Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

Alternatives: 9

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 91.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 10000000272564224:\\ \;\;\;\;\frac{1}{\frac{1}{1 - \frac{-0.5 \cdot \left(\frac{x}{s} \cdot x\right) - x}{s}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s} - \frac{2}{x}}{x}\right) \cdot x\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 10000000272564224.0)
   (/ 1.0 (+ (/ 1.0 (- 1.0 (/ (- (* -0.5 (* (/ x s) x)) x) s))) 1.0))
   (/ 1.0 (* (* (- (/ 0.5 (* s s)) (/ (- (/ 1.0 s) (/ 2.0 x)) x)) x) x))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 10000000272564224.0f) {
		tmp = 1.0f / ((1.0f / (1.0f - (((-0.5f * ((x / s) * x)) - x) / s))) + 1.0f);
	} else {
		tmp = 1.0f / ((((0.5f / (s * s)) - (((1.0f / s) - (2.0f / x)) / x)) * x) * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (exp((-x / s)) <= 10000000272564224.0e0) then
        tmp = 1.0e0 / ((1.0e0 / (1.0e0 - ((((-0.5e0) * ((x / s) * x)) - x) / s))) + 1.0e0)
    else
        tmp = 1.0e0 / ((((0.5e0 / (s * s)) - (((1.0e0 / s) - (2.0e0 / x)) / x)) * x) * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(10000000272564224.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(x / s) * x)) - x) / s))) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) - Float32(Float32(Float32(Float32(1.0) / s) - Float32(Float32(2.0) / x)) / x)) * x) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp((-x / s)) <= single(10000000272564224.0))
		tmp = single(1.0) / ((single(1.0) / (single(1.0) - (((single(-0.5) * ((x / s) * x)) - x) / s))) + single(1.0));
	else
		tmp = single(1.0) / ((((single(0.5) / (s * s)) - (((single(1.0) / s) - (single(2.0) / x)) / x)) * x) * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 1.00000003e16

   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. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 96.0

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

   7. Applied rewrites 96.0%

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

   if 1.00000003e16 < (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 x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \frac{1}{{x}^{2} \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{s} - 2 \cdot \frac{1}{x}}{x} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 84.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 10000000272564224:\\ \;\;\;\;\frac{1}{\frac{1}{1 - \frac{-0.5 \cdot \left(\frac{x}{s} \cdot x\right) - x}{s}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s} - \frac{2}{x}}{x}\right) \cdot x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 1.9999999949504854e-6)
   0.5
   (/ 1.0 (+ (- 1.0 (/ x s)) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 1.9999999949504854e-6f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((1.0f - (x / s)) + 1.0f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (exp((-x / s)) <= 1.9999999949504854e-6) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((1.0e0 - (x / s)) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(1.9999999949504854e-6))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) - Float32(x / s)) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp((-x / s)) <= single(1.9999999949504854e-6))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(1.0) - (x / s)) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 1.99999999e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

      \[\leadsto \color{blue}{0.5} \]

   if 1.99999999e-6 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 59.3

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

   5. Applied rewrites 59.3%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 1.9999999949504854e-6) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 1.9999999949504854e-6f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (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 (exp((-x / s)) <= 1.9999999949504854e-6) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(1.9999999949504854e-6))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp((-x / s)) <= single(1.9999999949504854e-6))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 1.99999999e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

      \[\leadsto \color{blue}{0.5} \]

   if 1.99999999e-6 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 59.2

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

   5. Applied rewrites 59.2%

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

Alternative 5: 90.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 600:\\ \;\;\;\;\frac{1}{\frac{1}{1 - \frac{-0.5 \cdot \left(\frac{x}{s} \cdot x\right) - x}{s}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s}}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 600.0)
   (/ 1.0 (+ (/ 1.0 (- 1.0 (/ (- (* -0.5 (* (/ x s) x)) x) s))) 1.0))
   (/ 1.0 (* (* x x) (- (/ 0.5 (* s s)) (/ (/ 1.0 s) x))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 600.0f) {
		tmp = 1.0f / ((1.0f / (1.0f - (((-0.5f * ((x / s) * x)) - x) / s))) + 1.0f);
	} else {
		tmp = 1.0f / ((x * x) * ((0.5f / (s * s)) - ((1.0f / s) / x)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 600.0e0) then
        tmp = 1.0e0 / ((1.0e0 / (1.0e0 - ((((-0.5e0) * ((x / s) * x)) - x) / s))) + 1.0e0)
    else
        tmp = 1.0e0 / ((x * x) * ((0.5e0 / (s * s)) - ((1.0e0 / s) / x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(600.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(x / s) * x)) - x) / s))) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * x) * Float32(Float32(Float32(0.5) / Float32(s * s)) - Float32(Float32(Float32(1.0) / s) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(600.0))
		tmp = single(1.0) / ((single(1.0) / (single(1.0) - (((single(-0.5) * ((x / s) * x)) - x) / s))) + single(1.0));
	else
		tmp = single(1.0) / ((x * x) * ((single(0.5) / (s * s)) - ((single(1.0) / s) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 600

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 93.9

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

   7. Applied rewrites 93.9%

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

   if 600 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{{x}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{s}^{2}} - \frac{1}{s \cdot x}\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.5%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 600:\\ \;\;\;\;\frac{1}{\frac{1}{1 - \frac{-0.5 \cdot \left(\frac{x}{s} \cdot x\right) - x}{s}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s}}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 600:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s}}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 600.0)
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
   (/ 1.0 (* (* x x) (- (/ 0.5 (* s s)) (/ (/ 1.0 s) x))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 600.0f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else {
		tmp = 1.0f / ((x * x) * ((0.5f / (s * s)) - ((1.0f / s) / x)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 600.0e0) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else
        tmp = 1.0e0 / ((x * x) * ((0.5e0 / (s * s)) - ((1.0e0 / s) / x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(600.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0))) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * x) * Float32(Float32(Float32(0.5) / Float32(s * s)) - Float32(Float32(Float32(1.0) / s) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(600.0))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	else
		tmp = single(1.0) / ((x * x) * ((single(0.5) / (s * s)) - ((single(1.0) / s) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 600

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 91.7

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

   7. Applied rewrites 91.7%

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

   if 600 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{{x}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{s}^{2}} - \frac{1}{s \cdot x}\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.5%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 600:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s}}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 200000:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(0.5 \cdot x - s\right) \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 200000.0)
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
   (/ 1.0 (/ (* (- (* 0.5 x) s) x) (* s s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 200000.0f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else {
		tmp = 1.0f / ((((0.5f * x) - s) * x) / (s * 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 / s) <= 200000.0e0) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else
        tmp = 1.0e0 / ((((0.5e0 * x) - s) * x) / (s * s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(200000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0))) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(0.5) * x) - s) * x) / Float32(s * s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(200000.0))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	else
		tmp = single(1.0) / ((((single(0.5) * x) - s) * x) / (s * s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 2e5

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 89.7

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

   7. Applied rewrites 89.7%

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

   if 2e5 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.3%

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

   7. Applied rewrites 6.3%

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

   8. Taylor expanded in s around 0

      \[\leadsto \frac{1}{\frac{-1 \cdot \left(s \cdot x\right) + \frac{1}{2} \cdot {x}^{2}}{\color{blue}{{s}^{2}}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.2%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 200000:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(0.5 \cdot x - s\right) \cdot x}{s \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -0.03999999910593033:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -0.03999999910593033)
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
   (/ 1.0 (+ (- 1.0 (/ x s)) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -0.03999999910593033f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else {
		tmp = 1.0f / ((1.0f - (x / s)) + 1.0f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-0.03999999910593033e0)) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else
        tmp = 1.0e0 / ((1.0e0 - (x / s)) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-0.03999999910593033))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0))) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) - Float32(x / s)) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-0.03999999910593033))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	else
		tmp = single(1.0) / ((single(1.0) - (x / s)) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -0.0399999991

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 93.5

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

   7. Applied rewrites 93.5%

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

   if -0.0399999991 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 59.6

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

   5. Applied rewrites 59.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -0.03999999910593033:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.3% 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 x around 0

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

5. Applied rewrites 36.8%

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

Reproduce

herbie shell --seed 2024296 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))