Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 7.7s

Alternatives: 6

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

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

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(x / Float32(-s)))))
end

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 44.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9999999827968:\\ \;\;\;\;\frac{1}{2 - \frac{1}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9999999827968.0)
   (/ 1.0 (- 2.0 (/ 1.0 (/ s x))))
   (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ 2.0 (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9999999827968.0f) {
		tmp = 1.0f / (2.0f - (1.0f / (s / x)));
	} else {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / (2.0f + (x / s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9999999827968.0e0)) then
        tmp = 1.0e0 / (2.0e0 - (1.0e0 / (s / x)))
    else
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / (2.0e0 + (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9999999827968.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(Float32(1.0) / Float32(s / x))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(2.0) + Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9999999827968.0))
		tmp = single(1.0) / (single(2.0) - (single(1.0) / (s / x)));
	else
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / (single(2.0) + (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9999999830000

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.7%

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

      1. neg-mul-1 78.7%

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

      2. unsub-neg 78.7%

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

   5. Simplified 78.7%

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

      1. clear-num 78.7%

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

      2. associate-/r/ 78.7%

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

   7. Applied egg-rr 78.7%

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

      1. associate-*l/ 78.7%

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

      2. *-un-lft-identity 78.7%

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

      3. clear-num 78.7%

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

   9. Applied egg-rr 78.7%

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

   if -9999999830000 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 36.4%

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

      1. neg-mul-1 36.4%

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

      2. unsub-neg 36.4%

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

   5. Simplified 36.4%

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

      1. clear-num 36.4%

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

      2. associate-/r/ 36.4%

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

   7. Applied egg-rr 36.4%

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

      1. sub-neg 36.4%

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

      2. flip-+ 45.0%

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

      3. metadata-eval 45.0%

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

      4. associate-*l/ 45.0%

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

      5. *-un-lft-identity 45.0%

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

      6. associate-*l/ 45.0%

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

      7. *-un-lft-identity 45.0%

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

      8. distribute-neg-frac2 45.0%

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

      9. distribute-neg-frac2 45.0%

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

      10. associate-*l/ 45.0%

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

      11. *-un-lft-identity 45.0%

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

      12. distribute-neg-frac2 45.0%

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

   9. Applied egg-rr 45.0%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9999999827968:\\ \;\;\;\;\frac{1}{2 - \frac{1}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 40.5% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(3 - \frac{x}{s}\right) + -1} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ (- 3.0 (/ x s)) -1.0)))

C

float code(float x, float s) {
	return 1.0f / ((3.0f - (x / s)) + -1.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((3.0e0 - (x / s)) + (-1.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(Float32(3.0) - Float32(x / s)) + Float32(-1.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((single(3.0) - (x / s)) + single(-1.0));
end

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 41.6%

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

   1. neg-mul-1 41.6%

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

   2. unsub-neg 41.6%

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

5. Simplified 41.6%

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

   1. clear-num 41.6%

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

   2. associate-/r/ 41.6%

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

7. Applied egg-rr 41.6%

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

   1. expm1-log1p-u 39.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{1}{s} \cdot x\right)\right)}} \]

   2. associate-*l/ 39.6%

      \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \color{blue}{\frac{1 \cdot x}{s}}\right)\right)} \]

   3. *-un-lft-identity 39.6%

      \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{\color{blue}{x}}{s}\right)\right)} \]

9. Applied egg-rr 39.6%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{x}{s}\right)\right)}} \]
10. Step-by-step derivation

    1. expm1-undefine 39.6%

       \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(2 - \frac{x}{s}\right)} - 1}} \]

    2. sub-neg 39.6%

       \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(2 - \frac{x}{s}\right)} + \left(-1\right)}} \]

    3. log1p-undefine 39.6%

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

    4. rem-exp-log 41.5%

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

    5. associate-+r- 41.6%

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

    6. metadata-eval 41.6%

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

    7. metadata-eval 41.6%

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

11. Simplified 41.6%

    \[\leadsto \frac{1}{\color{blue}{\left(3 - \frac{x}{s}\right) + -1}} \]
12. Add Preprocessing

Alternative 4: 40.5% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - \frac{x}{s}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (- 2.0 (/ x s))))

C

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

Fortran

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)))
end

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 41.6%

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

   1. neg-mul-1 41.6%

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

   2. unsub-neg 41.6%

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

5. Simplified 41.6%

   \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]
6. Add Preprocessing

Alternative 5: 29.2% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ 0.5 + \frac{0.25}{\frac{s}{x}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (+ 0.5 (/ 0.25 (/ s x))))

C

float code(float x, float s) {
	return 0.5f + (0.25f / (s / x));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 + (0.25e0 / (s / x))
end function

Julia

function code(x, s)
	return Float32(Float32(0.5) + Float32(Float32(0.25) / Float32(s / x)))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) + (single(0.25) / (s / x));
end

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 32.0%

   \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
4. Step-by-step derivation

   1. associate-*r/ 32.0%

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

5. Simplified 32.0%

   \[\leadsto \color{blue}{0.5 + \frac{0.25 \cdot x}{s}} \]
6. Step-by-step derivation

   1. *-commutative 32.0%

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

   2. associate-/l* 32.0%

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

7. Applied egg-rr 32.0%

   \[\leadsto 0.5 + \color{blue}{x \cdot \frac{0.25}{s}} \]
8. Step-by-step derivation

   1. *-commutative 32.0%

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

   2. div-inv 32.0%

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

   3. associate-*l* 32.0%

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

   4. associate-*l/ 32.0%

      \[\leadsto 0.5 + 0.25 \cdot \color{blue}{\frac{1 \cdot x}{s}} \]

   5. *-un-lft-identity 32.0%

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

   6. clear-num 32.0%

      \[\leadsto 0.5 + 0.25 \cdot \color{blue}{\frac{1}{\frac{s}{x}}} \]

   7. div-inv 32.0%

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

9. Applied egg-rr 32.0%

   \[\leadsto 0.5 + \color{blue}{\frac{0.25}{\frac{s}{x}}} \]
10. Add Preprocessing

Alternative 6: 29.2% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ 0.5 + x \cdot \frac{0.25}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (+ 0.5 (* x (/ 0.25 s))))

C

float code(float x, float s) {
	return 0.5f + (x * (0.25f / s));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 + (x * (0.25e0 / s))
end function

Julia

function code(x, s)
	return Float32(Float32(0.5) + Float32(x * Float32(Float32(0.25) / s)))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) + (x * (single(0.25) / s));
end

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 32.0%

   \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
4. Step-by-step derivation

   1. associate-*r/ 32.0%

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

5. Simplified 32.0%

   \[\leadsto \color{blue}{0.5 + \frac{0.25 \cdot x}{s}} \]
6. Step-by-step derivation

   1. *-commutative 32.0%

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

   2. associate-/l* 32.0%

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

7. Applied egg-rr 32.0%

   \[\leadsto 0.5 + \color{blue}{x \cdot \frac{0.25}{s}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024179 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))