Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 8.9s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% 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: 80.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-x}{s}}\\ \mathbf{if}\;t\_0 \leq 0.0020000000949949026:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, \frac{x}{s}, -1\right)}{s}, x, 1\right), 1, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x) s))))
   (if (<= t_0 0.0020000000949949026)
     (/ 1.0 (fma (fma (/ (fma 0.5 (/ x s) -1.0) s) x 1.0) 1.0 1.0))
     (if (<= t_0 5.0)
       (+ (* 0.25 (/ x s)) 0.5)
       (/ 1.0 (* (* (/ 0.5 (* s s)) x) x))))))

C

float code(float x, float s) {
	float t_0 = expf((-x / s));
	float tmp;
	if (t_0 <= 0.0020000000949949026f) {
		tmp = 1.0f / fmaf(fmaf((fmaf(0.5f, (x / s), -1.0f) / s), x, 1.0f), 1.0f, 1.0f);
	} else if (t_0 <= 5.0f) {
		tmp = (0.25f * (x / s)) + 0.5f;
	} else {
		tmp = 1.0f / (((0.5f / (s * s)) * x) * x);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-x) / s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(1.0) / fma(fma(Float32(fma(Float32(0.5), Float32(x / s), Float32(-1.0)) / s), x, Float32(1.0)), Float32(1.0), Float32(1.0)));
	elseif (t_0 <= Float32(5.0))
		tmp = Float32(Float32(Float32(0.25) * Float32(x / s)) + Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) * x));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 0.00200000009

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{s} + \left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)\right)}} \]

   5. Applied rewrites 27.9%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), x, 1\right)}} \]

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 28.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 98.8

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

   10. Applied rewrites 98.8%

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

   if 0.00200000009 < (exp.f32 (/.f32 (neg.f32 x) s)) < 5

   1. Initial program 99.5%

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

      1. lift-/.f32 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 N/A

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

      8. lift-+.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f32 N/A

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

      11. metadata-eval 99.4

          \[\leadsto {\left({\left(e^{\frac{-x}{s}} + 1\right)}^{2}\right)}^{\color{blue}{-0.5}} \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{{\left({\left(e^{\frac{-x}{s}} + 1\right)}^{2}\right)}^{-0.5}} \]

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 83.2

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

   7. Applied rewrites 82.5%

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

   9. Applied rewrites 93.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 88.1%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, \frac{x}{s}, -1\right)}{s}, x, 1\right), 1, 1\right)}\\ \mathbf{elif}\;e^{\frac{-x}{s}} \leq 5:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-x}{s}}\\ \mathbf{if}\;t\_0 \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(0.5, \frac{x}{s}, -1\right)}{s}, 1\right) + 1}\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x) s))))
   (if (<= t_0 1.999999987845058e-8)
     (/ 1.0 (+ (fma x (/ (fma 0.5 (/ x s) -1.0) s) 1.0) 1.0))
     (if (<= t_0 5.0)
       (+ (* 0.25 (/ x s)) 0.5)
       (/ 1.0 (* (* (/ 0.5 (* s s)) x) x))))))

C

float code(float x, float s) {
	float t_0 = expf((-x / s));
	float tmp;
	if (t_0 <= 1.999999987845058e-8f) {
		tmp = 1.0f / (fmaf(x, (fmaf(0.5f, (x / s), -1.0f) / s), 1.0f) + 1.0f);
	} else if (t_0 <= 5.0f) {
		tmp = (0.25f * (x / s)) + 0.5f;
	} else {
		tmp = 1.0f / (((0.5f / (s * s)) * x) * x);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-x) / s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.999999987845058e-8))
		tmp = Float32(Float32(1.0) / Float32(fma(x, Float32(fma(Float32(0.5), Float32(x / s), Float32(-1.0)) / s), Float32(1.0)) + Float32(1.0)));
	elseif (t_0 <= Float32(5.0))
		tmp = Float32(Float32(Float32(0.25) * Float32(x / s)) + Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) * x));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 1.99999999e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{s} + \left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)\right)}} \]

   5. Applied rewrites 27.9%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), x, 1\right)}} \]

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 28.1%

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

   10. Applied rewrites 28.1%

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

   if 1.99999999e-8 < (exp.f32 (/.f32 (neg.f32 x) s)) < 5

   1. Initial program 99.5%

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

      1. lift-/.f32 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 N/A

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

      8. lift-+.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f32 N/A

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

      11. metadata-eval 99.4

          \[\leadsto {\left({\left(e^{\frac{-x}{s}} + 1\right)}^{2}\right)}^{\color{blue}{-0.5}} \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{{\left({\left(e^{\frac{-x}{s}} + 1\right)}^{2}\right)}^{-0.5}} \]

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 81.0

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

   7. Applied rewrites 80.3%

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

   9. Applied rewrites 91.2%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 88.1%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(0.5, \frac{x}{s}, -1\right)}{s}, 1\right) + 1}\\ \mathbf{elif}\;e^{\frac{-x}{s}} \leq 5:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x}{s}}{s}\\ \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.949999988079071:\\ \;\;\;\;\frac{1}{\left(0.5 \cdot \left(t\_0 \cdot x\right) + 2\right) - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), t\_0, \frac{-1}{s}\right), x, 1\right), 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (/ x s) s)))
   (if (<= (/ 1.0 (+ (exp (/ (- x) s)) 1.0)) 0.949999988079071)
     (/ 1.0 (- (+ (* 0.5 (* t_0 x)) 2.0) (/ x s)))
     (/
      1.0
      (fma
       (fma (fma (fma -0.16666666666666666 (/ x s) 0.5) t_0 (/ -1.0 s)) x 1.0)
       1.0
       1.0)))))

C

float code(float x, float s) {
	float t_0 = (x / s) / s;
	float tmp;
	if ((1.0f / (expf((-x / s)) + 1.0f)) <= 0.949999988079071f) {
		tmp = 1.0f / (((0.5f * (t_0 * x)) + 2.0f) - (x / s));
	} else {
		tmp = 1.0f / fmaf(fmaf(fmaf(fmaf(-0.16666666666666666f, (x / s), 0.5f), t_0, (-1.0f / s)), x, 1.0f), 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(x / s) / s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(exp(Float32(Float32(-x) / s)) + Float32(1.0))) <= Float32(0.949999988079071))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) * Float32(t_0 * x)) + Float32(2.0)) - Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / fma(fma(fma(fma(Float32(-0.16666666666666666), Float32(x / s), Float32(0.5)), t_0, Float32(Float32(-1.0) / s)), x, Float32(1.0)), 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)))) < 0.949999988

   1. Initial program 99.7%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 41.7%

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

   7. Applied rewrites 85.8%

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

   9. Applied rewrites 86.2%

      \[\leadsto \frac{1}{\left(2 + \left(x \cdot \frac{\frac{x}{s}}{s}\right) \cdot 0.5\right) + \frac{-x}{s}} \]

   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 s around inf

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{s} + \left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)\right)}} \]

   5. Applied rewrites 27.9%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), x, 1\right)}} \]
   6. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{1 + \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(\frac{-1}{6}, \frac{x}{s}, \frac{1}{2}\right), \frac{-1}{s}\right), x, 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(\frac{-1}{6}, \frac{x}{s}, \frac{1}{2}\right), \frac{-1}{s}\right), x, 1\right) + 1}} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{1}{\color{blue}{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(\frac{-1}{6}, \frac{x}{s}, \frac{1}{2}\right), \frac{-1}{s}\right), x, 1\right)} + 1} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(\frac{-1}{6}, \frac{x}{s}, \frac{1}{2}\right), \frac{-1}{s}\right), x, 1\right) \cdot 1} + 1} \]

      5. lower-fma.f32 98.8

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), x, 1\right), 1, 1\right)}} \]

   7. Applied rewrites 98.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{\frac{x}{s}}{s}, \frac{-1}{s}\right), x, 1\right), 1, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.949999988079071:\\ \;\;\;\;\frac{1}{\left(0.5 \cdot \left(\frac{\frac{x}{s}}{s} \cdot x\right) + 2\right) - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{\frac{x}{s}}{s}, \frac{-1}{s}\right), x, 1\right), 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, \frac{x}{s}, -1\right)}{s}, x, 1\right), 1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(0.5 \cdot \left(\frac{\frac{x}{s}}{s} \cdot x\right) + 2\right) - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 0.0020000000949949026)
   (/ 1.0 (fma (fma (/ (fma 0.5 (/ x s) -1.0) s) x 1.0) 1.0 1.0))
   (/ 1.0 (- (+ (* 0.5 (* (/ (/ x s) s) x)) 2.0) (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 0.0020000000949949026f) {
		tmp = 1.0f / fmaf(fmaf((fmaf(0.5f, (x / s), -1.0f) / s), x, 1.0f), 1.0f, 1.0f);
	} else {
		tmp = 1.0f / (((0.5f * (((x / s) / s) * x)) + 2.0f) - (x / s));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(1.0) / fma(fma(Float32(fma(Float32(0.5), Float32(x / s), Float32(-1.0)) / s), x, Float32(1.0)), Float32(1.0), Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) * Float32(Float32(Float32(x / s) / s) * x)) + 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)) < 0.00200000009

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{s} + \left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)\right)}} \]

   5. Applied rewrites 27.9%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), x, 1\right)}} \]

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 28.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 98.8

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

   10. Applied rewrites 98.8%

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

   if 0.00200000009 < (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 s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 41.7%

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

   7. Applied rewrites 85.8%

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

   9. Applied rewrites 86.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, \frac{x}{s}, -1\right)}{s}, x, 1\right), 1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(0.5 \cdot \left(\frac{\frac{x}{s}}{s} \cdot x\right) + 2\right) - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.10000000149011612:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 0.10000000149011612)
   0.5
   (/ 1.0 (+ (- 1.0 (/ x s)) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 0.10000000149011612f) {
		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)) <= 0.10000000149011612e0) 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(0.10000000149011612))
		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(0.10000000149011612))
		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)) < 0.100000001

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 28.1%

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

   if 0.100000001 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 62.9

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

   5. Applied rewrites 62.9%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.10000000149011612:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.10000000149011612:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 0.10000000149011612) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 0.10000000149011612f) {
		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)) <= 0.10000000149011612e0) 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(0.10000000149011612))
		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(0.10000000149011612))
		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)) < 0.100000001

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 28.1%

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

   if 0.100000001 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 62.9

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

   5. Applied rewrites 62.9%

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

Alternative 8: 63.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -150000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{-1}{s}, x, 1\right) + 1}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -150000.0)
     (/ 1.0 (+ (fma (/ -1.0 s) x 1.0) 1.0))
     (if (<= t_0 2.0)
       (+ (* 0.25 (/ x s)) 0.5)
       (/ 1.0 (* (* (/ 0.5 (* s s)) x) x))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -150000.0f) {
		tmp = 1.0f / (fmaf((-1.0f / s), x, 1.0f) + 1.0f);
	} else if (t_0 <= 2.0f) {
		tmp = (0.25f * (x / s)) + 0.5f;
	} else {
		tmp = 1.0f / (((0.5f / (s * s)) * x) * x);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-150000.0))
		tmp = Float32(Float32(1.0) / Float32(fma(Float32(Float32(-1.0) / s), x, Float32(1.0)) + Float32(1.0)));
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(Float32(0.25) * Float32(x / s)) + Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) * x));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -1.5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{s} + \left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)\right)}} \]

   5. Applied rewrites 27.9%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), x, 1\right)}} \]

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 28.9%

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

   if -1.5e5 < (/.f32 (neg.f32 x) s) < 2

   1. Initial program 99.5%

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

      1. lift-/.f32 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 N/A

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

      8. lift-+.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f32 N/A

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

      11. metadata-eval 99.4

          \[\leadsto {\left({\left(e^{\frac{-x}{s}} + 1\right)}^{2}\right)}^{\color{blue}{-0.5}} \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{{\left({\left(e^{\frac{-x}{s}} + 1\right)}^{2}\right)}^{-0.5}} \]

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 77.1

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

   7. Applied rewrites 76.5%

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

   9. Applied rewrites 85.9%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 88.1%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -150000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{-1}{s}, x, 1\right) + 1}\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.6% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -2.0)
   (/ 1.0 (+ (fma (/ -1.0 s) x 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 / s), x, 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(Float32(-1.0) / s), x, 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 99.9%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + \left(-1 \cdot \frac{x}{s} + \left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)\right)}} \]

   5. Applied rewrites 28.1%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), x, 1\right)}} \]

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 28.9%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 62.9

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

   5. Applied rewrites 62.9%

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

4. Final simplification 49.6%

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

Alternative 10: 35.2% accurate, 128.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 36.5%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))