Logistic function

Percentage Accurate: 99.8% → 99.7%

Time: 9.8s

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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left({\left(1 + e^{\frac{-x}{s}}\right)}^{2}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (pow (pow (+ 1.0 (exp (/ (- x) s))) 2.0) -0.5))

C

float code(float x, float s) {
	return powf(powf((1.0f + expf((-x / s))), 2.0f), -0.5f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = ((1.0e0 + exp((-x / s))) ** 2.0e0) ** (-0.5e0)
end function

Julia

function code(x, s)
	return (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) ^ Float32(2.0)) ^ Float32(-0.5)
end

MATLAB

function tmp = code(x, s)
	tmp = ((single(1.0) + exp((-x / s))) ^ single(2.0)) ^ single(-0.5);
end

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
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 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

   \[\leadsto {\left({\left(1 + e^{\frac{-x}{s}}\right)}^{2}\right)}^{-0.5} \]
6. Add Preprocessing

Alternative 2: 56.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-x}{s}}\\ t_1 := 0.25 \cdot \frac{x}{s}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(1, t\_1, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;0.5 + t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x) s))) (t_1 (* 0.25 (/ x s))))
   (if (<= t_0 0.0)
     (fma 1.0 t_1 0.5)
     (if (<= t_0 5.0) (+ 0.5 t_1) (/ 1.0 (* (* (/ 0.5 (* s s)) x) x))))))

C

float code(float x, float s) {
	float t_0 = expf((-x / s));
	float t_1 = 0.25f * (x / s);
	float tmp;
	if (t_0 <= 0.0f) {
		tmp = fmaf(1.0f, t_1, 0.5f);
	} else if (t_0 <= 5.0f) {
		tmp = 0.5f + t_1;
	} else {
		tmp = 1.0f / (((0.5f / (s * s)) * x) * x);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-x) / s))
	t_1 = Float32(Float32(0.25) * Float32(x / s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0))
		tmp = fma(Float32(1.0), t_1, Float32(0.5));
	elseif (t_0 <= Float32(5.0))
		tmp = Float32(Float32(0.5) + t_1);
	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)) < 0.0

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 100.0

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

   4. Applied rewrites 100.0%

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

   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 28.1

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

   7. Applied rewrites 28.1%

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

   9. Applied rewrites 27.9%

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

   if 0.0 < (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 99.6

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

   4. Applied rewrites 99.6%

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

   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 87.3

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

   7. Applied rewrites 87.3%

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

   9. Applied rewrites 95.2%

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

   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 6.7%

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

   6. 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 82.2%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(1, 0.25 \cdot \frac{x}{s}, 0.5\right)\\ \mathbf{elif}\;e^{\frac{-x}{s}} \leq 5:\\ \;\;\;\;0.5 + 0.25 \cdot \frac{x}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{{\left(e^{2}\right)}^{\left(\frac{x}{s} \cdot -0.5\right)} + 1} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ (pow (exp 2.0) (* (/ x s) -0.5)) 1.0)))

C

float code(float x, float s) {
	return 1.0f / (powf(expf(2.0f), ((x / s) * -0.5f)) + 1.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((exp(2.0e0) ** ((x / s) * (-0.5e0))) + 1.0e0)
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32((exp(Float32(2.0)) ^ Float32(Float32(x / s) * Float32(-0.5))) + Float32(1.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((exp(single(2.0)) ^ ((x / s) * single(-0.5))) + single(1.0));
end

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. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lift-E.f32 N/A

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

   5. lift-E.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. e-exp-1 N/A

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

   8. e-exp-1 N/A

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

   9. prod-exp N/A

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

   10. metadata-eval N/A

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

   11. lower-exp.f32 N/A

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

   12. lift-/.f32 N/A

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

   13. lift-neg.f32 N/A

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

   14. distribute-frac-neg N/A

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

   15. lift-/.f32 N/A

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

   16. mul-1-neg N/A

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

   17. *-commutative N/A

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

   18. associate-/l* N/A

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

   19. metadata-eval N/A

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

   20. lower-*.f32 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

   \[\leadsto \frac{1}{{\left(e^{2}\right)}^{\left(\frac{x}{s} \cdot -0.5\right)} + 1} \]
8. Add Preprocessing

Alternative 4: 27.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \frac{x}{s}\\ \mathbf{if}\;e^{\frac{-x}{s}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (* 0.25 (/ x s))))
   (if (<= (exp (/ (- x) s)) 0.0) (fma 1.0 t_0 0.5) (+ 0.5 t_0))))

C

float code(float x, float s) {
	float t_0 = 0.25f * (x / s);
	float tmp;
	if (expf((-x / s)) <= 0.0f) {
		tmp = fmaf(1.0f, t_0, 0.5f);
	} else {
		tmp = 0.5f + t_0;
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 0.0

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 100.0

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

   4. Applied rewrites 100.0%

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

   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 28.1

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

   7. Applied rewrites 28.1%

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

   9. Applied rewrites 27.9%

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

   if 0.0 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.7%

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

      1. lift-/.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 99.8

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

   4. Applied rewrites 99.8%

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

   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 40.2

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

   7. Applied rewrites 40.2%

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

   9. Applied rewrites 42.0%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(1, 0.25 \cdot \frac{x}{s}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.25 \cdot \frac{x}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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

Derivation

1. Initial program 99.8%

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

Alternative 6: 54.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-2.0))
		tmp = fma(Float32(1.0), Float32(Float32(0.25) * Float32(x / s)), Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(Float32(x / s) / s) * Float32(0.5)) * x) + Float32(2.0)) - Float32(x / s)));
	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. 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 100.0

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

   4. Applied rewrites 100.0%

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

   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 28.1

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

   7. Applied rewrites 28.1%

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

   9. Applied rewrites 27.9%

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

   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 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 40.3%

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

   7. Applied rewrites 83.7%

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

4. Final simplification 65.5%

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

Alternative 7: 54.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-2.0))
		tmp = fma(Float32(1.0), Float32(Float32(0.25) * Float32(x / s)), Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(Float32(0.5) / s) * x) / s) * x) + Float32(Float32(2.0) - Float32(x / s))));
	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. 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 100.0

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

   4. Applied rewrites 100.0%

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

   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 28.1

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

   7. Applied rewrites 28.1%

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

   9. Applied rewrites 27.9%

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

   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 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 40.3%

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

   7. Applied rewrites 40.3%

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

   9. Applied rewrites 83.7%

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

4. Final simplification 65.5%

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

Alternative 8: 40.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -2.0f) {
		tmp = fmaf(1.0f, (0.25f * (x / s)), 0.5f);
	} 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 = fma(Float32(1.0), Float32(Float32(0.25) * Float32(x / s)), Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) - Float32(x / s)) + Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 100.0

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

   4. Applied rewrites 100.0%

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

   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 28.1

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

   7. Applied rewrites 28.1%

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

   9. Applied rewrites 27.9%

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

   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 s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 64.0

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

   5. Applied rewrites 64.0%

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

4. Final simplification 52.3%

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

Alternative 9: 40.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 100.0

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

   4. Applied rewrites 100.0%

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

   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 28.1

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

   7. Applied rewrites 28.1%

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

   9. Applied rewrites 27.9%

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

   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 s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 64.0

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

   5. Applied rewrites 64.0%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;\mathsf{fma}\left(1, 0.25 \cdot \frac{x}{s}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
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 36.3%

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

Reproduce

herbie shell --seed 2024243 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))