Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 8.6s

Alternatives: 15

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, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{hypot}\left(1, {\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)}\right)\right)}^{-2} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (pow (hypot 1.0 (pow (exp -0.5) (/ x s))) -2.0))

C

float code(float x, float s) {
	return powf(hypotf(1.0f, powf(expf(-0.5f), (x / s))), -2.0f);
}

Julia

function code(x, s)
	return hypot(Float32(1.0), (exp(Float32(-0.5)) ^ Float32(x / s))) ^ Float32(-2.0)
end

MATLAB

function tmp = code(x, s)
	tmp = hypot(single(1.0), (exp(single(-0.5)) ^ (x / s))) ^ single(-2.0);
end

Derivation

1. Initial program 99.6%

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

   1. distribute-frac-neg 99.6%

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

   2. exp-neg 99.6%

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

4. Applied egg-rr 99.6%

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

6. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, {\left(e^{\frac{x}{s}}\right)}^{-0.5}\right)\right)}^{-1} \cdot {\left(\mathsf{hypot}\left(1, {\left(e^{\frac{x}{s}}\right)}^{-0.5}\right)\right)}^{-1}} \]
7. Step-by-step derivation

   1. pow-sqr 99.7%

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

   2. metadata-eval 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in x around inf 99.5%

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

    1. metadata-eval 99.5%

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

    2. unpow-1 99.5%

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

    3. metadata-eval 99.5%

       \[\leadsto {\left(\sqrt{1 \cdot 1 + {\left(e^{\frac{x}{s}}\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right)}^{-2} \]

    4. pow-sqr 99.5%

       \[\leadsto {\left(\sqrt{1 \cdot 1 + \color{blue}{{\left(e^{\frac{x}{s}}\right)}^{-0.5} \cdot {\left(e^{\frac{x}{s}}\right)}^{-0.5}}}\right)}^{-2} \]

    5. hypot-undefine 99.7%

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

    6. exp-prod 99.7%

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

    7. *-commutative 99.7%

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

    8. exp-prod 99.8%

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

11. Simplified 99.8%

    \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(1, {\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)}\right)\right)}}^{-2} \]
12. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{hypot}\left(1, e^{-0.5 \cdot \frac{x}{s}}\right)\right)}^{-2} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (pow (hypot 1.0 (exp (* -0.5 (/ x s)))) -2.0))

C

float code(float x, float s) {
	return powf(hypotf(1.0f, expf((-0.5f * (x / s)))), -2.0f);
}

Julia

function code(x, s)
	return hypot(Float32(1.0), exp(Float32(Float32(-0.5) * Float32(x / s)))) ^ Float32(-2.0)
end

MATLAB

function tmp = code(x, s)
	tmp = hypot(single(1.0), exp((single(-0.5) * (x / s)))) ^ single(-2.0);
end

Derivation

1. Initial program 99.6%

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

   1. distribute-frac-neg 99.6%

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

   2. exp-neg 99.6%

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

4. Applied egg-rr 99.6%

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

6. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, {\left(e^{\frac{x}{s}}\right)}^{-0.5}\right)\right)}^{-1} \cdot {\left(\mathsf{hypot}\left(1, {\left(e^{\frac{x}{s}}\right)}^{-0.5}\right)\right)}^{-1}} \]
7. Step-by-step derivation

   1. pow-sqr 99.7%

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

   2. metadata-eval 99.7%

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

8. Simplified 99.7%

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

   1. add-exp-log 99.7%

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

   2. log-pow 99.7%

      \[\leadsto {\left(\mathsf{hypot}\left(1, e^{\color{blue}{-0.5 \cdot \log \left(e^{\frac{x}{s}}\right)}}\right)\right)}^{-2} \]

   3. add-log-exp 99.7%

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

10. Applied egg-rr 99.7%

    \[\leadsto {\left(\mathsf{hypot}\left(1, \color{blue}{e^{-0.5 \cdot \frac{x}{s}}}\right)\right)}^{-2} \]
11. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\left(e^{\frac{x}{-s}} + 1\right)}^{-2}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (sqrt (pow (+ (exp (/ x (- s))) 1.0) -2.0)))

C

float code(float x, float s) {
	return sqrtf(powf((expf((x / -s)) + 1.0f), -2.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = sqrt(((exp((x / -s)) + 1.0e0) ** (-2.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. distribute-frac-neg 99.6%

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

   2. exp-neg 99.6%

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

4. Applied egg-rr 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

   \[\leadsto \sqrt{{\left(e^{\frac{x}{-s}} + 1\right)}^{-2}} \]
8. Add Preprocessing

Alternative 4: 88.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 0.5:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{elif}\;t\_0 \leq 499999992153374700:\\ \;\;\;\;\frac{-\frac{{s}^{2}}{s}}{x}\\ \mathbf{elif}\;t\_0 \leq 1.9999999867631625 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 0.5)
     (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
     (if (<= t_0 499999992153374700.0)
       (/ (- (/ (pow s 2.0) s)) x)
       (if (<= t_0 1.9999999867631625e+37)
         (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ x s) 2.0)))
         (/ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= 0.5f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else if (t_0 <= 499999992153374700.0f) {
		tmp = -(powf(s, 2.0f) / s) / x;
	} else if (t_0 <= 1.9999999867631625e+37f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} 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) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (t_0 <= 0.5e0) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else if (t_0 <= 499999992153374700.0e0) then
        tmp = -((s ** 2.0e0) / s) / x
    else if (t_0 <= 1.9999999867631625e+37) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / (x / s)
    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(0.5))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0))) + Float32(1.0)));
	elseif (t_0 <= Float32(499999992153374700.0))
		tmp = Float32(Float32(-Float32((s ^ Float32(2.0)) / s)) / x);
	elseif (t_0 <= Float32(1.9999999867631625e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(0.5))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	elseif (t_0 <= single(499999992153374700.0))
		tmp = -((s ^ single(2.0)) / s) / x;
	elseif (t_0 <= single(1.9999999867631625e+37))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

2. if (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 94.4%

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

      1. +-commutative 94.4%

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

   7. Simplified 94.4%

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

   if 0.5 < (/.f32 (neg.f32 x) s) < 4.99999992e17

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0 9.3%

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

      1. neg-mul-1 9.3%

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

      2. unsub-neg 9.3%

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

   5. Simplified 9.3%

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

   6. Taylor expanded in x around inf 9.1%

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

      1. associate-*r/ 9.1%

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

      2. neg-mul-1 9.1%

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

   8. Simplified 9.1%

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

      1. neg-sub0 9.1%

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

      2. flip-- 39.8%

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

      3. metadata-eval 39.8%

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

      4. pow2 39.8%

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

      5. add-sqr-sqrt 39.8%

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

      6. sqrt-unprod 5.9%

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

      7. sqr-neg 5.9%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt 38.4%

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

      10. sub-neg 38.4%

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

      11. neg-sub0 38.4%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 5.9%

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

      14. sqr-neg 5.9%

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

      15. sqrt-unprod 39.8%

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

      16. add-sqr-sqrt 39.8%

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

   10. Applied egg-rr 39.8%

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

       1. sub0-neg 39.8%

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

   12. Simplified 39.8%

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

   if 4.99999992e17 < (/.f32 (neg.f32 x) s) < 1.99999999e37

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 13.6%

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

      1. neg-mul-1 13.6%

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

      2. unsub-neg 13.6%

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

   5. Simplified 13.6%

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

      1. *-un-lft-identity 13.6%

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

      2. cancel-sign-sub-inv 13.6%

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

      3. metadata-eval 13.6%

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

      4. add-log-exp 100.0%

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

      5. pow-exp 100.0%

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

      6. flip-+ -0.0%

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

      7. metadata-eval -0.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{4} - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right) \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      8. pow-exp -0.0%

         \[\leadsto \frac{1}{\frac{4 - \log \color{blue}{\left(e^{-1 \cdot \frac{x}{s}}\right)} \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      9. add-log-exp -0.0%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\left(-1 \cdot \frac{x}{s}\right)} \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      10. neg-mul-1 -0.0%

          \[\leadsto \frac{1}{\frac{4 - \color{blue}{\left(-\frac{x}{s}\right)} \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      11. pow-exp -0.0%

          \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \log \color{blue}{\left(e^{-1 \cdot \frac{x}{s}}\right)}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      12. add-log-exp 0.1%

          \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \color{blue}{\left(-1 \cdot \frac{x}{s}\right)}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      13. neg-mul-1 0.1%

          \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \color{blue}{\left(-\frac{x}{s}\right)}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      14. distribute-neg-frac2 0.1%

          \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{-s}} \cdot \left(-\frac{x}{s}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      15. distribute-neg-frac2 0.1%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \color{blue}{\frac{x}{-s}}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      16. pow-exp 0.1%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \log \color{blue}{\left(e^{-1 \cdot \frac{x}{s}}\right)}}} \]

   7. Applied egg-rr 97.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \frac{x}{-s}}}} \]

   if 1.99999999e37 < (/.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 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 87.6%

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

      1. associate-*r/ 87.6%

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

      2. neg-mul-1 87.6%

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

   8. Simplified 87.6%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.5:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{elif}\;\frac{x}{-s} \leq 499999992153374700:\\ \;\;\;\;\frac{-\frac{{s}^{2}}{s}}{x}\\ \mathbf{elif}\;\frac{x}{-s} \leq 1.9999999867631625 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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(x / Float32(-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 99.6%

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

3. Final simplification 99.6%

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

Alternative 6: 84.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -0.0005499999970197678:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{elif}\;t\_0 \leq 1.9999999867631625 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -0.0005499999970197678)
     (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
     (if (<= t_0 1.9999999867631625e+37)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ x s) 2.0)))
       (/ 1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= -0.0005499999970197678f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else if (t_0 <= 1.9999999867631625e+37f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} 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) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (t_0 <= (-0.0005499999970197678e0)) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else if (t_0 <= 1.9999999867631625e+37) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / (x / s)
    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(-0.0005499999970197678))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0))) + Float32(1.0)));
	elseif (t_0 <= Float32(1.9999999867631625e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(-0.0005499999970197678))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	elseif (t_0 <= single(1.9999999867631625e+37))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -5.5e-4

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 94.3%

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

      1. +-commutative 94.3%

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

   7. Simplified 94.3%

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

   if -5.5e-4 < (/.f32 (neg.f32 x) s) < 1.99999999e37

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 41.6%

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

      1. neg-mul-1 41.6%

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

      2. unsub-neg 41.6%

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

   5. Simplified 41.6%

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

      1. *-un-lft-identity 41.6%

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

      2. cancel-sign-sub-inv 41.6%

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

      3. metadata-eval 41.6%

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

      4. add-log-exp 95.0%

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

      5. pow-exp 95.0%

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

      6. flip-+ 35.0%

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

      7. metadata-eval 35.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{4} - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right) \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      8. pow-exp 35.0%

         \[\leadsto \frac{1}{\frac{4 - \log \color{blue}{\left(e^{-1 \cdot \frac{x}{s}}\right)} \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      9. add-log-exp 35.0%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\left(-1 \cdot \frac{x}{s}\right)} \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      10. neg-mul-1 35.0%

          \[\leadsto \frac{1}{\frac{4 - \color{blue}{\left(-\frac{x}{s}\right)} \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      11. pow-exp 35.0%

          \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \log \color{blue}{\left(e^{-1 \cdot \frac{x}{s}}\right)}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      12. add-log-exp 36.0%

          \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \color{blue}{\left(-1 \cdot \frac{x}{s}\right)}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      13. neg-mul-1 36.0%

          \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \color{blue}{\left(-\frac{x}{s}\right)}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      14. distribute-neg-frac2 36.0%

          \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{-s}} \cdot \left(-\frac{x}{s}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      15. distribute-neg-frac2 36.0%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \color{blue}{\frac{x}{-s}}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      16. pow-exp 36.0%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \log \color{blue}{\left(e^{-1 \cdot \frac{x}{s}}\right)}}} \]

   7. Applied egg-rr 68.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \frac{x}{-s}}}} \]

   if 1.99999999e37 < (/.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 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 87.6%

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

      1. associate-*r/ 87.6%

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

      2. neg-mul-1 87.6%

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

   8. Simplified 87.6%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.0005499999970197678:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{elif}\;\frac{x}{-s} \leq 1.9999999867631625 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.0005499999970197678:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x - s \cdot 2}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -0.0005499999970197678)
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
   (/ -1.0 (/ (- x (* s 2.0)) s))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -0.0005499999970197678f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else {
		tmp = -1.0f / ((x - (s * 2.0f)) / 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) <= (-0.0005499999970197678e0)) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else
        tmp = (-1.0e0) / ((x - (s * 2.0e0)) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(-0.0005499999970197678))
		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 - Float32(s * Float32(2.0))) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(-0.0005499999970197678))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	else
		tmp = single(-1.0) / ((x - (s * single(2.0))) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -5.5e-4

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 94.3%

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

      1. +-commutative 94.3%

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

   7. Simplified 94.3%

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

   if -5.5e-4 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 56.3%

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

      1. neg-mul-1 56.3%

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

      2. unsub-neg 56.3%

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

   5. Simplified 56.3%

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

   6. Taylor expanded in s around 0 56.3%

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

      1. *-commutative 56.3%

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

   8. Simplified 56.3%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.0005499999970197678:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x - s \cdot 2}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.1% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 28.2%

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

   if -1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 56.6%

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

      1. neg-mul-1 56.6%

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

      2. unsub-neg 56.6%

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

   5. Simplified 56.6%

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

   6. Taylor expanded in s around 0 56.6%

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

      1. *-commutative 56.6%

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

   8. Simplified 56.6%

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

4. Final simplification 45.9%

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

Alternative 9: 49.1% accurate, 6.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -1.0f) {
		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) <= (-1.0e0)) 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(-1.0))
		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(-1.0))
		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) < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 28.2%

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

   if -1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 56.6%

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

      1. neg-mul-1 56.6%

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

      2. unsub-neg 56.6%

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

   5. Simplified 56.6%

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

      1. expm1-log1p-u 56.5%

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

   7. Applied egg-rr 56.5%

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

      1. expm1-undefine 56.5%

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

      2. sub-neg 56.5%

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

      3. log1p-undefine 56.4%

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

      4. rem-exp-log 56.6%

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

      5. associate-+r- 56.6%

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

      6. metadata-eval 56.6%

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

      7. metadata-eval 56.6%

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

   9. Simplified 56.6%

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

4. Final simplification 45.9%

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

Alternative 10: 49.1% accurate, 7.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -1.0f) {
		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) <= (-1.0e0)) 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(-1.0))
		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(-1.0))
		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) < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 28.2%

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

   if -1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 56.6%

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

      1. neg-mul-1 56.6%

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

      2. unsub-neg 56.6%

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

   5. Simplified 56.6%

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

4. Final simplification 45.9%

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

Alternative 11: 47.6% 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 46.4%

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

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

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 42.1%

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

      1. neg-mul-1 42.1%

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

      2. unsub-neg 42.1%

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

   5. Simplified 42.1%

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

   6. Taylor expanded in x around inf 42.1%

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

      1. mul-1-neg 42.1%

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

      2. distribute-frac-neg2 42.1%

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

   8. Simplified 42.1%

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

4. Final simplification 44.5%

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

Alternative 12: 47.1% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999717180685e-10) (/ 1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999717180685e-10f) {
		tmp = 1.0f / (x / s);
	} 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 <= (-9.999999717180685e-10)) then
        tmp = 1.0e0 / (x / s)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999717180685e-10))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999717180685e-10))
		tmp = single(1.0) / (x / s);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999972e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 48.7%

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

      1. neg-mul-1 48.7%

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

      2. unsub-neg 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in x around inf 43.5%

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

      1. associate-*r/ 43.5%

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

      2. neg-mul-1 43.5%

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

   8. Simplified 43.5%

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

      1. clear-num 48.6%

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

      2. inv-pow 48.6%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 63.9%

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

      5. sqr-neg 63.9%

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

      6. sqrt-unprod 48.2%

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

      7. add-sqr-sqrt 48.2%

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

   10. Applied egg-rr 48.2%

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

       1. unpow-1 48.2%

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

   12. Simplified 48.2%

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

   if -9.99999972e-10 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 42.0%

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

Alternative 13: 45.4% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.9999998245167e-14) (/ s (- x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.9999998245167e-14f) {
		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 <= (-9.9999998245167e-14)) 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(-9.9999998245167e-14))
		tmp = Float32(s / Float32(-x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.9999998245167e-14))
		tmp = s / -x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999982e-14

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 46.0%

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

      1. neg-mul-1 46.0%

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

      2. unsub-neg 46.0%

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

   5. Simplified 46.0%

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

   6. Taylor expanded in x around inf 41.0%

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

      1. associate-*r/ 41.0%

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

      2. neg-mul-1 41.0%

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

   8. Simplified 41.0%

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

   if -9.99999982e-14 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 43.6%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.7% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999717180685e-10) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999717180685e-10f) {
		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 <= (-9.999999717180685e-10)) 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(-9.999999717180685e-10))
		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(-9.999999717180685e-10))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999972e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 48.7%

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

      1. neg-mul-1 48.7%

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

      2. unsub-neg 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in x around inf 43.5%

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

      1. associate-*r/ 43.5%

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

      2. neg-mul-1 43.5%

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

   8. Simplified 43.5%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 61.3%

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

      3. sqr-neg 61.3%

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

      4. sqrt-unprod 43.1%

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

      5. add-sqr-sqrt 43.1%

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

      6. *-un-lft-identity 43.1%

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

   10. Applied egg-rr 43.1%

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

       1. *-lft-identity 43.1%

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

   12. Simplified 43.1%

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

   if -9.99999972e-10 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 42.0%

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

Alternative 15: 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 99.6%

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

3. Taylor expanded in x around 0 29.0%

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

Reproduce

herbie shell --seed 2024177 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))