Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 8.1s

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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{\frac{x}{-s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (exp (- (log1p (exp (/ x (- s)))))))

C

float code(float x, float s) {
	return expf(-log1pf(expf((x / -s))));
}

Julia

function code(x, s)
	return exp(Float32(-log1p(exp(Float32(x / Float32(-s))))))
end

Derivation

1. Initial program 99.9%

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

   1. div-inv 99.9%

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

   2. exp-prod 82.9%

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

   3. neg-mul-1 82.9%

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

   4. exp-prod 82.9%

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

   5. pow-pow 99.9%

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

   6. div-inv 99.9%

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

4. Applied egg-rr 99.9%

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

   1. add-exp-log 99.9%

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

   2. log-rec 99.9%

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

   3. log1p-expm1-u 99.9%

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

   4. log1p-define 100.0%

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

   5. expm1-log1p-u 100.0%

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

   6. pow-exp 100.0%

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

   7. neg-mul-1 100.0%

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

   8. distribute-neg-frac2 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(e^{\frac{x}{-s}}\right)}} \]
7. Add Preprocessing

Alternative 2: 61.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 49999999215337470:\\ \;\;\;\;\frac{\frac{{s}^{2}}{-s}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-1 - \frac{x}{s}\right)}{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 2.0)
     0.5
     (if (<= t_0 49999999215337470.0)
       (/ (/ (pow s 2.0) (- s)) x)
       (/ 1.0 (/ (+ 9.0 (* (+ 1.0 (/ x s)) (- -1.0 (/ x s)))) 4.0))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= 2.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 49999999215337470.0f) {
		tmp = (powf(s, 2.0f) / -s) / x;
	} else {
		tmp = 1.0f / ((9.0f + ((1.0f + (x / s)) * (-1.0f - (x / s)))) / 4.0f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (t_0 <= 2.0e0) then
        tmp = 0.5e0
    else if (t_0 <= 49999999215337470.0e0) then
        tmp = ((s ** 2.0e0) / -s) / x
    else
        tmp = 1.0e0 / ((9.0e0 + ((1.0e0 + (x / s)) * ((-1.0e0) - (x / s)))) / 4.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(2.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(49999999215337470.0))
		tmp = Float32(Float32((s ^ Float32(2.0)) / Float32(-s)) / x);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(9.0) + Float32(Float32(Float32(1.0) + Float32(x / s)) * Float32(Float32(-1.0) - Float32(x / s)))) / Float32(4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(2.0))
		tmp = single(0.5);
	elseif (t_0 <= single(49999999215337470.0))
		tmp = ((s ^ single(2.0)) / -s) / x;
	else
		tmp = single(1.0) / ((single(9.0) + ((single(1.0) + (x / s)) * (single(-1.0) - (x / s)))) / single(4.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 49.5%

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

   if 2 < (/.f32 (neg.f32 x) s) < 4.99999992e16

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 7.5%

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

      1. neg-mul-1 7.5%

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

      2. unsub-neg 7.5%

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

   5. Simplified 7.5%

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

   6. Taylor expanded in x around inf 7.5%

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

      1. associate-*r/ 7.5%

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

      2. neg-mul-1 7.5%

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

   8. Simplified 7.5%

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

      1. neg-sub0 7.5%

         \[\leadsto \frac{\color{blue}{0 - s}}{x} \]

      2. flip-- 57.0%

         \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}}}{x} \]

      3. metadata-eval 57.0%

         \[\leadsto \frac{\frac{\color{blue}{0} - s \cdot s}{0 + s}}{x} \]

      4. pow2 57.0%

         \[\leadsto \frac{\frac{0 - \color{blue}{{s}^{2}}}{0 + s}}{x} \]

      5. add-sqr-sqrt 57.0%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}}}{x} \]

      6. sqrt-unprod 3.4%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s \cdot s}}}}{x} \]

      7. sqr-neg 3.4%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}}}{x} \]

      8. sqrt-unprod -0.0%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}{x} \]

      9. add-sqr-sqrt 57.0%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\left(-s\right)}}}{x} \]

      10. sub-neg 57.0%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{0 - s}}}{x} \]

      11. neg-sub0 57.0%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{-s}}}{x} \]

      12. add-sqr-sqrt -0.0%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}{x} \]

      13. sqrt-unprod 3.4%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}}{x} \]

      14. sqr-neg 3.4%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\sqrt{\color{blue}{s \cdot s}}}}{x} \]

      15. sqrt-unprod 57.0%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}{x} \]

      16. add-sqr-sqrt 57.0%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{s}}}{x} \]

   10. Applied egg-rr 57.0%

       \[\leadsto \frac{\color{blue}{\frac{0 - {s}^{2}}{s}}}{x} \]
   11. Step-by-step derivation

       1. sub0-neg 57.0%

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

   12. Simplified 57.0%

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

   if 4.99999992e16 < (/.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 56.2%

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

      1. neg-mul-1 56.2%

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

      2. unsub-neg 56.2%

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

   5. Simplified 56.2%

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

      1. expm1-log1p-u 56.2%

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

   7. Applied egg-rr 56.2%

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

      1. expm1-undefine 56.2%

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

      2. sub-neg 56.2%

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

      3. log1p-undefine 56.2%

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

      4. rem-exp-log 56.2%

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

      5. associate-+r- 56.2%

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

      6. metadata-eval 56.2%

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

      7. metadata-eval 56.2%

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

   9. Simplified 56.2%

      \[\leadsto \frac{1}{\color{blue}{\left(3 - \frac{x}{s}\right) + -1}} \]
   10. Step-by-step derivation

       1. associate-+l- 56.2%

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

       2. flip-- 47.6%

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

       3. metadata-eval 47.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{9} - \left(\frac{x}{s} - -1\right) \cdot \left(\frac{x}{s} - -1\right)}{3 + \left(\frac{x}{s} - -1\right)}} \]

       4. sub-neg 47.6%

          \[\leadsto \frac{1}{\frac{9 - \color{blue}{\left(\frac{x}{s} + \left(--1\right)\right)} \cdot \left(\frac{x}{s} - -1\right)}{3 + \left(\frac{x}{s} - -1\right)}} \]

       5. metadata-eval 47.6%

          \[\leadsto \frac{1}{\frac{9 - \left(\frac{x}{s} + \color{blue}{1}\right) \cdot \left(\frac{x}{s} - -1\right)}{3 + \left(\frac{x}{s} - -1\right)}} \]

       6. sub-neg 47.6%

          \[\leadsto \frac{1}{\frac{9 - \left(\frac{x}{s} + 1\right) \cdot \color{blue}{\left(\frac{x}{s} + \left(--1\right)\right)}}{3 + \left(\frac{x}{s} - -1\right)}} \]

       7. metadata-eval 47.6%

          \[\leadsto \frac{1}{\frac{9 - \left(\frac{x}{s} + 1\right) \cdot \left(\frac{x}{s} + \color{blue}{1}\right)}{3 + \left(\frac{x}{s} - -1\right)}} \]

       8. sub-neg 47.6%

          \[\leadsto \frac{1}{\frac{9 - \left(\frac{x}{s} + 1\right) \cdot \left(\frac{x}{s} + 1\right)}{3 + \color{blue}{\left(\frac{x}{s} + \left(--1\right)\right)}}} \]

       9. metadata-eval 47.6%

          \[\leadsto \frac{1}{\frac{9 - \left(\frac{x}{s} + 1\right) \cdot \left(\frac{x}{s} + 1\right)}{3 + \left(\frac{x}{s} + \color{blue}{1}\right)}} \]

   11. Applied egg-rr 47.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{9 - \left(\frac{x}{s} + 1\right) \cdot \left(\frac{x}{s} + 1\right)}{3 + \left(\frac{x}{s} + 1\right)}}} \]
   12. Step-by-step derivation

       1. sub-neg 47.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{9 + \left(-\left(\frac{x}{s} + 1\right) \cdot \left(\frac{x}{s} + 1\right)\right)}}{3 + \left(\frac{x}{s} + 1\right)}} \]

       2. distribute-rgt-neg-in 47.6%

          \[\leadsto \frac{1}{\frac{9 + \color{blue}{\left(\frac{x}{s} + 1\right) \cdot \left(-\left(\frac{x}{s} + 1\right)\right)}}{3 + \left(\frac{x}{s} + 1\right)}} \]

       3. +-commutative 47.6%

          \[\leadsto \frac{1}{\frac{9 + \color{blue}{\left(1 + \frac{x}{s}\right)} \cdot \left(-\left(\frac{x}{s} + 1\right)\right)}{3 + \left(\frac{x}{s} + 1\right)}} \]

       4. +-commutative 47.6%

          \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-\color{blue}{\left(1 + \frac{x}{s}\right)}\right)}{3 + \left(\frac{x}{s} + 1\right)}} \]

       5. distribute-neg-in 47.6%

          \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \color{blue}{\left(\left(-1\right) + \left(-\frac{x}{s}\right)\right)}}{3 + \left(\frac{x}{s} + 1\right)}} \]

       6. metadata-eval 47.6%

          \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(\color{blue}{-1} + \left(-\frac{x}{s}\right)\right)}{3 + \left(\frac{x}{s} + 1\right)}} \]

       7. sub-neg 47.6%

          \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \color{blue}{\left(-1 - \frac{x}{s}\right)}}{3 + \left(\frac{x}{s} + 1\right)}} \]

       8. +-commutative 47.6%

          \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-1 - \frac{x}{s}\right)}{\color{blue}{\left(\frac{x}{s} + 1\right) + 3}}} \]

       9. associate-+l+ 47.6%

          \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-1 - \frac{x}{s}\right)}{\color{blue}{\frac{x}{s} + \left(1 + 3\right)}}} \]

       10. metadata-eval 47.6%

           \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-1 - \frac{x}{s}\right)}{\frac{x}{s} + \color{blue}{4}}} \]

   13. Simplified 47.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-1 - \frac{x}{s}\right)}{\frac{x}{s} + 4}}} \]

   14. Taylor expanded in x around 0 97.2%

       \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-1 - \frac{x}{s}\right)}{\color{blue}{4}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{x}{-s} \leq 49999999215337470:\\ \;\;\;\;\frac{\frac{{s}^{2}}{-s}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-1 - \frac{x}{s}\right)}{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 58.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-1 - \frac{x}{s}\right)}{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -2.0)
   0.5
   (/ 1.0 (/ (+ 9.0 (* (+ 1.0 (/ x s)) (- -1.0 (/ x s)))) 4.0))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -2.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((9.0f + ((1.0f + (x / s)) * (-1.0f - (x / s)))) / 4.0f);
	}
	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) <= (-2.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((9.0e0 + ((1.0e0 + (x / s)) * ((-1.0e0) - (x / s)))) / 4.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(-2.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(9.0) + Float32(Float32(Float32(1.0) + Float32(x / s)) * Float32(Float32(-1.0) - Float32(x / s)))) / Float32(4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(-2.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(9.0) + ((single(1.0) + (x / s)) * (single(-1.0) - (x / s)))) / single(4.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 28.1%

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

   if -2 < (/.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 62.4%

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

      1. neg-mul-1 62.4%

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

      2. unsub-neg 62.4%

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

   5. Simplified 62.4%

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

      1. expm1-log1p-u 62.3%

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

   7. Applied egg-rr 62.3%

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

      1. expm1-undefine 62.3%

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

      2. sub-neg 62.3%

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

      3. log1p-undefine 62.4%

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

      4. rem-exp-log 62.4%

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

      5. associate-+r- 62.5%

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

      6. metadata-eval 62.5%

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

      7. metadata-eval 62.5%

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

   9. Simplified 62.5%

      \[\leadsto \frac{1}{\color{blue}{\left(3 - \frac{x}{s}\right) + -1}} \]
   10. Step-by-step derivation

       1. associate-+l- 62.4%

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

       2. flip-- 59.2%

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

       3. metadata-eval 59.2%

          \[\leadsto \frac{1}{\frac{\color{blue}{9} - \left(\frac{x}{s} - -1\right) \cdot \left(\frac{x}{s} - -1\right)}{3 + \left(\frac{x}{s} - -1\right)}} \]

       4. sub-neg 59.2%

          \[\leadsto \frac{1}{\frac{9 - \color{blue}{\left(\frac{x}{s} + \left(--1\right)\right)} \cdot \left(\frac{x}{s} - -1\right)}{3 + \left(\frac{x}{s} - -1\right)}} \]

       5. metadata-eval 59.2%

          \[\leadsto \frac{1}{\frac{9 - \left(\frac{x}{s} + \color{blue}{1}\right) \cdot \left(\frac{x}{s} - -1\right)}{3 + \left(\frac{x}{s} - -1\right)}} \]

       6. sub-neg 59.2%

          \[\leadsto \frac{1}{\frac{9 - \left(\frac{x}{s} + 1\right) \cdot \color{blue}{\left(\frac{x}{s} + \left(--1\right)\right)}}{3 + \left(\frac{x}{s} - -1\right)}} \]

       7. metadata-eval 59.2%

          \[\leadsto \frac{1}{\frac{9 - \left(\frac{x}{s} + 1\right) \cdot \left(\frac{x}{s} + \color{blue}{1}\right)}{3 + \left(\frac{x}{s} - -1\right)}} \]

       8. sub-neg 59.2%

          \[\leadsto \frac{1}{\frac{9 - \left(\frac{x}{s} + 1\right) \cdot \left(\frac{x}{s} + 1\right)}{3 + \color{blue}{\left(\frac{x}{s} + \left(--1\right)\right)}}} \]

       9. metadata-eval 59.2%

          \[\leadsto \frac{1}{\frac{9 - \left(\frac{x}{s} + 1\right) \cdot \left(\frac{x}{s} + 1\right)}{3 + \left(\frac{x}{s} + \color{blue}{1}\right)}} \]

   11. Applied egg-rr 59.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{9 - \left(\frac{x}{s} + 1\right) \cdot \left(\frac{x}{s} + 1\right)}{3 + \left(\frac{x}{s} + 1\right)}}} \]
   12. Step-by-step derivation

       1. sub-neg 59.2%

          \[\leadsto \frac{1}{\frac{\color{blue}{9 + \left(-\left(\frac{x}{s} + 1\right) \cdot \left(\frac{x}{s} + 1\right)\right)}}{3 + \left(\frac{x}{s} + 1\right)}} \]

       2. distribute-rgt-neg-in 59.2%

          \[\leadsto \frac{1}{\frac{9 + \color{blue}{\left(\frac{x}{s} + 1\right) \cdot \left(-\left(\frac{x}{s} + 1\right)\right)}}{3 + \left(\frac{x}{s} + 1\right)}} \]

       3. +-commutative 59.2%

          \[\leadsto \frac{1}{\frac{9 + \color{blue}{\left(1 + \frac{x}{s}\right)} \cdot \left(-\left(\frac{x}{s} + 1\right)\right)}{3 + \left(\frac{x}{s} + 1\right)}} \]

       4. +-commutative 59.2%

          \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-\color{blue}{\left(1 + \frac{x}{s}\right)}\right)}{3 + \left(\frac{x}{s} + 1\right)}} \]

       5. distribute-neg-in 59.2%

          \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \color{blue}{\left(\left(-1\right) + \left(-\frac{x}{s}\right)\right)}}{3 + \left(\frac{x}{s} + 1\right)}} \]

       6. metadata-eval 59.2%

          \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(\color{blue}{-1} + \left(-\frac{x}{s}\right)\right)}{3 + \left(\frac{x}{s} + 1\right)}} \]

       7. sub-neg 59.2%

          \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \color{blue}{\left(-1 - \frac{x}{s}\right)}}{3 + \left(\frac{x}{s} + 1\right)}} \]

       8. +-commutative 59.2%

          \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-1 - \frac{x}{s}\right)}{\color{blue}{\left(\frac{x}{s} + 1\right) + 3}}} \]

       9. associate-+l+ 59.2%

          \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-1 - \frac{x}{s}\right)}{\color{blue}{\frac{x}{s} + \left(1 + 3\right)}}} \]

       10. metadata-eval 59.2%

           \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-1 - \frac{x}{s}\right)}{\frac{x}{s} + \color{blue}{4}}} \]

   13. Simplified 59.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-1 - \frac{x}{s}\right)}{\frac{x}{s} + 4}}} \]

   14. Taylor expanded in x around 0 76.2%

       \[\leadsto \frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-1 - \frac{x}{s}\right)}{\color{blue}{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{9 + \left(1 + \frac{x}{s}\right) \cdot \left(-1 - \frac{x}{s}\right)}{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -2.0) 0.5 (/ 1.0 (+ -1.0 (- 3.0 (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -2.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (-1.0f + (3.0f - (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 / -s) <= (-2.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((-1.0e0) + (3.0e0 - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(-2.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(3.0) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(-2.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(-1.0) + (single(3.0) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 28.1%

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

   if -2 < (/.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 62.4%

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

      1. neg-mul-1 62.4%

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

      2. unsub-neg 62.4%

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

   5. Simplified 62.4%

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

      1. expm1-log1p-u 62.3%

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

   7. Applied egg-rr 62.3%

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

      1. expm1-undefine 62.3%

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

      2. sub-neg 62.3%

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

      3. log1p-undefine 62.4%

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

      4. rem-exp-log 62.4%

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

      5. associate-+r- 62.5%

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

      6. metadata-eval 62.5%

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

      7. metadata-eval 62.5%

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

   9. Simplified 62.5%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.2% accurate, 7.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -2.0f) {
		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 ((x / -s) <= (-2.0e0)) 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 (Float32(x / Float32(-s)) <= Float32(-2.0))
		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 ((x / -s) <= single(-2.0))
		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 (/.f32 (neg.f32 x) s) < -2

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 28.1%

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

   if -2 < (/.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 62.4%

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

      1. neg-mul-1 62.4%

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

      2. unsub-neg 62.4%

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

   5. Simplified 62.4%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.5% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 49.5%

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

   if 2 < (/.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 39.8%

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

      1. neg-mul-1 39.8%

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

      2. unsub-neg 39.8%

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

   5. Simplified 39.8%

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

   6. Taylor expanded in x around inf 36.8%

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

      1. associate-*r/ 36.8%

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

      2. neg-mul-1 36.8%

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

   8. Simplified 36.8%

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

      1. clear-num 39.8%

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

      2. inv-pow 39.8%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 71.2%

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

      5. sqr-neg 71.2%

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

      6. sqrt-unprod 39.8%

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

      7. add-sqr-sqrt 39.8%

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

   10. Applied egg-rr 39.8%

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

       1. unpow-1 39.8%

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

   12. Simplified 39.8%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.1% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0015999999595806003:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0015999999595806003) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0015999999595806003f) {
		tmp = s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.0015999999595806003e0)) then
        tmp = s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0015999999595806003))
		tmp = Float32(s / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0015999999595806003))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -0.00159999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 54.0%

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

      1. neg-mul-1 54.0%

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

      2. unsub-neg 54.0%

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

   5. Simplified 54.0%

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

   6. Taylor expanded in x around inf 49.8%

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

      1. associate-*r/ 49.8%

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

      2. neg-mul-1 49.8%

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

   8. Simplified 49.8%

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

      1. div-inv 49.8%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 61.8%

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

      4. sqr-neg 61.8%

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

      5. sqrt-unprod 49.8%

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

      6. add-sqr-sqrt 49.8%

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

   10. Applied egg-rr 49.8%

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

       1. associate-*r/ 49.8%

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

       2. *-rgt-identity 49.8%

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

   12. Simplified 49.8%

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

   if -0.00159999996 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 43.8%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 35.0% accurate, 108.0× 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 99.9%

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

3. Taylor expanded in x around 0 35.5%

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

Reproduce

herbie shell --seed 2024149 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))