1}\\ \mathbf{if}\;t\_0 \leq 0.009999999776482582:\\ \;\;\;\;\frac{1}{\left(\frac{x}{s \cdot s} \cdot x\right) \cdot 0.5 + 1}\\ \mathbf{elif}\;t\_0 \leq 0.949999988079071:\\ \;\;\;\;\frac{0.25}{s} \cdot x + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ (exp (/ (- x) s)) 1.0))))
   (if (<= t_0 0.009999999776482582)
     (/ 1.0 (+ (* (* (/ x (* s s)) x) 0.5) 1.0))
     (if (<= t_0 0.949999988079071)
       (+ (* (/ 0.25 s) x) 0.5)
       (/ 1.0 (fma (- 1.0 (/ x s)) 1.0 1.0))))))

C

float code(float x, float s) {
	float t_0 = 1.0f / (expf((-x / s)) + 1.0f);
	float tmp;
	if (t_0 <= 0.009999999776482582f) {
		tmp = 1.0f / ((((x / (s * s)) * x) * 0.5f) + 1.0f);
	} else if (t_0 <= 0.949999988079071f) {
		tmp = ((0.25f / s) * x) + 0.5f;
	} else {
		tmp = 1.0f / fmaf((1.0f - (x / s)), 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) / Float32(exp(Float32(Float32(-x) / s)) + Float32(1.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.009999999776482582))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x / Float32(s * s)) * x) * Float32(0.5)) + Float32(1.0)));
	elseif (t_0 <= Float32(0.949999988079071))
		tmp = Float32(Float32(Float32(Float32(0.25) / s) * x) + Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(1.0) - Float32(x / s)), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.00999999978

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 6.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 78.7%

       \[\leadsto \frac{1}{1 + \left(x \cdot \frac{x}{s \cdot s}\right) \cdot \color{blue}{0.5}} \]

   if 0.00999999978 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.949999988

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{x \cdot \log \mathsf{E}\left(\right)}{s} + \frac{1}{2}} \]

      2. log-E N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f32 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 87.1

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

   7. Applied rewrites 85.8%

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

   9. Applied rewrites 96.8%

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

   if 0.949999988 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   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. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   7. Applied rewrites 99.3%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.009999999776482582:\\ \;\;\;\;\frac{1}{\left(\frac{x}{s \cdot s} \cdot x\right) \cdot 0.5 + 1}\\ \mathbf{elif}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.949999988079071:\\ \;\;\;\;\frac{0.25}{s} \cdot x + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{s}\\ \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.6000000238418579:\\ \;\;\;\;\frac{1}{t\_0 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(t\_0, 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- 1.0 (/ x s))))
   (if (<= (/ 1.0 (+ (exp (/ (- x) s)) 1.0)) 0.6000000238418579)
     (/ 1.0 (+ t_0 1.0))
     (/ 1.0 (fma t_0 1.0 1.0)))))

C

float code(float x, float s) {
	float t_0 = 1.0f - (x / s);
	float tmp;
	if ((1.0f / (expf((-x / s)) + 1.0f)) <= 0.6000000238418579f) {
		tmp = 1.0f / (t_0 + 1.0f);
	} else {
		tmp = 1.0f / fmaf(t_0, 1.0f, 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(1.0) / Float32(exp(Float32(Float32(-x) / s)) + Float32(1.0))) <= Float32(0.6000000238418579))
		tmp = Float32(Float32(1.0) / Float32(t_0 + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / fma(t_0, Float32(1.0), Float32(1.0)));
	end
	return 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 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 60.4

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

   5. Applied rewrites 60.4%

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

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

   1. Initial program 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 5.0

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

   5. Applied rewrites 5.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 - \frac{x}{s}\right)}} \]
   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. *-commutative N/A

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

      5. lower-fma.f32 98.6

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

   7. Applied rewrites 97.5%

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

4. Final simplification 73.5%

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

Alternative 4: 49.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 0.20000000298023224)
   0.5
   (/ 1.0 (+ (- 1.0 (/ x s)) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 0.20000000298023224f) {
		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 (exp((-x / s)) <= 0.20000000298023224e0) 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 (exp(Float32(Float32(-x) / s)) <= Float32(0.20000000298023224))
		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 (exp((-x / s)) <= single(0.20000000298023224))
		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 (exp.f32 (/.f32 (neg.f32 x) s)) < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

   if 0.200000003 < (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 60.4

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

   5. Applied rewrites 60.4%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 0.20000000298023224) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 0.20000000298023224f) {
		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 (exp((-x / s)) <= 0.20000000298023224e0) 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 (exp(Float32(Float32(-x) / s)) <= Float32(0.20000000298023224))
		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 (exp((-x / s)) <= single(0.20000000298023224))
		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 (exp.f32 (/.f32 (neg.f32 x) s)) < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

   if 0.200000003 < (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 60.4

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

   5. Applied rewrites 60.4%

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

Alternative 6: 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 7: 80.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -2.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), \frac{x}{s}, 1\right), 1, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\frac{0.25}{s} \cdot x + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\frac{0.5}{s \cdot s} - \frac{\frac{-1}{x} - \frac{-1}{s}}{x}\right) \cdot x\right) \cdot x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.5)
     (/ 1.0 (fma (fma (fma (/ 0.5 s) x -1.0) (/ x s) 1.0) 1.0 1.0))
     (if (<= t_0 0.5)
       (+ (* (/ 0.25 s) x) 0.5)
       (/
        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 t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.5f) {
		tmp = 1.0f / fmaf(fmaf(fmaf((0.5f / s), x, -1.0f), (x / s), 1.0f), 1.0f, 1.0f);
	} else if (t_0 <= 0.5f) {
		tmp = ((0.25f / s) * x) + 0.5f;
	} else {
		tmp = 1.0f / (((((0.5f / (s * s)) - (((-1.0f / x) - (-1.0f / s)) / x)) * x) * x) + 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.5))
		tmp = Float32(Float32(1.0) / fma(fma(fma(Float32(Float32(0.5) / s), x, Float32(-1.0)), Float32(x / s), Float32(1.0)), Float32(1.0), Float32(1.0)));
	elseif (t_0 <= Float32(0.5))
		tmp = Float32(Float32(Float32(Float32(0.25) / s) * x) + Float32(0.5));
	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

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -2.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   7. Applied rewrites 99.3%

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

   if -2.5 < (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{x \cdot \log \mathsf{E}\left(\right)}{s} + \frac{1}{2}} \]

      2. log-E N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f32 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 87.1

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

   7. Applied rewrites 85.8%

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

   9. Applied rewrites 96.8%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.6%

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

   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 81.4%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), \frac{x}{s}, 1\right), 1, 1\right)}\\ \mathbf{elif}\;\frac{-x}{s} \leq 0.5:\\ \;\;\;\;\frac{0.25}{s} \cdot x + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\frac{0.5}{s \cdot s} - \frac{\frac{-1}{x} - \frac{-1}{s}}{x}\right) \cdot x\right) \cdot x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -2.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), \frac{x}{s}, 1\right), 1, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\frac{0.25}{s} \cdot x + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{x}}{s}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.5)
     (/ 1.0 (fma (fma (fma (/ 0.5 s) x -1.0) (/ x s) 1.0) 1.0 1.0))
     (if (<= t_0 0.5)
       (+ (* (/ 0.25 s) x) 0.5)
       (/ 1.0 (+ (* (* x x) (- (/ 0.5 (* s s)) (/ (/ 1.0 x) s))) 1.0))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.5f) {
		tmp = 1.0f / fmaf(fmaf(fmaf((0.5f / s), x, -1.0f), (x / s), 1.0f), 1.0f, 1.0f);
	} else if (t_0 <= 0.5f) {
		tmp = ((0.25f / s) * x) + 0.5f;
	} else {
		tmp = 1.0f / (((x * x) * ((0.5f / (s * s)) - ((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(-2.5))
		tmp = Float32(Float32(1.0) / fma(fma(fma(Float32(Float32(0.5) / s), x, Float32(-1.0)), Float32(x / s), Float32(1.0)), Float32(1.0), Float32(1.0)));
	elseif (t_0 <= Float32(0.5))
		tmp = Float32(Float32(Float32(Float32(0.25) / s) * x) + Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(x * x) * Float32(Float32(Float32(0.5) / Float32(s * s)) - Float32(Float32(Float32(1.0) / x) / s))) + Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -2.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   7. Applied rewrites 99.3%

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

   if -2.5 < (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{x \cdot \log \mathsf{E}\left(\right)}{s} + \frac{1}{2}} \]

      2. log-E N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f32 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 87.1

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

   7. Applied rewrites 87.1%

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

   9. Applied rewrites 96.8%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 79.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), \frac{x}{s}, 1\right), 1, 1\right)}\\ \mathbf{elif}\;\frac{-x}{s} \leq 0.5:\\ \;\;\;\;\frac{0.25}{s} \cdot x + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{x}}{s}\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -2.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), \frac{x}{s}, 1\right), 1, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\frac{0.25}{s} \cdot x + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{x}{s \cdot s} \cdot x\right) \cdot 0.5 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.5)
     (/ 1.0 (fma (fma (fma (/ 0.5 s) x -1.0) (/ x s) 1.0) 1.0 1.0))
     (if (<= t_0 0.5)
       (+ (* (/ 0.25 s) x) 0.5)
       (/ 1.0 (+ (* (* (/ x (* s s)) x) 0.5) 1.0))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.5f) {
		tmp = 1.0f / fmaf(fmaf(fmaf((0.5f / s), x, -1.0f), (x / s), 1.0f), 1.0f, 1.0f);
	} else if (t_0 <= 0.5f) {
		tmp = ((0.25f / s) * x) + 0.5f;
	} else {
		tmp = 1.0f / ((((x / (s * s)) * x) * 0.5f) + 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.5))
		tmp = Float32(Float32(1.0) / fma(fma(fma(Float32(Float32(0.5) / s), x, Float32(-1.0)), Float32(x / s), Float32(1.0)), Float32(1.0), Float32(1.0)));
	elseif (t_0 <= Float32(0.5))
		tmp = Float32(Float32(Float32(Float32(0.25) / s) * x) + Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x / Float32(s * s)) * x) * Float32(0.5)) + Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -2.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   7. Applied rewrites 99.3%

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

   if -2.5 < (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{x \cdot \log \mathsf{E}\left(\right)}{s} + \frac{1}{2}} \]

      2. log-E N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f32 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 87.1

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

   7. Applied rewrites 85.8%

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

   9. Applied rewrites 96.8%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 6.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 78.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), \frac{x}{s}, 1\right), 1, 1\right)}\\ \mathbf{elif}\;\frac{-x}{s} \leq 0.5:\\ \;\;\;\;\frac{0.25}{s} \cdot x + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{x}{s \cdot s} \cdot x\right) \cdot 0.5 + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.5% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -2.0)
   (/ 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 ((-x / s) <= -2.0f) {
		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 (Float32(Float32(-x) / s) <= Float32(-2.0))
		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 (/.f32 (neg.f32 x) s) < -2

   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 5.0

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

   5. Applied rewrites 5.0%

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

   7. Applied rewrites 28.1%

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

   9. Applied rewrites 28.9%

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

   11. Applied rewrites 28.9%

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

   if -2 < (/.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 60.4

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

   5. Applied rewrites 60.4%

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

4. Final simplification 49.2%

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

Alternative 11: 47.5% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -2.0)
   (/ 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) <= -2.0f) {
		tmp = 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(-2.0))
		tmp = Float32(Float32(1.0) / Float32(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) < -2

   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 5.0

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

   5. Applied rewrites 5.0%

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

   7. Applied rewrites 28.1%

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

   9. Applied rewrites 28.9%

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

   11. Applied rewrites 28.1%

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

   if -2 < (/.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 60.4

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

   5. Applied rewrites 60.4%

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

4. Final simplification 49.2%

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

Alternative 12: 47.2% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -2.0)
   (/ 1.0 (+ (fma -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 ((-x / s) <= -2.0f) {
		tmp = 1.0f / (fmaf(-1.0f, (x / 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(-2.0))
		tmp = Float32(Float32(1.0) / Float32(fma(Float32(-1.0), 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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. 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 5.0

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

   5. Applied rewrites 5.0%

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

   7. Applied rewrites 28.1%

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

   9. Applied rewrites 28.9%

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

   11. Applied rewrites 28.1%

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

   if -2 < (/.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 60.4

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

   5. Applied rewrites 60.4%

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

4. Final simplification 49.2%

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

Alternative 13: 35.0% 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 36.6%

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

Reproduce

herbie shell --seed 2024284 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))