Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 13.7s

Alternatives: 16

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, 0.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(-1.0f), (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((-1.0e0)) ** (x / s)))
end function

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(single(-1.0)) ^ (x / 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 85.1%

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

   3. neg-mul-1 85.1%

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

   4. exp-prod 85.1%

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

   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. Final simplification 99.9%

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

Alternative 2: 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 3: 85.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -0.006000000052154064:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t\_0 \leq 4.999999840142846 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s} \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -0.006000000052154064)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 4.999999840142846e+37)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ x s) 2.0)))
       (/ 1.0 (* (/ x s) 3.0))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= -0.006000000052154064f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 4.999999840142846e+37f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / ((x / s) * 3.0f);
	}
	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 <= (-0.006000000052154064e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 4.999999840142846e+37) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / ((x / s) * 3.0e0)
    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(-0.006000000052154064))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(4.999999840142846e+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(Float32(x / s) * Float32(3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(-0.006000000052154064))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(4.999999840142846e+37))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / ((x / s) * single(3.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -0.00600000005

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      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 89.7%

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

   if -0.00600000005 < (/.f32 (neg.f32 x) s) < 4.99999984e37

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 59.5%

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

      1. mul-1-neg 59.5%

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

      2. unsub-neg 59.5%

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

   5. Simplified 59.5%

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

      1. *-un-lft-identity 59.5%

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

      2. cancel-sign-sub-inv 59.5%

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

      3. metadata-eval 59.5%

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

      4. add-log-exp 95.3%

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

      5. pow-exp 95.3%

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

      6. flip-+ 55.1%

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

      7. metadata-eval 55.1%

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

      8. pow-exp 55.1%

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

      9. add-log-exp 55.1%

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

      10. neg-mul-1 55.1%

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

      11. pow-exp 55.1%

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

      12. add-log-exp 55.7%

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

      13. neg-mul-1 55.7%

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

      14. distribute-neg-frac2 55.7%

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

      15. distribute-neg-frac2 55.7%

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

      16. pow-exp 55.7%

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

   7. Applied egg-rr 77.3%

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

   if 4.99999984e37 < (/.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 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unsub-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in x around inf 98.4%

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

      1. associate-*r/ 98.4%

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

      2. metadata-eval 98.4%

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

   8. Simplified 98.4%

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

      1. add-sqr-sqrt -0.0%

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

      2. *-un-lft-identity -0.0%

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

      3. prod-diff -0.0%

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

      4. associate-/r/ -0.0%

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

      5. clear-num -0.0%

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

      6. fma-define -0.0%

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

      7. add-sqr-sqrt 98.4%

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

      8. distribute-frac-neg2 98.4%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 100.0%

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

      11. sqr-neg 100.0%

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

      12. sqrt-unprod 98.4%

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

      13. add-sqr-sqrt 98.4%

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

   10. Applied egg-rr 100.0%

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

       1. +-commutative 100.0%

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

       2. fma-undefine 100.0%

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

       3. *-rgt-identity 100.0%

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

       4. associate-+l+ 100.0%

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

       5. +-commutative 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around inf 100.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.006000000052154064:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 4.999999840142846 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s} \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot 2 - x\right) \cdot \frac{1}{x \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 0.0020000000949949026)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (* x (* (- (* s 2.0) x) (/ 1.0 (* x s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 0.0020000000949949026f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (x * (((s * 2.0f) - x) * (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) <= 0.0020000000949949026e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (x * (((s * 2.0e0) - x) * (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(0.0020000000949949026))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(Float32(s * Float32(2.0)) - x) * 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(0.0020000000949949026))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (x * (((s * single(2.0)) - x) * (single(1.0) / (x * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.00200000009

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      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 92.7%

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

   if 0.00200000009 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 54.5%

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

      1. mul-1-neg 54.5%

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

      2. unsub-neg 54.5%

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

   5. Simplified 54.5%

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

   6. Taylor expanded in x around inf 54.5%

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

      1. associate-*r/ 54.5%

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

      2. metadata-eval 54.5%

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

   8. Simplified 54.5%

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

      1. frac-sub 59.5%

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

      2. div-inv 65.3%

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

      3. *-commutative 65.3%

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

      4. *-rgt-identity 65.3%

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

   10. Applied egg-rr 65.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot 2 - x\right) \cdot \frac{1}{x \cdot s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot 2 - x\right) \cdot \frac{x}{x \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 0.0020000000949949026)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (* (- (* s 2.0) x) (/ x (* x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 0.0020000000949949026f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (((s * 2.0f) - x) * (x / (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) <= 0.0020000000949949026e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (((s * 2.0e0) - x) * (x / (x * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(s * Float32(2.0)) - x) * Float32(x / Float32(x * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(0.0020000000949949026))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (((s * single(2.0)) - x) * (x / (x * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.00200000009

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      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 92.7%

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

   if 0.00200000009 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 54.5%

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

      1. mul-1-neg 54.5%

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

      2. unsub-neg 54.5%

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

   5. Simplified 54.5%

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

   6. Taylor expanded in x around inf 54.5%

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

      1. associate-*r/ 54.5%

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

      2. metadata-eval 54.5%

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

   8. Simplified 54.5%

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

      1. frac-sub 59.5%

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

      2. associate-*r/ 60.0%

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

      3. *-commutative 60.0%

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

      4. *-rgt-identity 60.0%

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

   10. Applied egg-rr 60.0%

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

       1. *-commutative 60.0%

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

       2. associate-/l* 59.5%

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

       3. *-commutative 59.5%

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

   12. Simplified 59.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot 2 - x\right) \cdot \frac{x}{x \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 10:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot 3}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 10.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (/ (* x 3.0) s))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 10.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / ((x * 3.0f) / 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) <= 10.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / ((x * 3.0e0) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(10.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(3.0)) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(10.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / ((x * single(3.0)) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 10

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. 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 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 90.3%

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

   if 10 < (/.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 55.7%

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

      1. mul-1-neg 55.7%

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

      2. unsub-neg 55.7%

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

   5. Simplified 55.7%

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

   6. Taylor expanded in x around inf 55.7%

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

      1. associate-*r/ 55.7%

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

      2. metadata-eval 55.7%

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

   8. Simplified 55.7%

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

      1. add-sqr-sqrt -0.0%

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

      2. *-un-lft-identity -0.0%

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

      3. prod-diff -0.0%

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

      4. associate-/r/ -0.0%

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

      5. clear-num -0.0%

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

      6. fma-define -0.0%

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

      7. add-sqr-sqrt 55.7%

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

      8. distribute-frac-neg2 55.7%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 76.7%

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

      11. sqr-neg 76.7%

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

      12. sqrt-unprod 55.7%

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

      13. add-sqr-sqrt 55.7%

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

   10. Applied egg-rr 56.8%

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

       1. +-commutative 56.8%

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

       2. fma-undefine 56.8%

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

       3. *-rgt-identity 56.8%

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

       4. associate-+l+ 56.8%

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

       5. +-commutative 56.8%

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

   12. Simplified 56.8%

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

   13. Taylor expanded in x around inf 56.8%

       \[\leadsto \frac{1}{\color{blue}{3 \cdot \frac{x}{s}}} \]
   14. Step-by-step derivation

       1. associate-*r/ 56.8%

          \[\leadsto \frac{1}{\color{blue}{\frac{3 \cdot x}{s}}} \]

       2. *-commutative 56.8%

          \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot 3}}{s}} \]

   15. Simplified 56.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 10:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot 3}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(s \cdot 2 - x\right)}{x \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.999999936531045e-21)
   (/ 1.0 (/ (* x (- (* s 2.0) x)) (* x s)))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.999999936531045e-21f) {
		tmp = 1.0f / ((x * ((s * 2.0f) - x)) / (x * s));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (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 <= (-1.999999936531045e-21)) then
        tmp = 1.0e0 / ((x * ((s * 2.0e0) - x)) / (x * s))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.999999936531045e-21))
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(s * Float32(2.0)) - x)) / Float32(x * s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.999999936531045e-21))
		tmp = single(1.0) / ((x * ((s * single(2.0)) - x)) / (x * s));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999e-21

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 57.3%

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

      1. mul-1-neg 57.3%

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

      2. unsub-neg 57.3%

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

   5. Simplified 57.3%

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

   6. Taylor expanded in x around inf 57.3%

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

      1. associate-*r/ 57.3%

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

      2. metadata-eval 57.3%

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

   8. Simplified 57.3%

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

      1. *-commutative 57.3%

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

      2. frac-sub 60.3%

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

      3. associate-*l/ 62.9%

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

      4. *-commutative 62.9%

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

      5. *-rgt-identity 62.9%

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

   10. Applied egg-rr 62.9%

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

   if -1.9999999e-21 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      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 91.5%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(s \cdot 2 - x\right)}{x \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.6% accurate, 6.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -1.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} 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 = 1.0e0 / (x * (2.0e0 / x))
    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(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	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(1.0) / (x * (single(2.0) / x));
	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 99.9%

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

   3. Taylor expanded in x around 0 5.5%

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

      1. mul-1-neg 5.5%

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

      2. unsub-neg 5.5%

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

   5. Simplified 5.5%

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

   6. Taylor expanded in x around inf 5.5%

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

      1. associate-*r/ 5.5%

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

      2. metadata-eval 5.5%

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

   8. Simplified 5.5%

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

   9. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. unsub-neg 70.6%

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

   5. Simplified 70.6%

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

      1. expm1-log1p-u 70.5%

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

   7. Applied egg-rr 70.5%

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

      1. expm1-undefine 70.6%

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

      2. sub-neg 70.6%

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

      3. log1p-undefine 70.6%

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

      4. rem-exp-log 70.6%

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

      5. associate-+r- 70.6%

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

      6. metadata-eval 70.6%

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

      7. metadata-eval 70.6%

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

   9. Simplified 70.6%

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

4. Final simplification 53.1%

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

Alternative 9: 48.4% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s} \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 10.0) 0.5 (/ 1.0 (* (/ x s) 3.0))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 10.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((x / s) * 3.0f);
	}
	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) <= 10.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((x / s) * 3.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(10.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x / s) * Float32(3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(10.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((x / s) * single(3.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 10

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 49.4%

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

   if 10 < (/.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 55.7%

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

      1. mul-1-neg 55.7%

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

      2. unsub-neg 55.7%

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

   5. Simplified 55.7%

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

   6. Taylor expanded in x around inf 55.7%

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

      1. associate-*r/ 55.7%

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

      2. metadata-eval 55.7%

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

   8. Simplified 55.7%

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

      1. add-sqr-sqrt -0.0%

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

      2. *-un-lft-identity -0.0%

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

      3. prod-diff -0.0%

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

      4. associate-/r/ -0.0%

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

      5. clear-num -0.0%

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

      6. fma-define -0.0%

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

      7. add-sqr-sqrt 55.7%

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

      8. distribute-frac-neg2 55.7%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 76.7%

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

      11. sqr-neg 76.7%

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

      12. sqrt-unprod 55.7%

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

      13. add-sqr-sqrt 55.7%

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

   10. Applied egg-rr 56.8%

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

       1. +-commutative 56.8%

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

       2. fma-undefine 56.8%

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

       3. *-rgt-identity 56.8%

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

       4. associate-+l+ 56.8%

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

       5. +-commutative 56.8%

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

   12. Simplified 56.8%

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

   13. Taylor expanded in x around inf 56.8%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s} \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{3}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 10.0) 0.5 (/ 1.0 (* x (/ 3.0 s)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 10.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (x * (3.0f / 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) <= 10.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (x * (3.0e0 / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(10.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(3.0) / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(10.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (x * (single(3.0) / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 10

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 49.4%

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

   if 10 < (/.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 55.7%

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

      1. mul-1-neg 55.7%

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

      2. unsub-neg 55.7%

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

   5. Simplified 55.7%

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

   6. Taylor expanded in x around inf 55.7%

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

      1. associate-*r/ 55.7%

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

      2. metadata-eval 55.7%

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

   8. Simplified 55.7%

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

      1. add-sqr-sqrt -0.0%

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

      2. *-un-lft-identity -0.0%

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

      3. prod-diff -0.0%

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

      4. associate-/r/ -0.0%

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

      5. clear-num -0.0%

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

      6. fma-define -0.0%

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

      7. add-sqr-sqrt 55.7%

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

      8. distribute-frac-neg2 55.7%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 76.7%

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

      11. sqr-neg 76.7%

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

      12. sqrt-unprod 55.7%

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

      13. add-sqr-sqrt 55.7%

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

   10. Applied egg-rr 56.8%

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

       1. +-commutative 56.8%

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

       2. fma-undefine 56.8%

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

       3. *-rgt-identity 56.8%

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

       4. associate-+l+ 56.8%

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

       5. +-commutative 56.8%

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

   12. Simplified 56.8%

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

   13. Taylor expanded in s around 0 56.8%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{3}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.6% accurate, 7.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -1.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} 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 = 1.0e0 / (x * (2.0e0 / x))
    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(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	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(1.0) / (x * (single(2.0) / x));
	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 99.9%

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

   3. Taylor expanded in x around 0 5.5%

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

      1. mul-1-neg 5.5%

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

      2. unsub-neg 5.5%

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

   5. Simplified 5.5%

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

   6. Taylor expanded in x around inf 5.5%

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

      1. associate-*r/ 5.5%

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

      2. metadata-eval 5.5%

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

   8. Simplified 5.5%

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

   9. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. unsub-neg 70.6%

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

   5. Simplified 70.6%

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

4. Final simplification 53.0%

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

Alternative 12: 48.1% 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 99.9%

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

   3. Taylor expanded in x around 0 50.1%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 54.2%

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

      1. mul-1-neg 54.2%

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

      2. unsub-neg 54.2%

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

   5. Simplified 54.2%

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

   6. Taylor expanded in x around inf 54.2%

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

      1. associate-*r/ 54.2%

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

      2. mul-1-neg 54.2%

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

   8. Simplified 54.2%

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

4. Final simplification 51.5%

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

Alternative 13: 46.8% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;0.3333333333333333 \cdot \frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.20000000298023224) (* 0.3333333333333333 (/ s x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.20000000298023224f) {
		tmp = 0.3333333333333333f * (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 <= (-0.20000000298023224e0)) then
        tmp = 0.3333333333333333e0 * (s / x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.20000000298023224))
		tmp = Float32(Float32(0.3333333333333333) * 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(-0.20000000298023224))
		tmp = single(0.3333333333333333) * (s / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 72.3%

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

      1. mul-1-neg 72.3%

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

      2. unsub-neg 72.3%

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

   5. Simplified 72.3%

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

   6. Taylor expanded in x around inf 72.3%

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

      1. associate-*r/ 72.3%

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

      2. metadata-eval 72.3%

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

   8. Simplified 72.3%

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

      1. add-sqr-sqrt -0.0%

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

      2. *-un-lft-identity -0.0%

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

      3. prod-diff -0.0%

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

      4. associate-/r/ -0.0%

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

      5. clear-num -0.0%

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

      6. fma-define -0.0%

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

      7. add-sqr-sqrt 72.3%

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

      8. distribute-frac-neg2 72.3%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 75.9%

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

      11. sqr-neg 75.9%

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

      12. sqrt-unprod 72.3%

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

      13. add-sqr-sqrt 72.3%

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

   10. Applied egg-rr 73.7%

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

       1. +-commutative 73.7%

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

       2. fma-undefine 73.7%

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

       3. *-rgt-identity 73.7%

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

       4. associate-+l+ 73.7%

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

       5. +-commutative 73.7%

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

   12. Simplified 73.7%

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

   13. Taylor expanded in x around inf 67.1%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{s}{x}} \]
   14. Step-by-step derivation

       1. *-commutative 67.1%

          \[\leadsto \color{blue}{\frac{s}{x} \cdot 0.3333333333333333} \]

   15. Simplified 67.1%

       \[\leadsto \color{blue}{\frac{s}{x} \cdot 0.3333333333333333} \]

   if -0.200000003 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 44.5%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;0.3333333333333333 \cdot \frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.8% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;\frac{s \cdot 0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.20000000298023224) (/ (* s 0.3333333333333333) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.20000000298023224f) {
		tmp = (s * 0.3333333333333333f) / 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 <= (-0.20000000298023224e0)) then
        tmp = (s * 0.3333333333333333e0) / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.20000000298023224))
		tmp = Float32(Float32(s * Float32(0.3333333333333333)) / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.20000000298023224))
		tmp = (s * single(0.3333333333333333)) / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 72.3%

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

      1. mul-1-neg 72.3%

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

      2. unsub-neg 72.3%

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

   5. Simplified 72.3%

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

   6. Taylor expanded in x around inf 72.3%

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

      1. associate-*r/ 72.3%

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

      2. metadata-eval 72.3%

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

   8. Simplified 72.3%

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

      1. add-sqr-sqrt -0.0%

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

      2. *-un-lft-identity -0.0%

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

      3. prod-diff -0.0%

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

      4. associate-/r/ -0.0%

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

      5. clear-num -0.0%

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

      6. fma-define -0.0%

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

      7. add-sqr-sqrt 72.3%

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

      8. distribute-frac-neg2 72.3%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 75.9%

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

      11. sqr-neg 75.9%

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

      12. sqrt-unprod 72.3%

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

      13. add-sqr-sqrt 72.3%

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

   10. Applied egg-rr 73.7%

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

       1. +-commutative 73.7%

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

       2. fma-undefine 73.7%

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

       3. *-rgt-identity 73.7%

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

       4. associate-+l+ 73.7%

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

       5. +-commutative 73.7%

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

   12. Simplified 73.7%

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

   13. Taylor expanded in x around inf 67.1%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{s}{x}} \]
   14. Step-by-step derivation

       1. associate-*r/ 67.1%

          \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot s}{x}} \]

       2. *-commutative 67.1%

          \[\leadsto \frac{\color{blue}{s \cdot 0.3333333333333333}}{x} \]

   15. Simplified 67.1%

       \[\leadsto \color{blue}{\frac{s \cdot 0.3333333333333333}{x}} \]

   if -0.200000003 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 44.5%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;\frac{s \cdot 0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.6% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000116860974e-7) (/ s (- x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000116860974e-7f) {
		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 <= (-1.0000000116860974e-7)) 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(-1.0000000116860974e-7))
		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(-1.0000000116860974e-7))
		tmp = s / -x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000001e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 64.6%

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

      1. mul-1-neg 64.6%

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

      2. unsub-neg 64.6%

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

   5. Simplified 64.6%

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

   6. Taylor expanded in x around inf 58.2%

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

      1. associate-*r/ 58.2%

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

      2. neg-mul-1 58.2%

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

   8. Simplified 58.2%

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

   if -1.00000001e-7 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 46.3%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.2% 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 35.3%

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

4. Final simplification 35.3%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))