Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 16.3s

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: 95.7% accurate, 0.9× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{2}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (if (<= (/ (- x) s) 20.0)
  (/ 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) <= 20.0f) {
		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) <= 20.0e0) 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(20.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / 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(20.0))
		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) < 20

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 61.1%

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

   if 20 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 40.9%

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

   5. Taylor expanded in undef-var around zero

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

   7. Applied rewrites 40.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{0}{2} \]

   10. Applied rewrites 40.9%

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

Alternative 2: 69.6% accurate, 1.4× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20:\\ \;\;\;\;\frac{0.5 \cdot s}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{2}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (if (<= (/ (- x) s) 20.0) (/ (* 0.5 s) s) (/ 0.0 2.0)))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 20.0f) {
		tmp = (0.5f * s) / 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) <= 20.0e0) then
        tmp = (0.5e0 * s) / 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(20.0))
		tmp = Float32(Float32(Float32(0.5) * s) / 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(20.0))
		tmp = (single(0.5) * s) / 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) < 20

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 29.4%

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

   5. Taylor expanded in s around 0

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

   7. Applied rewrites 29.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{2} \cdot s}{s} \]

   10. Applied rewrites 35.0%

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

   if 20 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 40.9%

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

   5. Taylor expanded in undef-var around zero

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

   7. Applied rewrites 40.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{0}{2} \]

   10. Applied rewrites 40.9%

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

Alternative 3: 69.6% accurate, 1.7× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 12.155284881591797:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{2}\\ \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (if (<= (/ (- x) s) 12.155284881591797) 0.5 (/ 0.0 2.0)))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 12.155284881591797f) {
		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) <= 12.155284881591797e0) 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(12.155284881591797))
		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(12.155284881591797))
		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) < 12.1552849

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{2} \]

   4. Applied rewrites 35.0%

      \[\leadsto 0.5 \]

   if 12.1552849 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 40.9%

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

   5. Taylor expanded in undef-var around zero

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

   7. Applied rewrites 40.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{0}{2} \]

   10. Applied rewrites 40.9%

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

Alternative 4: 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

   \[\leadsto \frac{1}{2} \]

4. Applied rewrites 35.0%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2026089 +o generate:egglog
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))