Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 14.2s

Alternatives: 14

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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. div-inv 99.7%

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

   2. exp-prod 76.4%

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

   3. neg-mul-1 76.4%

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

   4. exp-prod 76.4%

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

   5. pow-pow 99.7%

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

   6. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

   1. add-exp-log 99.7%

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

   2. log-rec 99.7%

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

   3. log1p-define 99.8%

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

   4. add-exp-log 99.8%

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

   5. log-pow 99.8%

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

   6. rem-log-exp 99.8%

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

   7. pow-exp 99.8%

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

   8. inv-pow 99.8%

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

   9. add-exp-log 99.8%

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

   10. neg-log 99.8%

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

   11. add-log-exp 99.8%

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

   12. distribute-neg-frac2 99.8%

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

6. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(-0.5f), (2.0f * (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((-0.5e0)) ** (2.0e0 * (x / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. div-inv 99.7%

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

   2. exp-prod 76.4%

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

   3. neg-mul-1 76.4%

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

   4. exp-prod 76.4%

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

   5. pow-pow 99.7%

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

   6. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

   1. add-sqr-sqrt 99.7%

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

   2. unpow-prod-down 99.7%

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

6. Applied egg-rr 99.7%

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

   1. pow-sqr 99.7%

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

   2. *-commutative 99.7%

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

8. Simplified 99.7%

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

   1. +-commutative 99.7%

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

   2. add-sqr-sqrt 99.7%

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

   3. fma-define 99.7%

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

   4. pow-unpow 99.7%

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

   5. sqrt-pow1 99.7%

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

   6. metadata-eval 99.7%

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

   7. pow1 99.7%

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

   8. pow1/2 99.7%

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

   9. pow-exp 99.7%

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

   10. metadata-eval 99.7%

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

   11. pow-unpow 99.7%

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

   12. sqrt-pow1 99.7%

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

   13. metadata-eval 99.7%

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

   14. pow1 99.7%

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

   15. pow1/2 99.7%

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

   16. pow-exp 99.7%

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

   17. metadata-eval 99.7%

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

10. Applied egg-rr 99.7%

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

    1. fma-undefine 99.7%

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

    2. pow-sqr 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

    \[\leadsto \frac{1}{1 + {\left(e^{-0.5}\right)}^{\left(2 \cdot \frac{x}{s}\right)}} \]
14. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 4: 92.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;\frac{x - s}{x}\\ \mathbf{elif}\;t\_0 \leq 0.0010000000474974513:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{s}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -20.0)
     (/ (- x s) x)
     (if (<= t_0 0.0010000000474974513)
       (+ 0.5 (/ (* x 0.25) s))
       (+ -1.0 (- 1.0 (/ s x)))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= -20.0f) {
		tmp = (x - s) / x;
	} else if (t_0 <= 0.0010000000474974513f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = -1.0f + (1.0f - (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (t_0 <= (-20.0e0)) then
        tmp = (x - s) / x
    else if (t_0 <= 0.0010000000474974513e0) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = (-1.0e0) + (1.0e0 - (s / x))
    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(-20.0))
		tmp = Float32(Float32(x - s) / x);
	elseif (t_0 <= Float32(0.0010000000474974513))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(1.0) - Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(-20.0))
		tmp = (x - s) / x;
	elseif (t_0 <= single(0.0010000000474974513))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(-1.0) + (single(1.0) - (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   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 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. exp-1-e 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 94.3%

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

       1. +-commutative 94.3%

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

       2. log-E 94.3%

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

       3. *-rgt-identity 94.3%

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

   11. Simplified 94.3%

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

   12. Taylor expanded in x around inf 94.3%

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

       1. mul-1-neg 94.3%

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

       2. unsub-neg 94.3%

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

   14. Simplified 94.3%

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

   15. Taylor expanded in x around 0 94.4%

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

   if -20 < (/.f32 (neg.f32 x) s) < 0.00100000005

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 96.6%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   4. Step-by-step derivation

      1. associate-*r/ 96.6%

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

      2. *-commutative 96.6%

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

   5. Simplified 96.6%

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

   if 0.00100000005 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 40.6%

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

      1. mul-1-neg 40.6%

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

      2. unsub-neg 40.6%

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

   5. Simplified 40.6%

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

   6. Taylor expanded in x around inf 37.1%

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

      1. associate-*r/ 37.1%

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

      2. mul-1-neg 37.1%

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

   8. Simplified 37.1%

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

      1. expm1-log1p-u 37.1%

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

      2. expm1-undefine 78.6%

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

      3. associate-/r/ 78.6%

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

      4. add-sqr-sqrt 78.6%

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

      5. sqrt-unprod 76.1%

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

      6. sqr-neg 76.1%

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

      7. sqrt-unprod -0.0%

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

      8. add-sqr-sqrt 76.7%

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

      9. frac-2neg 76.7%

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

      10. metadata-eval 76.7%

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

      11. add-sqr-sqrt 76.7%

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

      12. sqrt-unprod 74.6%

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

      13. sqr-neg 74.6%

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

      14. sqrt-unprod -0.0%

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

      15. add-sqr-sqrt 78.6%

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

   10. Applied egg-rr 78.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1}{x} \cdot s\right)} - 1} \]
   11. Step-by-step derivation

       1. sub-neg 78.6%

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

       2. log1p-undefine 78.6%

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

       3. rem-exp-log 78.6%

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

       4. associate-*l/ 78.6%

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

       5. associate-*r/ 78.6%

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

       6. mul-1-neg 78.6%

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

       7. unsub-neg 78.6%

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

       8. metadata-eval 78.6%

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

   12. Simplified 78.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -20:\\ \;\;\;\;\frac{x - s}{x}\\ \mathbf{elif}\;\frac{x}{-s} \leq 0.0010000000474974513:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{s}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;\frac{x - s}{x}\\ \mathbf{elif}\;t\_0 \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -20.0)
     (/ (- x s) x)
     (if (<= t_0 4.999999873689376e-6)
       (+ 0.5 (/ (* x 0.25) s))
       (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= -20.0f) {
		tmp = (x - s) / x;
	} else if (t_0 <= 4.999999873689376e-6f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} 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 <= (-20.0e0)) then
        tmp = (x - s) / x
    else if (t_0 <= 4.999999873689376e-6) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    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(-20.0))
		tmp = Float32(Float32(x - s) / x);
	elseif (t_0 <= Float32(4.999999873689376e-6))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	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(-20.0))
		tmp = (x - s) / x;
	elseif (t_0 <= single(4.999999873689376e-6))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	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) < -20

   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 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. exp-1-e 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 94.3%

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

       1. +-commutative 94.3%

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

       2. log-E 94.3%

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

       3. *-rgt-identity 94.3%

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

   11. Simplified 94.3%

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

   12. Taylor expanded in x around inf 94.3%

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

       1. mul-1-neg 94.3%

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

       2. unsub-neg 94.3%

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

   14. Simplified 94.3%

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

   15. Taylor expanded in x around 0 94.4%

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

   if -20 < (/.f32 (neg.f32 x) s) < 4.99999987e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 96.4%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   4. Step-by-step derivation

      1. associate-*r/ 96.4%

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

      2. *-commutative 96.4%

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

   5. Simplified 96.4%

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

   if 4.99999987e-6 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 42.2%

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

      1. mul-1-neg 42.2%

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

      2. unsub-neg 42.2%

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

   5. Simplified 42.2%

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

   6. Taylor expanded in x around inf 36.5%

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

      1. associate-*r/ 36.5%

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

      2. mul-1-neg 36.5%

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

   8. Simplified 36.5%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -20:\\ \;\;\;\;\frac{x - s}{x}\\ \mathbf{elif}\;\frac{x}{-s} \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -80:\\ \;\;\;\;\frac{x - s}{x}\\ \mathbf{elif}\;t\_0 \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -80.0)
     (/ (- x s) x)
     (if (<= t_0 4.999999980020986e-13) 0.5 (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= -80.0f) {
		tmp = (x - s) / x;
	} else if (t_0 <= 4.999999980020986e-13f) {
		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) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (t_0 <= (-80.0e0)) then
        tmp = (x - s) / x
    else if (t_0 <= 4.999999980020986e-13) then
        tmp = 0.5e0
    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(-80.0))
		tmp = Float32(Float32(x - s) / x);
	elseif (t_0 <= Float32(4.999999980020986e-13))
		tmp = Float32(0.5);
	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(-80.0))
		tmp = (x - s) / x;
	elseif (t_0 <= single(4.999999980020986e-13))
		tmp = single(0.5);
	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) < -80

   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 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. exp-1-e 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 95.9%

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

       1. +-commutative 95.9%

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

       2. log-E 95.9%

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

       3. *-rgt-identity 95.9%

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

   11. Simplified 95.9%

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

   12. Taylor expanded in x around inf 95.8%

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

       1. mul-1-neg 95.8%

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

       2. unsub-neg 95.8%

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

   14. Simplified 95.8%

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

   15. Taylor expanded in x around 0 95.9%

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

   if -80 < (/.f32 (neg.f32 x) s) < 4.99999998e-13

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 78.6%

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

   if 4.99999998e-13 < (/.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 43.2%

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

      1. mul-1-neg 43.2%

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

      2. unsub-neg 43.2%

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

   5. Simplified 43.2%

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

   6. Taylor expanded in x around inf 36.1%

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

      1. associate-*r/ 36.1%

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

      2. mul-1-neg 36.1%

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

   8. Simplified 36.1%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -80:\\ \;\;\;\;\frac{x - s}{x}\\ \mathbf{elif}\;\frac{x}{-s} \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.1% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\left(2 + \frac{1}{1 + \frac{x}{s}}\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{s}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 4.999999873689376e-5)
   (/ 1.0 (+ (+ 2.0 (/ 1.0 (+ 1.0 (/ x s)))) -1.0))
   (+ -1.0 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 4.999999873689376e-5f) {
		tmp = 1.0f / ((2.0f + (1.0f / (1.0f + (x / s)))) + -1.0f);
	} else {
		tmp = -1.0f + (1.0f - (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 4.999999873689376e-5) then
        tmp = 1.0e0 / ((2.0e0 + (1.0e0 / (1.0e0 + (x / s)))) + (-1.0e0))
    else
        tmp = (-1.0e0) + (1.0e0 - (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(4.999999873689376e-5))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(2.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))) + Float32(-1.0)));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(1.0) - Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(4.999999873689376e-5))
		tmp = single(1.0) / ((single(2.0) + (single(1.0) / (single(1.0) + (x / s)))) + single(-1.0));
	else
		tmp = single(-1.0) + (single(1.0) - (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 4.99999987e-5

   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. Step-by-step derivation

      1. *-un-lft-identity 99.9%

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

      2. exp-prod 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. exp-1-e 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 93.6%

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

       1. +-commutative 93.6%

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

       2. log-E 93.6%

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

       3. *-rgt-identity 93.6%

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

   11. Simplified 93.6%

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

       1. expm1-log1p-u 93.6%

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

       2. expm1-undefine 93.6%

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

   13. Applied egg-rr 93.6%

       \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(1 + \frac{1}{\frac{x}{s} + 1}\right)} - 1}} \]
   14. Step-by-step derivation

       1. sub-neg 93.6%

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

       2. log1p-undefine 93.6%

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

       3. rem-exp-log 93.6%

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

       4. associate-+r+ 93.6%

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

       5. metadata-eval 93.6%

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

       6. +-commutative 93.6%

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

       7. metadata-eval 93.6%

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

   15. Simplified 93.6%

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

   if 4.99999987e-5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.5%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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. mul-1-neg 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. Taylor expanded in x around inf 36.7%

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

      1. associate-*r/ 36.7%

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

      2. mul-1-neg 36.7%

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

   8. Simplified 36.7%

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

      1. expm1-log1p-u 36.7%

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

      2. expm1-undefine 77.5%

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

      3. associate-/r/ 77.5%

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

      4. add-sqr-sqrt 77.5%

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

      5. sqrt-unprod 74.7%

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

      6. sqr-neg 74.7%

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

      7. sqrt-unprod -0.0%

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

      8. add-sqr-sqrt 75.3%

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

      9. frac-2neg 75.3%

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

      10. metadata-eval 75.3%

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

      11. add-sqr-sqrt 75.3%

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

      12. sqrt-unprod 73.1%

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

      13. sqr-neg 73.1%

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

      14. sqrt-unprod -0.0%

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

      15. add-sqr-sqrt 77.5%

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

   10. Applied egg-rr 77.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1}{x} \cdot s\right)} - 1} \]
   11. Step-by-step derivation

       1. sub-neg 77.5%

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

       2. log1p-undefine 77.5%

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

       3. rem-exp-log 77.5%

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

       4. associate-*l/ 77.5%

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

       5. associate-*r/ 77.5%

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

       6. mul-1-neg 77.5%

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

       7. unsub-neg 77.5%

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

       8. metadata-eval 77.5%

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

   12. Simplified 77.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\left(2 + \frac{1}{1 + \frac{x}{s}}\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{s}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{s}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 4.999999873689376e-5)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (+ -1.0 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 4.999999873689376e-5f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = -1.0f + (1.0f - (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 4.999999873689376e-5) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = (-1.0e0) + (1.0e0 - (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(4.999999873689376e-5))
		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(1.0) - Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(4.999999873689376e-5))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(-1.0) + (single(1.0) - (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 4.99999987e-5

   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 93.6%

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

   if 4.99999987e-5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.5%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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. mul-1-neg 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. Taylor expanded in x around inf 36.7%

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

      1. associate-*r/ 36.7%

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

      2. mul-1-neg 36.7%

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

   8. Simplified 36.7%

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

      1. expm1-log1p-u 36.7%

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

      2. expm1-undefine 77.5%

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

      3. associate-/r/ 77.5%

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

      4. add-sqr-sqrt 77.5%

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

      5. sqrt-unprod 74.7%

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

      6. sqr-neg 74.7%

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

      7. sqrt-unprod -0.0%

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

      8. add-sqr-sqrt 75.3%

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

      9. frac-2neg 75.3%

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

      10. metadata-eval 75.3%

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

      11. add-sqr-sqrt 75.3%

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

      12. sqrt-unprod 73.1%

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

      13. sqr-neg 73.1%

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

      14. sqrt-unprod -0.0%

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

      15. add-sqr-sqrt 77.5%

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

   10. Applied egg-rr 77.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1}{x} \cdot s\right)} - 1} \]
   11. Step-by-step derivation

       1. sub-neg 77.5%

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

       2. log1p-undefine 77.5%

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

       3. rem-exp-log 77.5%

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

       4. associate-*l/ 77.5%

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

       5. associate-*r/ 77.5%

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

       6. mul-1-neg 77.5%

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

       7. unsub-neg 77.5%

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

       8. metadata-eval 77.5%

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

   12. Simplified 77.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{s}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -399999991808:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -399999991808.0)
   (/ 1.0 (/ x s))
   (if (<= x 4.999999918875795e-18) 0.5 (/ (- x s) x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -399999991808.0f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 4.999999918875795e-18f) {
		tmp = 0.5f;
	} else {
		tmp = (x - s) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-399999991808.0e0)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 4.999999918875795e-18) then
        tmp = 0.5e0
    else
        tmp = (x - s) / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-399999991808.0))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(4.999999918875795e-18))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(x - s) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-399999991808.0))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(4.999999918875795e-18))
		tmp = single(0.5);
	else
		tmp = (x - s) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -399999992000

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 74.8%

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

      1. mul-1-neg 74.8%

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

      2. unsub-neg 74.8%

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

   5. Simplified 74.8%

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

   6. Taylor expanded in x around inf 74.8%

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

      1. associate-*r/ 74.8%

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

      2. mul-1-neg 74.8%

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

   8. Simplified 74.8%

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

      1. distribute-frac-neg 74.8%

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

      2. distribute-frac-neg2 74.8%

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

      3. inv-pow 74.8%

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

      4. add-sqr-sqrt 74.8%

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

      5. sqrt-pow1 74.8%

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

      6. add-sqr-sqrt -0.0%

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

      7. sqrt-unprod 71.1%

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

      8. sqr-neg 71.1%

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

      9. sqrt-unprod 71.1%

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

      10. add-sqr-sqrt 71.1%

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

      11. metadata-eval 71.1%

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

      12. sqrt-pow1 71.1%

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

      13. add-sqr-sqrt -0.0%

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

      14. sqrt-unprod 71.1%

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

      15. sqr-neg 71.1%

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

      16. sqrt-unprod 71.1%

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

      17. add-sqr-sqrt 71.1%

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

      18. metadata-eval 71.1%

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

   10. Applied egg-rr 71.1%

       \[\leadsto \color{blue}{{\left(\frac{x}{s}\right)}^{-0.5} \cdot {\left(\frac{x}{s}\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. pow-sqr 74.8%

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

       2. metadata-eval 74.8%

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

       3. unpow-1 74.8%

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

   12. Simplified 74.8%

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

   if -399999992000 < x < 4.99999992e-18

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 38.7%

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

   if 4.99999992e-18 < 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 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. exp-1-e 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 95.2%

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

       1. +-commutative 95.2%

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

       2. log-E 95.2%

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

       3. *-rgt-identity 95.2%

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

   11. Simplified 95.2%

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

   12. Taylor expanded in x around inf 88.2%

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

       1. mul-1-neg 88.2%

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

       2. unsub-neg 88.2%

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

   14. Simplified 88.2%

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

   15. Taylor expanded in x around 0 88.2%

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

Alternative 10: 67.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -399999991808:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -399999991808.0)
   (/ 1.0 (/ x s))
   (if (<= x 4.999999918875795e-18) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -399999991808.0f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 4.999999918875795e-18f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f - (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-399999991808.0e0)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 4.999999918875795e-18) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-399999991808.0))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(4.999999918875795e-18))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-399999991808.0))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(4.999999918875795e-18))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -399999992000

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 74.8%

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

      1. mul-1-neg 74.8%

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

      2. unsub-neg 74.8%

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

   5. Simplified 74.8%

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

   6. Taylor expanded in x around inf 74.8%

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

      1. associate-*r/ 74.8%

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

      2. mul-1-neg 74.8%

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

   8. Simplified 74.8%

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

      1. distribute-frac-neg 74.8%

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

      2. distribute-frac-neg2 74.8%

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

      3. inv-pow 74.8%

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

      4. add-sqr-sqrt 74.8%

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

      5. sqrt-pow1 74.8%

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

      6. add-sqr-sqrt -0.0%

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

      7. sqrt-unprod 71.1%

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

      8. sqr-neg 71.1%

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

      9. sqrt-unprod 71.1%

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

      10. add-sqr-sqrt 71.1%

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

      11. metadata-eval 71.1%

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

      12. sqrt-pow1 71.1%

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

      13. add-sqr-sqrt -0.0%

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

      14. sqrt-unprod 71.1%

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

      15. sqr-neg 71.1%

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

      16. sqrt-unprod 71.1%

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

      17. add-sqr-sqrt 71.1%

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

      18. metadata-eval 71.1%

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

   10. Applied egg-rr 71.1%

       \[\leadsto \color{blue}{{\left(\frac{x}{s}\right)}^{-0.5} \cdot {\left(\frac{x}{s}\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. pow-sqr 74.8%

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

       2. metadata-eval 74.8%

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

       3. unpow-1 74.8%

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

   12. Simplified 74.8%

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

   if -399999992000 < x < 4.99999992e-18

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 38.7%

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

   if 4.99999992e-18 < 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 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. exp-1-e 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 95.2%

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

       1. +-commutative 95.2%

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

       2. log-E 95.2%

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

       3. *-rgt-identity 95.2%

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

   11. Simplified 95.2%

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

   12. Taylor expanded in x around inf 88.2%

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

       1. mul-1-neg 88.2%

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

       2. unsub-neg 88.2%

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

   14. Simplified 88.2%

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

Alternative 11: 66.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -20000000:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -20000000.0)
   (* s (/ -1.0 x))
   (if (<= x 1.0000000195414814e-25) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -20000000.0f) {
		tmp = s * (-1.0f / x);
	} else if (x <= 1.0000000195414814e-25f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f - (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-20000000.0e0)) then
        tmp = s * ((-1.0e0) / x)
    else if (x <= 1.0000000195414814e-25) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-20000000.0))
		tmp = Float32(s * Float32(Float32(-1.0) / x));
	elseif (x <= Float32(1.0000000195414814e-25))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-20000000.0))
		tmp = s * (single(-1.0) / x);
	elseif (x <= single(1.0000000195414814e-25))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2e7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 68.1%

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

      1. mul-1-neg 68.1%

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

      2. unsub-neg 68.1%

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

   5. Simplified 68.1%

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

   6. Taylor expanded in x around inf 68.1%

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

      1. associate-*r/ 68.1%

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

      2. mul-1-neg 68.1%

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

   8. Simplified 68.1%

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

      1. associate-/r/ 60.4%

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

      2. add-sqr-sqrt 60.4%

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

      3. sqrt-unprod 75.9%

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

      4. sqr-neg 75.9%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 60.4%

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

      7. frac-2neg 60.4%

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

      8. metadata-eval 60.4%

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

      9. add-sqr-sqrt 60.4%

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

      10. sqrt-unprod 75.9%

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

      11. sqr-neg 75.9%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 60.4%

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

   10. Applied egg-rr 60.4%

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

   if -2e7 < x < 1.00000002e-25

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 38.9%

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

   if 1.00000002e-25 < 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 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. exp-prod 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. exp-1-e 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 95.3%

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

       1. +-commutative 95.3%

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

       2. log-E 95.3%

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

       3. *-rgt-identity 95.3%

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

   11. Simplified 95.3%

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

   12. Taylor expanded in x around inf 83.4%

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

       1. mul-1-neg 83.4%

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

       2. unsub-neg 83.4%

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

   14. Simplified 83.4%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20000000:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.9% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -20000000:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (if (<= x -20000000.0) (* s (/ -1.0 x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -20000000.0f) {
		tmp = s * (-1.0f / 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 <= (-20000000.0e0)) then
        tmp = s * ((-1.0e0) / x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-20000000.0))
		tmp = Float32(s * Float32(Float32(-1.0) / x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-20000000.0))
		tmp = s * (single(-1.0) / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2e7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 68.1%

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

      1. mul-1-neg 68.1%

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

      2. unsub-neg 68.1%

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

   5. Simplified 68.1%

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

   6. Taylor expanded in x around inf 68.1%

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

      1. associate-*r/ 68.1%

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

      2. mul-1-neg 68.1%

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

   8. Simplified 68.1%

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

      1. associate-/r/ 60.4%

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

      2. add-sqr-sqrt 60.4%

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

      3. sqrt-unprod 75.9%

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

      4. sqr-neg 75.9%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 60.4%

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

      7. frac-2neg 60.4%

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

      8. metadata-eval 60.4%

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

      9. add-sqr-sqrt 60.4%

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

      10. sqrt-unprod 75.9%

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

      11. sqr-neg 75.9%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 60.4%

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

   10. Applied egg-rr 60.4%

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

   if -2e7 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 36.2%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20000000:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.9% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -20000000:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (if (<= x -20000000.0) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -20000000.0f) {
		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 <= (-20000000.0e0)) 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(-20000000.0))
		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(-20000000.0))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2e7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 68.1%

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

      1. mul-1-neg 68.1%

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

      2. unsub-neg 68.1%

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

   5. Simplified 68.1%

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

   6. Taylor expanded in x around inf 68.1%

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

      1. associate-*r/ 68.1%

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

      2. mul-1-neg 68.1%

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

   8. Simplified 68.1%

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

      1. associate-/r/ 60.4%

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

      2. add-sqr-sqrt 60.4%

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

      3. sqrt-unprod 75.9%

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

      4. sqr-neg 75.9%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 60.4%

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

      7. frac-2neg 60.4%

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

      8. metadata-eval 60.4%

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

      9. add-sqr-sqrt 60.4%

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

      10. sqrt-unprod 75.9%

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

      11. sqr-neg 75.9%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 60.4%

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

   10. Applied egg-rr 60.4%

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

       1. associate-*l/ 60.4%

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

       2. neg-mul-1 60.4%

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

       3. add-sqr-sqrt -0.0%

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

       4. sqrt-unprod 65.3%

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

       5. sqr-neg 65.3%

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

       6. sqrt-unprod 60.4%

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

       7. add-sqr-sqrt 60.4%

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

   12. Applied egg-rr 60.4%

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

   if -2e7 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 36.2%

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

Alternative 14: 35.1% 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 99.7%

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

3. Taylor expanded in x around 0 30.9%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))