Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 2.3s

Alternatives: 7

Speedup: 0.8×

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: 99.8% accurate, 0.8× speedup

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

Math

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

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (pow E (/ (- x) s)))))

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(Float32(-x) / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (single(2.71828182845904523536) ^ (-x / s)));
end

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.7%

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

5. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 0.9× speedup

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

Math

\[\frac{1}{1 + \sqrt{e^{-2 \cdot \frac{x}{s}}}} \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (sqrt (exp (* -2.0 (/ x s)))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + sqrtf(expf((-2.0f * (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 + sqrt(exp(((-2.0e0) * (x / s)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.7%

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

5. Applied rewrites 99.7%

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

Alternative 3: 95.7% accurate, 0.9× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 10:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-\left(1 + \frac{x}{s}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{2}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (if (<= (/ (- x) s) 10.0)
  (/ 1.0 (+ 1.0 (/ -1.0 (- (+ 1.0 (/ x s))))))
  (/ 0.0 2.0)))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 10.0f) {
		tmp = 1.0f / (1.0f + (-1.0f / -(1.0f + (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) <= 10.0e0) then
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / -(1.0e0 + (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(10.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(-Float32(Float32(1.0) + Float32(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(10.0))
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / -(single(1.0) + (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) < 10

   1. Initial program 99.8%

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

   3. Applied rewrites 99.7%

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 61.4%

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

   if 10 < (/.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 40.7%

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

   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: 69.3% accurate, 1.2× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 10:\\ \;\;\;\;\frac{0.5 \cdot s}{s} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{2}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (if (<= (/ (- x) s) 10.0) (* (/ (* 0.5 s) s) 1.0) (/ 0.0 2.0)))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 10.0f) {
		tmp = ((0.5f * s) / s) * 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) :: tmp
    if ((-x / s) <= 10.0e0) then
        tmp = ((0.5e0 * s) / s) * 1.0e0
    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(10.0))
		tmp = Float32(Float32(Float32(Float32(0.5) * s) / s) * Float32(1.0));
	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(10.0))
		tmp = ((single(0.5) * s) / s) * 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) < 10

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 29.4%

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

   6. Applied rewrites 29.3%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 35.0%

      \[\leadsto \frac{1}{\frac{s}{0.5 \cdot s}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.0%

       \[\leadsto \frac{0.5 \cdot s}{s} \cdot 1 \]

   if 10 < (/.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 40.7%

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

   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 5: 69.3% accurate, 1.7× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{2}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (if (<= (/ (- x) s) 10.0) 0.5 (/ 0.0 2.0)))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 10.0f) {
		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) <= 10.0e0) 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(10.0))
		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(10.0))
		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) < 10

   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 35.0%

      \[\leadsto 0.5 \]

   if 10 < (/.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 40.7%

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

   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 6: 35.0% 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 35.0%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2026070 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))