Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 9.0s

Alternatives: 21

Speedup: 1.0×

Specification

\[0 \leq s \land s \leq 1.0651631\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. div-inv 99.8%

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

   2. exp-prod 83.2%

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

   3. neg-mul-1 83.2%

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

   4. exp-prod 83.2%

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

   5. pow-pow 99.8%

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

   6. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (1.0f + (1.0f / powf(((float) M_E), (x / s))));
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / (Float32(exp(1)) ^ Float32(x / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(2.71828182845904523536) ^ (x / s))));
end

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(-Float32(x / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (single(2.71828182845904523536) ^ -(x / s)));
end

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

6. Applied egg-rr 99.8%

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

   1. pow-flip 99.8%

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

   2. exp-1-e 99.8%

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

   3. distribute-neg-frac2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (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 + (1.0e0 / exp((x / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

4. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{e^{\frac{x}{s}}}}} \]
5. Add Preprocessing

Alternative 5: 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 99.8%

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

3. Final simplification 99.8%

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

Alternative 6: 89.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 40:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t\_0 \leq 19999999961012896000:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{2}{x} + \left(\frac{1}{s} + \frac{s + s}{s \cdot s}\right)\right)}\\ \mathbf{elif}\;t\_0 \leq 9.999999680285692 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 40.0)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 19999999961012896000.0)
       (/ 1.0 (* x (+ (/ 2.0 x) (+ (/ 1.0 s) (/ (+ s s) (* s s))))))
       (if (<= t_0 9.999999680285692e+37)
         (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ x s) 2.0)))
         (/ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= 40.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 19999999961012896000.0f) {
		tmp = 1.0f / (x * ((2.0f / x) + ((1.0f / s) + ((s + s) / (s * s)))));
	} else if (t_0 <= 9.999999680285692e+37f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / (x / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (t_0 <= 40.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 19999999961012896000.0e0) then
        tmp = 1.0e0 / (x * ((2.0e0 / x) + ((1.0e0 / s) + ((s + s) / (s * s)))))
    else if (t_0 <= 9.999999680285692e+37) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(40.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(19999999961012896000.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(Float32(2.0) / x) + Float32(Float32(Float32(1.0) / s) + Float32(Float32(s + s) / Float32(s * s))))));
	elseif (t_0 <= Float32(9.999999680285692e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(40.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(19999999961012896000.0))
		tmp = single(1.0) / (x * ((single(2.0) / x) + ((single(1.0) / s) + ((s + s) / (s * s)))));
	elseif (t_0 <= single(9.999999680285692e+37))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

2. if (/.f32 (neg.f32 x) s) < 40

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 93.8%

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

      1. +-commutative 93.8%

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

   7. Simplified 93.8%

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

   if 40 < (/.f32 (neg.f32 x) s) < 2e19

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 7.8%

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

      1. neg-mul-1 7.8%

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

      2. unsub-neg 7.8%

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

   5. Simplified 7.8%

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

   6. Taylor expanded in x around inf 7.8%

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

      1. associate-*r/ 7.8%

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

      2. metadata-eval 7.8%

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

   8. Simplified 7.8%

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

      1. *-un-lft-identity 7.8%

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

      2. add-sqr-sqrt -0.0%

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

      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. distribute-frac-neg2 -0.0%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod -0.0%

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

      9. sqr-neg -0.0%

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

      10. sqrt-unprod -0.0%

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

      11. add-sqr-sqrt -0.0%

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

      12. fma-define -0.0%

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

      13. add-sqr-sqrt 7.6%

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

   10. Applied egg-rr 7.6%

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

       1. associate-+l+ 7.6%

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

       2. fma-undefine 7.6%

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

       3. *-rgt-identity 7.6%

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

   12. Simplified 7.6%

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

       1. frac-2neg 7.6%

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

       2. metadata-eval 7.6%

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

       3. frac-add 52.0%

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

       4. *-un-lft-identity 52.0%

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

   14. Applied egg-rr 52.0%

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

   if 2e19 < (/.f32 (neg.f32 x) s) < 9.99999968e37

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 14.7%

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

      1. neg-mul-1 14.7%

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

      2. unsub-neg 14.7%

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

   5. Simplified 14.7%

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

      1. *-un-lft-identity 14.7%

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

      2. cancel-sign-sub-inv 14.7%

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

      3. metadata-eval 14.7%

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

      4. add-log-exp 100.0%

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

      5. pow-exp 100.0%

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

      6. flip-+ -0.0%

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

      7. metadata-eval -0.0%

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

      8. pow-exp -0.0%

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

      9. add-log-exp -0.0%

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

      10. neg-mul-1 -0.0%

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

      11. pow-exp -0.0%

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

      12. add-log-exp -0.0%

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

      13. neg-mul-1 -0.0%

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

      14. distribute-neg-frac2 -0.0%

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

      15. distribute-neg-frac2 -0.0%

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

      16. pow-exp -0.0%

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

   7. Applied egg-rr 100.0%

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

   if 9.99999968e37 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 94.1%

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

      1. associate-*r/ 94.1%

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

      2. neg-mul-1 94.1%

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

   8. Simplified 94.1%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 40:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 19999999961012896000:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{2}{x} + \left(\frac{1}{s} + \frac{s + s}{s \cdot s}\right)\right)}\\ \mathbf{elif}\;\frac{x}{-s} \leq 9.999999680285692 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 40:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t\_0 \leq 19999999961012896000:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(x - s \cdot 2\right) \cdot \frac{-1}{x \cdot s}\right)}\\ \mathbf{elif}\;t\_0 \leq 9.999999680285692 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 40.0)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 19999999961012896000.0)
       (/ 1.0 (* x (* (- x (* s 2.0)) (/ -1.0 (* x s)))))
       (if (<= t_0 9.999999680285692e+37)
         (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ x s) 2.0)))
         (/ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= 40.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 19999999961012896000.0f) {
		tmp = 1.0f / (x * ((x - (s * 2.0f)) * (-1.0f / (x * s))));
	} else if (t_0 <= 9.999999680285692e+37f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / (x / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (t_0 <= 40.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 19999999961012896000.0e0) then
        tmp = 1.0e0 / (x * ((x - (s * 2.0e0)) * ((-1.0e0) / (x * s))))
    else if (t_0 <= 9.999999680285692e+37) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(40.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(19999999961012896000.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(x - Float32(s * Float32(2.0))) * Float32(Float32(-1.0) / Float32(x * s)))));
	elseif (t_0 <= Float32(9.999999680285692e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(40.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(19999999961012896000.0))
		tmp = single(1.0) / (x * ((x - (s * single(2.0))) * (single(-1.0) / (x * s))));
	elseif (t_0 <= single(9.999999680285692e+37))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

2. if (/.f32 (neg.f32 x) s) < 40

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 93.8%

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

      1. +-commutative 93.8%

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

   7. Simplified 93.8%

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

   if 40 < (/.f32 (neg.f32 x) s) < 2e19

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 7.8%

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

      1. neg-mul-1 7.8%

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

      2. unsub-neg 7.8%

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

   5. Simplified 7.8%

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

   6. Taylor expanded in x around inf 7.8%

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

      1. associate-*r/ 7.8%

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

      2. metadata-eval 7.8%

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

   8. Simplified 7.8%

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

      1. frac-sub 44.0%

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

      2. div-inv 47.9%

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

      3. *-rgt-identity 47.9%

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

      4. *-commutative 47.9%

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

   10. Applied egg-rr 47.9%

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

   if 2e19 < (/.f32 (neg.f32 x) s) < 9.99999968e37

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 14.7%

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

      1. neg-mul-1 14.7%

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

      2. unsub-neg 14.7%

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

   5. Simplified 14.7%

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

      1. *-un-lft-identity 14.7%

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

      2. cancel-sign-sub-inv 14.7%

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

      3. metadata-eval 14.7%

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

      4. add-log-exp 100.0%

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

      5. pow-exp 100.0%

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

      6. flip-+ -0.0%

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

      7. metadata-eval -0.0%

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

      8. pow-exp -0.0%

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

      9. add-log-exp -0.0%

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

      10. neg-mul-1 -0.0%

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

      11. pow-exp -0.0%

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

      12. add-log-exp -0.0%

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

      13. neg-mul-1 -0.0%

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

      14. distribute-neg-frac2 -0.0%

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

      15. distribute-neg-frac2 -0.0%

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

      16. pow-exp -0.0%

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

   7. Applied egg-rr 100.0%

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

   if 9.99999968e37 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 94.1%

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

      1. associate-*r/ 94.1%

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

      2. neg-mul-1 94.1%

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

   8. Simplified 94.1%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 40:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 19999999961012896000:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(x - s \cdot 2\right) \cdot \frac{-1}{x \cdot s}\right)}\\ \mathbf{elif}\;\frac{x}{-s} \leq 9.999999680285692 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;0.9583333333333334\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -10.0)
     0.9583333333333334
     (if (<= t_0 0.5) (+ 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 <= -10.0f) {
		tmp = 0.9583333333333334f;
	} else if (t_0 <= 0.5f) {
		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 <= (-10.0e0)) then
        tmp = 0.9583333333333334e0
    else if (t_0 <= 0.5e0) 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(-10.0))
		tmp = Float32(0.9583333333333334);
	elseif (t_0 <= Float32(0.5))
		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(-10.0))
		tmp = single(0.9583333333333334);
	elseif (t_0 <= single(0.5))
		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) < -10

   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. Taylor expanded in s around inf 0.4%

      \[\leadsto \color{blue}{\left(0.5 + -1 \cdot \frac{-0.25 \cdot \left(-1 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + \left(-0.5 \cdot {x}^{3} + 0.16666666666666666 \cdot {x}^{3}\right)\right) + \left(-0.125 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + 0.5 \cdot \left(x \cdot \left(-0.25 \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{2}\right)\right)\right)}{{s}^{3}}\right) - \left(-0.25 \cdot \frac{x}{s} + \left(-0.25 \cdot \frac{-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}}{{s}^{2}} + -0.125 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} \]

   9. Simplified 39.3%

      \[\leadsto \color{blue}{0.9583333333333334} \]

   if -10 < (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 97.2%

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

      1. associate-*r/ 97.2%

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

   5. Simplified 97.2%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 37.4%

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

      1. neg-mul-1 37.4%

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

      2. unsub-neg 37.4%

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

   5. Simplified 37.4%

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

   6. Taylor expanded in x around inf 37.4%

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

      1. mul-1-neg 37.4%

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

      2. distribute-frac-neg2 37.4%

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

   8. Simplified 37.4%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -10:\\ \;\;\;\;0.9583333333333334\\ \mathbf{elif}\;\frac{x}{-s} \leq 0.5:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.8% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;0.9583333333333334\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -10.0)
     0.9583333333333334
     (if (<= t_0 0.5) 0.5 (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= -10.0f) {
		tmp = 0.9583333333333334f;
	} else if (t_0 <= 0.5f) {
		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 <= (-10.0e0)) then
        tmp = 0.9583333333333334e0
    else if (t_0 <= 0.5e0) 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(-10.0))
		tmp = Float32(0.9583333333333334);
	elseif (t_0 <= Float32(0.5))
		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(-10.0))
		tmp = single(0.9583333333333334);
	elseif (t_0 <= single(0.5))
		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) < -10

   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. Taylor expanded in s around inf 0.4%

      \[\leadsto \color{blue}{\left(0.5 + -1 \cdot \frac{-0.25 \cdot \left(-1 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + \left(-0.5 \cdot {x}^{3} + 0.16666666666666666 \cdot {x}^{3}\right)\right) + \left(-0.125 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + 0.5 \cdot \left(x \cdot \left(-0.25 \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{2}\right)\right)\right)}{{s}^{3}}\right) - \left(-0.25 \cdot \frac{x}{s} + \left(-0.25 \cdot \frac{-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}}{{s}^{2}} + -0.125 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} \]

   9. Simplified 39.3%

      \[\leadsto \color{blue}{0.9583333333333334} \]

   if -10 < (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 90.5%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 37.4%

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

      1. neg-mul-1 37.4%

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

      2. unsub-neg 37.4%

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

   5. Simplified 37.4%

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

   6. Taylor expanded in x around inf 37.4%

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

      1. mul-1-neg 37.4%

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

      2. distribute-frac-neg2 37.4%

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

   8. Simplified 37.4%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -10:\\ \;\;\;\;0.9583333333333334\\ \mathbf{elif}\;\frac{x}{-s} \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-4 + x \cdot \left(x \cdot -0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 50.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (+ -4.0 (* x (* x -0.16666666666666666))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 50.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (-4.0f + (x * (x * -0.16666666666666666f)));
	}
	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) <= 50.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / ((-4.0e0) + (x * (x * (-0.16666666666666666e0))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(50.0))
		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(-4.0) + Float32(x * Float32(x * Float32(-0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(50.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (single(-4.0) + (x * (x * single(-0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.7%

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

      1. distribute-frac-neg 99.7%

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

      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 93.3%

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

      1. +-commutative 93.3%

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

   7. Simplified 93.3%

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

   if 50 < (/.f32 (neg.f32 x) s)

   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. Taylor expanded in x around 0 95.1%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{{s}^{3}} + 0.5 \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]

   9. Simplified 43.7%

      \[\leadsto \frac{1}{\color{blue}{-4 + x \cdot \left(x \cdot -0.16666666666666666\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-4 + x \cdot \left(x \cdot -0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.7% 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 99.9%

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

   3. Taylor expanded in x around 0 41.8%

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

      1. neg-mul-1 41.8%

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

      2. unsub-neg 41.8%

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

   5. Simplified 41.8%

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

   6. Taylor expanded in x around inf 41.8%

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

      1. associate-*r/ 41.8%

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

      2. metadata-eval 41.8%

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

   8. Simplified 41.8%

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

      1. *-commutative 41.8%

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

      2. frac-sub 45.9%

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

      3. associate-*l/ 60.6%

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

      4. *-rgt-identity 60.6%

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

      5. *-commutative 60.6%

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

   10. Applied egg-rr 60.6%

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

   if -1.9999999e-21 < x

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 91.5%

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

      1. +-commutative 91.5%

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

   7. Simplified 91.5%

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

4. Final simplification 80.7%

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

Alternative 12: 76.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\left(x \cdot s\right) \cdot \frac{\frac{-1}{x}}{x - s \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.000000229068525e-19)
   (* (* x s) (/ (/ -1.0 x) (- x (* s 2.0))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.000000229068525e-19f) {
		tmp = (x * s) * ((-1.0f / x) / (x - (s * 2.0f)));
	} 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 <= (-5.000000229068525e-19)) then
        tmp = (x * s) * (((-1.0e0) / x) / (x - (s * 2.0e0)))
    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(-5.000000229068525e-19))
		tmp = Float32(Float32(x * s) * Float32(Float32(Float32(-1.0) / x) / Float32(x - Float32(s * Float32(2.0)))));
	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(-5.000000229068525e-19))
		tmp = (x * s) * ((single(-1.0) / x) / (x - (s * single(2.0))));
	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 < -5.00000023e-19

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 41.7%

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

      1. neg-mul-1 41.7%

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

      2. unsub-neg 41.7%

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

   5. Simplified 41.7%

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

   6. Taylor expanded in x around inf 41.7%

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

      1. associate-*r/ 41.7%

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

      2. metadata-eval 41.7%

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

   8. Simplified 41.7%

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

      1. associate-/r* 39.8%

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

      2. frac-sub 40.9%

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

      3. associate-/r/ 53.5%

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

      4. *-rgt-identity 53.5%

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

      5. *-commutative 53.5%

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

   10. Applied egg-rr 53.5%

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

   if -5.00000023e-19 < x

   1. Initial program 99.7%

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

      1. distribute-frac-neg 99.7%

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

      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 90.1%

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

      1. +-commutative 90.1%

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

   7. Simplified 90.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\left(x \cdot s\right) \cdot \frac{\frac{-1}{x}}{x - s \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -10:\\ \;\;\;\;0.9583333333333334\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -10.0)
   0.9583333333333334
   (/ 1.0 (+ -1.0 (- 3.0 (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -10.0f) {
		tmp = 0.9583333333333334f;
	} 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) <= (-10.0e0)) then
        tmp = 0.9583333333333334e0
    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(-10.0))
		tmp = Float32(0.9583333333333334);
	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(-10.0))
		tmp = single(0.9583333333333334);
	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) < -10

   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. Taylor expanded in s around inf 0.4%

      \[\leadsto \color{blue}{\left(0.5 + -1 \cdot \frac{-0.25 \cdot \left(-1 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + \left(-0.5 \cdot {x}^{3} + 0.16666666666666666 \cdot {x}^{3}\right)\right) + \left(-0.125 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + 0.5 \cdot \left(x \cdot \left(-0.25 \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{2}\right)\right)\right)}{{s}^{3}}\right) - \left(-0.25 \cdot \frac{x}{s} + \left(-0.25 \cdot \frac{-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}}{{s}^{2}} + -0.125 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} \]

   9. Simplified 39.3%

      \[\leadsto \color{blue}{0.9583333333333334} \]

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 58.8%

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

      1. neg-mul-1 58.8%

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

      2. unsub-neg 58.8%

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

   5. Simplified 58.8%

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

      1. expm1-log1p-u 58.7%

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

   7. Applied egg-rr 58.7%

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

      1. expm1-undefine 58.7%

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

      2. sub-neg 58.7%

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

      3. log1p-undefine 58.7%

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

      4. rem-exp-log 58.8%

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

      5. associate-+r- 58.8%

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

      6. metadata-eval 58.8%

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

      7. metadata-eval 58.8%

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

   9. Simplified 58.8%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -10:\\ \;\;\;\;0.9583333333333334\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -10:\\ \;\;\;\;0.9583333333333334\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -10.0) 0.9583333333333334 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -10.0f) {
		tmp = 0.9583333333333334f;
	} 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) <= (-10.0e0)) then
        tmp = 0.9583333333333334e0
    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(-10.0))
		tmp = Float32(0.9583333333333334);
	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(-10.0))
		tmp = single(0.9583333333333334);
	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) < -10

   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. Taylor expanded in s around inf 0.4%

      \[\leadsto \color{blue}{\left(0.5 + -1 \cdot \frac{-0.25 \cdot \left(-1 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + \left(-0.5 \cdot {x}^{3} + 0.16666666666666666 \cdot {x}^{3}\right)\right) + \left(-0.125 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + 0.5 \cdot \left(x \cdot \left(-0.25 \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{2}\right)\right)\right)}{{s}^{3}}\right) - \left(-0.25 \cdot \frac{x}{s} + \left(-0.25 \cdot \frac{-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}}{{s}^{2}} + -0.125 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} \]

   9. Simplified 39.3%

      \[\leadsto \color{blue}{0.9583333333333334} \]

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 58.8%

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

      1. neg-mul-1 58.8%

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

      2. unsub-neg 58.8%

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

   5. Simplified 58.8%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -10:\\ \;\;\;\;0.9583333333333334\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.3% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.9583333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999969612645e-9)
   (/ 1.0 (/ x s))
   (if (<= x 1.9999999920083944e-12) 0.5 0.9583333333333334)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999969612645e-9f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 1.9999999920083944e-12f) {
		tmp = 0.5f;
	} else {
		tmp = 0.9583333333333334f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999969612645e-9)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 1.9999999920083944e-12) then
        tmp = 0.5e0
    else
        tmp = 0.9583333333333334e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999969612645e-9))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(1.9999999920083944e-12))
		tmp = Float32(0.5);
	else
		tmp = Float32(0.9583333333333334);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999969612645e-9))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(1.9999999920083944e-12))
		tmp = single(0.5);
	else
		tmp = single(0.9583333333333334);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -4.99999997e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 45.7%

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

      1. neg-mul-1 45.7%

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

      2. unsub-neg 45.7%

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

   5. Simplified 45.7%

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

   6. Taylor expanded in x around inf 43.5%

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

      1. associate-*r/ 43.5%

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

      2. neg-mul-1 43.5%

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

   8. Simplified 43.5%

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

      1. clear-num 45.7%

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

      2. inv-pow 45.7%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 59.6%

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

      5. sqr-neg 59.6%

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

      6. sqrt-unprod 45.7%

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

      7. add-sqr-sqrt 45.7%

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

   10. Applied egg-rr 45.7%

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

       1. unpow-1 45.7%

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

   12. Simplified 45.7%

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

   if -4.99999997e-9 < x < 1.99999999e-12

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 61.1%

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

   if 1.99999999e-12 < x

   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. Taylor expanded in s around inf 1.5%

      \[\leadsto \color{blue}{\left(0.5 + -1 \cdot \frac{-0.25 \cdot \left(-1 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + \left(-0.5 \cdot {x}^{3} + 0.16666666666666666 \cdot {x}^{3}\right)\right) + \left(-0.125 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + 0.5 \cdot \left(x \cdot \left(-0.25 \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{2}\right)\right)\right)}{{s}^{3}}\right) - \left(-0.25 \cdot \frac{x}{s} + \left(-0.25 \cdot \frac{-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}}{{s}^{2}} + -0.125 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} \]

   9. Simplified 39.2%

      \[\leadsto \color{blue}{0.9583333333333334} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 49.3% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{s}{x}\\ \mathbf{elif}\;x \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.9583333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999969612645e-9)
   (* 0.3333333333333333 (/ s x))
   (if (<= x 1.9999999920083944e-12) 0.5 0.9583333333333334)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999969612645e-9f) {
		tmp = 0.3333333333333333f * (s / x);
	} else if (x <= 1.9999999920083944e-12f) {
		tmp = 0.5f;
	} else {
		tmp = 0.9583333333333334f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999969612645e-9)) then
        tmp = 0.3333333333333333e0 * (s / x)
    else if (x <= 1.9999999920083944e-12) then
        tmp = 0.5e0
    else
        tmp = 0.9583333333333334e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999969612645e-9))
		tmp = Float32(Float32(0.3333333333333333) * Float32(s / x));
	elseif (x <= Float32(1.9999999920083944e-12))
		tmp = Float32(0.5);
	else
		tmp = Float32(0.9583333333333334);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999969612645e-9))
		tmp = single(0.3333333333333333) * (s / x);
	elseif (x <= single(1.9999999920083944e-12))
		tmp = single(0.5);
	else
		tmp = single(0.9583333333333334);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -4.99999997e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 45.7%

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

      1. neg-mul-1 45.7%

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

      2. unsub-neg 45.7%

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

   5. Simplified 45.7%

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

   6. Taylor expanded in x around inf 45.7%

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

      1. associate-*r/ 45.7%

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

      2. metadata-eval 45.7%

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

   8. Simplified 45.7%

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

      1. *-un-lft-identity 45.7%

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

      2. add-sqr-sqrt -0.0%

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

      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. distribute-frac-neg2 -0.0%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod -0.0%

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

      9. sqr-neg -0.0%

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

      10. sqrt-unprod -0.0%

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

      11. add-sqr-sqrt -0.0%

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

      12. fma-define -0.0%

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

      13. add-sqr-sqrt 45.7%

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

   10. Applied egg-rr 46.1%

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

       1. associate-+l+ 46.1%

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

       2. fma-undefine 46.1%

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

       3. *-rgt-identity 46.1%

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

   12. Simplified 46.1%

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

   13. Taylor expanded in x around inf 44.2%

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

   if -4.99999997e-9 < x < 1.99999999e-12

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 61.1%

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

   if 1.99999999e-12 < x

   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. Taylor expanded in s around inf 1.5%

      \[\leadsto \color{blue}{\left(0.5 + -1 \cdot \frac{-0.25 \cdot \left(-1 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + \left(-0.5 \cdot {x}^{3} + 0.16666666666666666 \cdot {x}^{3}\right)\right) + \left(-0.125 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + 0.5 \cdot \left(x \cdot \left(-0.25 \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{2}\right)\right)\right)}{{s}^{3}}\right) - \left(-0.25 \cdot \frac{x}{s} + \left(-0.25 \cdot \frac{-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}}{{s}^{2}} + -0.125 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} \]

   9. Simplified 39.2%

      \[\leadsto \color{blue}{0.9583333333333334} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 49.0% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{elif}\;x \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.9583333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999969612645e-9)
   (/ s x)
   (if (<= x 1.9999999920083944e-12) 0.5 0.9583333333333334)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999969612645e-9f) {
		tmp = s / x;
	} else if (x <= 1.9999999920083944e-12f) {
		tmp = 0.5f;
	} else {
		tmp = 0.9583333333333334f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999969612645e-9)) then
        tmp = s / x
    else if (x <= 1.9999999920083944e-12) then
        tmp = 0.5e0
    else
        tmp = 0.9583333333333334e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999969612645e-9))
		tmp = Float32(s / x);
	elseif (x <= Float32(1.9999999920083944e-12))
		tmp = Float32(0.5);
	else
		tmp = Float32(0.9583333333333334);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999969612645e-9))
		tmp = s / x;
	elseif (x <= single(1.9999999920083944e-12))
		tmp = single(0.5);
	else
		tmp = single(0.9583333333333334);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -4.99999997e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 45.7%

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

      1. neg-mul-1 45.7%

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

      2. unsub-neg 45.7%

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

   5. Simplified 45.7%

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

   6. Taylor expanded in x around inf 43.5%

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

      1. associate-*r/ 43.5%

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

      2. neg-mul-1 43.5%

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

   8. Simplified 43.5%

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

      1. div-inv 43.5%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 57.4%

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

      4. sqr-neg 57.4%

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

      5. sqrt-unprod 43.5%

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

      6. add-sqr-sqrt 43.5%

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

   10. Applied egg-rr 43.5%

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

       1. associate-*r/ 43.5%

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

       2. *-rgt-identity 43.5%

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

   12. Simplified 43.5%

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

   if -4.99999997e-9 < x < 1.99999999e-12

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 61.1%

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

   if 1.99999999e-12 < x

   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. Taylor expanded in s around inf 1.5%

      \[\leadsto \color{blue}{\left(0.5 + -1 \cdot \frac{-0.25 \cdot \left(-1 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + \left(-0.5 \cdot {x}^{3} + 0.16666666666666666 \cdot {x}^{3}\right)\right) + \left(-0.125 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + 0.5 \cdot \left(x \cdot \left(-0.25 \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{2}\right)\right)\right)}{{s}^{3}}\right) - \left(-0.25 \cdot \frac{x}{s} + \left(-0.25 \cdot \frac{-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}}{{s}^{2}} + -0.125 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} \]

   9. Simplified 39.2%

      \[\leadsto \color{blue}{0.9583333333333334} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 37.3% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.9583333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999920083944e-12) 0.5 0.9583333333333334))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999920083944e-12f) {
		tmp = 0.5f;
	} else {
		tmp = 0.9583333333333334f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.9999999920083944e-12) then
        tmp = 0.5e0
    else
        tmp = 0.9583333333333334e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999920083944e-12))
		tmp = Float32(0.5);
	else
		tmp = Float32(0.9583333333333334);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999920083944e-12))
		tmp = single(0.5);
	else
		tmp = single(0.9583333333333334);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 36.7%

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

   if 1.99999999e-12 < x

   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. Taylor expanded in s around inf 1.5%

      \[\leadsto \color{blue}{\left(0.5 + -1 \cdot \frac{-0.25 \cdot \left(-1 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + \left(-0.5 \cdot {x}^{3} + 0.16666666666666666 \cdot {x}^{3}\right)\right) + \left(-0.125 \cdot \left(x \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right)\right) + 0.5 \cdot \left(x \cdot \left(-0.25 \cdot \left(-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{2}\right)\right)\right)}{{s}^{3}}\right) - \left(-0.25 \cdot \frac{x}{s} + \left(-0.25 \cdot \frac{-1 \cdot {x}^{2} + 0.5 \cdot {x}^{2}}{{s}^{2}} + -0.125 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} \]

   9. Simplified 39.2%

      \[\leadsto \color{blue}{0.9583333333333334} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 34.2% 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 99.8%

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

3. Taylor expanded in x around 0 33.8%

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

Alternative 20: 19.5% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 0.3333333333333333)

C

float code(float x, float s) {
	return 0.3333333333333333f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.3333333333333333e0
end function

Julia

function code(x, s)
	return Float32(0.3333333333333333)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.3333333333333333);
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 36.7%

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

   1. neg-mul-1 36.7%

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

   2. unsub-neg 36.7%

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

5. Simplified 36.7%

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

6. Taylor expanded in x around inf 36.7%

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

   1. associate-*r/ 36.7%

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

   2. metadata-eval 36.7%

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

8. Simplified 36.7%

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

   1. *-un-lft-identity 36.7%

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

   2. add-sqr-sqrt 14.1%

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

   3. prod-diff 14.0%

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

   4. associate-/r/ 14.0%

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

   5. clear-num 14.0%

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

   6. distribute-frac-neg2 14.0%

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

   7. add-sqr-sqrt -0.0%

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

   8. sqrt-unprod 12.5%

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

   9. sqr-neg 12.5%

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

   10. sqrt-unprod 14.9%

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

   11. add-sqr-sqrt 14.9%

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

   12. fma-define 14.9%

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

   13. add-sqr-sqrt 36.9%

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

10. Applied egg-rr 36.7%

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

    1. associate-+l+ 36.8%

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

    2. fma-undefine 36.8%

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

    3. *-rgt-identity 36.8%

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

12. Simplified 36.8%

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

13. Taylor expanded in x around inf 18.9%

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

15. Simplified 19.7%

    \[\leadsto \color{blue}{0.3333333333333333} \]
16. Add Preprocessing

Alternative 21: 17.1% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ 0.1111111111111111 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 0.1111111111111111)

C

float code(float x, float s) {
	return 0.1111111111111111f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.1111111111111111e0
end function

Julia

function code(x, s)
	return Float32(0.1111111111111111)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.1111111111111111);
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 36.7%

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

   1. neg-mul-1 36.7%

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

   2. unsub-neg 36.7%

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

5. Simplified 36.7%

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

6. Taylor expanded in x around inf 36.7%

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

   1. associate-*r/ 36.7%

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

   2. metadata-eval 36.7%

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

8. Simplified 36.7%

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

   1. *-un-lft-identity 36.7%

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

   2. add-sqr-sqrt 14.1%

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

   3. prod-diff 14.0%

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

   4. associate-/r/ 14.0%

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

   5. clear-num 14.0%

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

   6. distribute-frac-neg2 14.0%

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

   7. add-sqr-sqrt -0.0%

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

   8. sqrt-unprod 12.5%

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

   9. sqr-neg 12.5%

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

   10. sqrt-unprod 14.9%

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

   11. add-sqr-sqrt 14.9%

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

   12. fma-define 14.9%

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

   13. add-sqr-sqrt 36.9%

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

10. Applied egg-rr 36.7%

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

    1. associate-+l+ 36.8%

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

    2. fma-undefine 36.8%

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

    3. *-rgt-identity 36.8%

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

12. Simplified 36.8%

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

13. Taylor expanded in s around 0 17.0%

    \[\leadsto \color{blue}{s \cdot \left(-0.2222222222222222 \cdot \frac{s}{{x}^{2}} + 0.3333333333333333 \cdot \frac{1}{x}\right)} \]

15. Simplified 17.3%

    \[\leadsto \color{blue}{0.1111111111111111} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))