Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 2.8s

Alternatives: 5

Speedup: 1.0×

Specification

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

Math

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

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 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)
use fmin_fmax_functions
    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

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

Math

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

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 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)
use fmin_fmax_functions
    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: 82.8% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} t_0 := 1 + \frac{x}{s}\\ \mathbf{if}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;\frac{t\_0}{t\_0 - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{2}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (let* ((t_0 (+ 1.0 (/ x s))))
  (if (<= (/ (- x) s) 50.0) (/ t_0 (- t_0 -1.0)) (/ 0.0 2.0))))

C

float code(float x, float s) {
	float t_0 = 1.0f + (x / s);
	float tmp;
	if ((-x / s) <= 50.0f) {
		tmp = t_0 / (t_0 - -1.0f);
	} else {
		tmp = 0.0f / 2.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = 1.0e0 + (x / s)
    if ((-x / s) <= 50.0e0) then
        tmp = t_0 / (t_0 - (-1.0e0))
    else
        tmp = 0.0e0 / 2.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) + Float32(x / s))
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(50.0))
		tmp = Float32(t_0 / Float32(t_0 - Float32(-1.0)));
	else
		tmp = Float32(Float32(0.0) / Float32(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = single(1.0) + (x / s);
	tmp = single(0.0);
	if ((-x / s) <= single(50.0))
		tmp = t_0 / (t_0 - single(-1.0));
	else
		tmp = single(0.0) / single(2.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.8%

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

   3. Applied rewrites 63.1%

      \[\leadsto \frac{\left(e^{\frac{-x}{s}} + 1\right) + \left(e^{\frac{-x}{s}} + 1\right) \cdot 0}{\left(e^{\frac{-x}{s}} + 1\right) \cdot \left(e^{\frac{-x}{s}} + 1\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.8%

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

   if 50 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 41.4%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{2 + -1 \cdot \frac{x}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.6%

      \[\leadsto \frac{0}{2 + -1 \cdot \frac{x}{s}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{0}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.6%

       \[\leadsto \frac{0}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 82.8% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;\frac{1 + \frac{x}{s}}{1 + \frac{s + x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{2}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (if (<= (/ (- x) s) 50.0)
  (/ (+ 1.0 (/ x s)) (+ 1.0 (/ (+ s x) s)))
  (/ 0.0 2.0)))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 50.0f) {
		tmp = (1.0f + (x / s)) / (1.0f + ((s + x) / s));
	} else {
		tmp = 0.0f / 2.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 50.0e0) then
        tmp = (1.0e0 + (x / s)) / (1.0e0 + ((s + x) / s))
    else
        tmp = 0.0e0 / 2.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(50.0))
		tmp = Float32(Float32(Float32(1.0) + Float32(x / s)) / Float32(Float32(1.0) + Float32(Float32(s + x) / s)));
	else
		tmp = Float32(Float32(0.0) / Float32(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(50.0))
		tmp = (single(1.0) + (x / s)) / (single(1.0) + ((s + x) / s));
	else
		tmp = single(0.0) / single(2.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.8%

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

   3. Applied rewrites 63.1%

      \[\leadsto \frac{\left(e^{\frac{-x}{s}} + 1\right) + \left(e^{\frac{-x}{s}} + 1\right) \cdot 0}{\left(e^{\frac{-x}{s}} + 1\right) \cdot \left(e^{\frac{-x}{s}} + 1\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.8%

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

   10. Applied rewrites 47.7%

       \[\leadsto \frac{1 + \frac{x}{s}}{1 + \frac{s + x}{s}} \]

   if 50 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 41.4%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{2 + -1 \cdot \frac{x}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.6%

      \[\leadsto \frac{0}{2 + -1 \cdot \frac{x}{s}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{0}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.6%

       \[\leadsto \frac{0}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 69.5% accurate, 1.7× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 37.29636764526367:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{2}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (if (<= (/ (- x) s) 37.29636764526367) 0.5 (/ 0.0 2.0)))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 37.29636764526367f) {
		tmp = 0.5f;
	} else {
		tmp = 0.0f / 2.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 37.29636764526367e0) then
        tmp = 0.5e0
    else
        tmp = 0.0e0 / 2.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(37.29636764526367))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(0.0) / Float32(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(37.29636764526367))
		tmp = single(0.5);
	else
		tmp = single(0.0) / single(2.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 37.2963676

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 35.2%

      \[\leadsto 0.5 \]

   if 37.2963676 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 41.4%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{2 + -1 \cdot \frac{x}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.6%

      \[\leadsto \frac{0}{2 + -1 \cdot \frac{x}{s}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{0}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.6%

       \[\leadsto \frac{0}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 35.2% accurate, 23.2× speedup

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

Math

\[0.5 \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  0.5)

C

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

Fortran

real(4) function code(x, s)
use fmin_fmax_functions
    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. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \]
3. Step-by-step derivation

4. Applied rewrites 35.2%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2026086 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))