Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 13.9s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. div-inv 99.6%

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

   2. exp-prod 80.1%

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

   3. neg-mul-1 80.1%

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

   4. exp-prod 80.1%

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

   5. pow-pow 99.5%

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

   6. div-inv 99.6%

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

4. Applied egg-rr 99.6%

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

   1. inv-pow 99.6%

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

   2. add-sqr-sqrt 99.6%

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

   3. add-sqr-sqrt 99.6%

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

   4. pow-exp 99.6%

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

   5. *-commutative 99.6%

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

   6. pow-exp 99.5%

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

   7. metadata-eval 99.5%

      \[\leadsto {\left(1 + {\left(e^{\frac{x}{s}}\right)}^{\color{blue}{\left(3 \cdot -0.3333333333333333\right)}}\right)}^{-1} \]

   8. pow-pow 99.2%

      \[\leadsto {\left(1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}\right)}^{-1} \]

   9. add-sqr-sqrt 98.6%

      \[\leadsto {\color{blue}{\left(\sqrt{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}} \cdot \sqrt{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}\right)}}^{-1} \]

   10. unpow-prod-down 98.6%

       \[\leadsto \color{blue}{{\left(\sqrt{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}\right)}^{-1} \cdot {\left(\sqrt{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}\right)}^{-1}} \]

6. Applied egg-rr 99.1%

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

   1. pow-sqr 99.7%

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

   2. metadata-eval 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in x around inf 99.7%

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

    1. exp-prod 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. div-inv 99.6%

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

   2. exp-prod 80.1%

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

   3. neg-mul-1 80.1%

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

   4. exp-prod 80.1%

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

   5. pow-pow 99.5%

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

   6. div-inv 99.6%

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

4. Applied egg-rr 99.6%

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

   1. inv-pow 99.6%

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

   2. add-sqr-sqrt 99.6%

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

   3. add-sqr-sqrt 99.6%

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

   4. pow-exp 99.6%

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

   5. *-commutative 99.6%

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

   6. pow-exp 99.5%

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

   7. metadata-eval 99.5%

      \[\leadsto {\left(1 + {\left(e^{\frac{x}{s}}\right)}^{\color{blue}{\left(3 \cdot -0.3333333333333333\right)}}\right)}^{-1} \]

   8. pow-pow 99.2%

      \[\leadsto {\left(1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}\right)}^{-1} \]

   9. add-sqr-sqrt 98.6%

      \[\leadsto {\color{blue}{\left(\sqrt{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}} \cdot \sqrt{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}\right)}}^{-1} \]

   10. unpow-prod-down 98.6%

       \[\leadsto \color{blue}{{\left(\sqrt{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}\right)}^{-1} \cdot {\left(\sqrt{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}\right)}^{-1}} \]

6. Applied egg-rr 99.1%

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

   1. pow-sqr 99.7%

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

   2. metadata-eval 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in x around inf 99.7%

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

10. Final simplification 99.7%

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

Alternative 3: 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 99.6%

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

   1. div-inv 99.6%

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

   2. exp-prod 80.1%

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

   3. neg-mul-1 80.1%

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

   4. exp-prod 80.1%

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

   5. pow-pow 99.5%

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

   6. div-inv 99.6%

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

4. Applied egg-rr 99.6%

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

   1. add-exp-log 99.6%

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

   2. log-rec 99.6%

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

   3. log1p-udef 99.6%

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

   4. pow-exp 99.7%

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

   5. neg-mul-1 99.7%

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

   6. distribute-neg-frac 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. div-inv 99.6%

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

   2. exp-prod 80.1%

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

   3. neg-mul-1 80.1%

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

   4. exp-prod 80.1%

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

   5. pow-pow 99.5%

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

   6. div-inv 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

   \[\leadsto \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}} \]
6. 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(Float32(-x) / s))))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 6: 57.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 1.0000000409184788 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -10.0)
     0.5
     (if (<= t_0 1.0000000409184788e+35)
       (/ 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 <= -10.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.0000000409184788e+35f) {
		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 <= (-10.0e0)) then
        tmp = 0.5e0
    else if (t_0 <= 1.0000000409184788e+35) 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(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-10.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.0000000409184788e+35))
		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(-10.0))
		tmp = single(0.5);
	elseif (t_0 <= single(1.0000000409184788e+35))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 28.1%

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

   if -10 < (/.f32 (neg.f32 x) s) < 1.00000004e35

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 55.1%

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

      1. mul-1-neg 55.1%

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

      2. unsub-neg 55.1%

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

   5. Simplified 55.1%

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

      1. sub-neg 55.1%

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

      2. neg-mul-1 55.1%

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

      3. add-log-exp 94.4%

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

      4. log-pow 94.8%

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

      5. metadata-eval 94.8%

         \[\leadsto \frac{1}{2 + \log \left({\left(e^{\frac{x}{s}}\right)}^{\color{blue}{\left(3 \cdot -0.3333333333333333\right)}}\right)} \]

      6. pow-pow 94.8%

         \[\leadsto \frac{1}{2 + \log \color{blue}{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}} \]

      7. flip-+ 50.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot 2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right) \cdot \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}{2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}}} \]

   7. Applied egg-rr 77.9%

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

      1. distribute-frac-neg 77.9%

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

      2. distribute-frac-neg 77.9%

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

      3. sqr-neg 77.9%

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

      4. clear-num 77.9%

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

      5. frac-times 80.0%

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

      6. *-un-lft-identity 80.0%

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

   9. Applied egg-rr 80.0%

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

       1. *-un-lft-identity 80.0%

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

       2. frac-times 77.9%

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

       3. clear-num 77.9%

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

   11. Applied egg-rr 77.9%

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

   if 1.00000004e35 < (/.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. mul-1-neg 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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. add-log-exp 100.0%

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

      4. log-pow 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \frac{1}{2 + \log \left({\left(e^{\frac{x}{s}}\right)}^{\color{blue}{\left(3 \cdot -0.3333333333333333\right)}}\right)} \]

      6. pow-pow 100.0%

         \[\leadsto \frac{1}{2 + \log \color{blue}{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}} \]

      7. flip-+ -0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot 2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right) \cdot \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}{2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}}} \]

   7. Applied egg-rr -0.0%

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

      1. distribute-frac-neg -0.0%

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

      2. distribute-frac-neg -0.0%

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

      3. sqr-neg -0.0%

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

      4. clear-num -0.0%

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

      5. frac-2neg -0.0%

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

      6. frac-times -0.0%

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

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

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

      8. add-sqr-sqrt -0.0%

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

      9. sqrt-unprod -0.0%

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

      10. sqr-neg -0.0%

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

      11. sqrt-unprod -0.0%

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

      12. add-sqr-sqrt -0.0%

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

   9. Applied egg-rr -0.0%

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

   10. Taylor expanded in x around inf 100.0%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 1.0000000409184788 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 1.0000000409184788 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -10.0)
     0.5
     (if (<= t_0 1.0000000409184788e+35)
       (/ 1.0 (/ (- 4.0 (/ x (* s (/ s x)))) (+ (/ 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 <= -10.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.0000000409184788e+35f) {
		tmp = 1.0f / ((4.0f - (x / (s * (s / x)))) / ((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 <= (-10.0e0)) then
        tmp = 0.5e0
    else if (t_0 <= 1.0000000409184788e+35) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-10.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.0000000409184788e+35))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s * Float32(s / x)))) / 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(-10.0))
		tmp = single(0.5);
	elseif (t_0 <= single(1.0000000409184788e+35))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 28.1%

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

   if -10 < (/.f32 (neg.f32 x) s) < 1.00000004e35

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 55.1%

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

      1. mul-1-neg 55.1%

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

      2. unsub-neg 55.1%

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

   5. Simplified 55.1%

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

      1. sub-neg 55.1%

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

      2. neg-mul-1 55.1%

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

      3. add-log-exp 94.4%

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

      4. log-pow 94.8%

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

      5. metadata-eval 94.8%

         \[\leadsto \frac{1}{2 + \log \left({\left(e^{\frac{x}{s}}\right)}^{\color{blue}{\left(3 \cdot -0.3333333333333333\right)}}\right)} \]

      6. pow-pow 94.8%

         \[\leadsto \frac{1}{2 + \log \color{blue}{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}} \]

      7. flip-+ 50.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot 2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right) \cdot \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}{2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}}} \]

   7. Applied egg-rr 77.9%

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

      1. distribute-frac-neg 77.9%

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

      2. distribute-frac-neg 77.9%

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

      3. sqr-neg 77.9%

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

      4. clear-num 77.9%

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

      5. frac-times 80.0%

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

      6. *-un-lft-identity 80.0%

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

   9. Applied egg-rr 80.0%

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

   if 1.00000004e35 < (/.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. mul-1-neg 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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. add-log-exp 100.0%

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

      4. log-pow 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \frac{1}{2 + \log \left({\left(e^{\frac{x}{s}}\right)}^{\color{blue}{\left(3 \cdot -0.3333333333333333\right)}}\right)} \]

      6. pow-pow 100.0%

         \[\leadsto \frac{1}{2 + \log \color{blue}{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}} \]

      7. flip-+ -0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot 2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right) \cdot \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}{2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}}} \]

   7. Applied egg-rr -0.0%

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

      1. distribute-frac-neg -0.0%

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

      2. distribute-frac-neg -0.0%

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

      3. sqr-neg -0.0%

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

      4. clear-num -0.0%

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

      5. frac-2neg -0.0%

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

      6. frac-times -0.0%

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

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

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

      8. add-sqr-sqrt -0.0%

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

      9. sqrt-unprod -0.0%

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

      10. sqr-neg -0.0%

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

      11. sqrt-unprod -0.0%

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

      12. add-sqr-sqrt -0.0%

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

   9. Applied egg-rr -0.0%

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

   10. Taylor expanded in x around inf 100.0%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 1.0000000409184788 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 1.0000000409184788 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 2.0)
     0.5
     (if (<= t_0 1.0000000409184788e+35)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (/ x s)))
       (/ 1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 2.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.0000000409184788e+35f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / (x / 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 <= 2.0e0) then
        tmp = 0.5e0
    else if (t_0 <= 1.0000000409184788e+35) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / (x / s))
    else
        tmp = 1.0e0 / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(2.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.0000000409184788e+35))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(x / 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(2.0))
		tmp = single(0.5);
	elseif (t_0 <= single(1.0000000409184788e+35))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / (x / 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) < 2

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 53.4%

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

   if 2 < (/.f32 (neg.f32 x) s) < 1.00000004e35

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0 10.7%

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

      1. mul-1-neg 10.7%

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

      2. unsub-neg 10.7%

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

   5. Simplified 10.7%

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

      1. sub-neg 10.7%

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

      2. neg-mul-1 10.7%

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

      3. add-log-exp 95.7%

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

      4. log-pow 96.4%

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

      5. metadata-eval 96.4%

         \[\leadsto \frac{1}{2 + \log \left({\left(e^{\frac{x}{s}}\right)}^{\color{blue}{\left(3 \cdot -0.3333333333333333\right)}}\right)} \]

      6. pow-pow 96.4%

         \[\leadsto \frac{1}{2 + \log \color{blue}{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}} \]

      7. flip-+ 0.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot 2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right) \cdot \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}{2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}}} \]

   7. Applied egg-rr 60.0%

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

   8. Taylor expanded in x around inf 60.0%

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

   if 1.00000004e35 < (/.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. mul-1-neg 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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. add-log-exp 100.0%

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

      4. log-pow 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \frac{1}{2 + \log \left({\left(e^{\frac{x}{s}}\right)}^{\color{blue}{\left(3 \cdot -0.3333333333333333\right)}}\right)} \]

      6. pow-pow 100.0%

         \[\leadsto \frac{1}{2 + \log \color{blue}{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}} \]

      7. flip-+ -0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot 2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right) \cdot \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}{2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}}} \]

   7. Applied egg-rr -0.0%

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

      1. distribute-frac-neg -0.0%

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

      2. distribute-frac-neg -0.0%

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

      3. sqr-neg -0.0%

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

      4. clear-num -0.0%

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

      5. frac-2neg -0.0%

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

      6. frac-times -0.0%

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

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

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

      8. add-sqr-sqrt -0.0%

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

      9. sqrt-unprod -0.0%

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

      10. sqr-neg -0.0%

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

      11. sqrt-unprod -0.0%

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

      12. add-sqr-sqrt -0.0%

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

   9. Applied egg-rr -0.0%

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

   10. Taylor expanded in x around inf 100.0%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 1.0000000409184788 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{0.3333333333333333}{\frac{s}{x \cdot -3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -10.0)
   0.5
   (/ 1.0 (+ 2.0 (/ 0.3333333333333333 (/ s (* x -3.0)))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -10.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + (0.3333333333333333f / (s / (x * -3.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-10.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 + (0.3333333333333333e0 / (s / (x * (-3.0e0)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-10.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(s / Float32(x * Float32(-3.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-10.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) + (single(0.3333333333333333) / (s / (x * single(-3.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 28.1%

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

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

   1. Initial program 99.4%

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

      1. distribute-frac-neg 99.4%

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

      2. exp-neg 99.3%

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

      3. div-inv 99.3%

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

      4. add-sqr-sqrt 17.0%

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

      5. sqrt-unprod 41.9%

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

      6. sqr-neg 41.9%

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

      7. sqrt-unprod 26.0%

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

      8. add-sqr-sqrt 40.3%

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

      9. div-inv 40.3%

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

      10. add-cbrt-cube 40.3%

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

      11. pow1/3 40.3%

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

      12. pow-flip 40.3%

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

   4. Applied egg-rr 98.8%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}} \]

   5. Taylor expanded in s around -inf 63.5%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{-2 \cdot x + -1 \cdot x}{s}\right)}} \]
   6. Step-by-step derivation

      1. distribute-rgt-out 63.5%

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

      2. metadata-eval 63.5%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{x \cdot \color{blue}{-3}}{s}\right)} \]

   7. Simplified 63.5%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}\right)}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 63.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}\right)}\right)\right)} \]

      2. expm1-udef 90.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}\right)}\right)} - 1} \]

      3. associate-+r+ 90.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left(1 + 1\right) + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}}}\right)} - 1 \]

      4. metadata-eval 90.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{2} + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}}\right)} - 1 \]

      5. associate-*r/ 90.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{2 + \color{blue}{\frac{0.3333333333333333 \cdot \left(x \cdot -3\right)}{s}}}\right)} - 1 \]

   9. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{2 + \frac{0.3333333333333333 \cdot \left(x \cdot -3\right)}{s}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 63.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{2 + \frac{0.3333333333333333 \cdot \left(x \cdot -3\right)}{s}}\right)\right)} \]

       2. expm1-log1p 63.5%

          \[\leadsto \color{blue}{\frac{1}{2 + \frac{0.3333333333333333 \cdot \left(x \cdot -3\right)}{s}}} \]

       3. associate-/l* 63.5%

          \[\leadsto \frac{1}{2 + \color{blue}{\frac{0.3333333333333333}{\frac{s}{x \cdot -3}}}} \]

   11. Simplified 63.5%

       \[\leadsto \color{blue}{\frac{1}{2 + \frac{0.3333333333333333}{\frac{s}{x \cdot -3}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{0.3333333333333333}{\frac{s}{x \cdot -3}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -10.0) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -10.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-10.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-10.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-10.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 28.1%

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

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

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 63.5%

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

      1. mul-1-neg 63.5%

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

      2. unsub-neg 63.5%

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

   5. Simplified 63.5%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s))) (if (<= t_0 2.0) 0.5 (/ 1.0 t_0))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 2.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

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

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(2.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(2.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 53.4%

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

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

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 40.4%

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

      1. mul-1-neg 40.4%

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

      2. unsub-neg 40.4%

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

   5. Simplified 40.4%

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

   6. Taylor expanded in x around inf 40.4%

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

      1. mul-1-neg 40.4%

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

      2. distribute-frac-neg 40.4%

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

   8. Simplified 40.4%

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

4. Final simplification 48.7%

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

Alternative 12: 47.1% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999747378752e-6) (/ 1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999747378752e-6f) {
		tmp = 1.0f / (x / s);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999747378752e-6)) then
        tmp = 1.0e0 / (x / s)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999747378752e-6))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999747378752e-6))
		tmp = single(1.0) / (x / s);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999975e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 52.1%

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

      1. mul-1-neg 52.1%

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

      2. unsub-neg 52.1%

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

   5. Simplified 52.1%

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

      1. sub-neg 52.1%

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

      2. neg-mul-1 52.1%

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

      3. add-log-exp 98.7%

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

      4. log-pow 98.7%

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

      5. metadata-eval 98.7%

         \[\leadsto \frac{1}{2 + \log \left({\left(e^{\frac{x}{s}}\right)}^{\color{blue}{\left(3 \cdot -0.3333333333333333\right)}}\right)} \]

      6. pow-pow 98.7%

         \[\leadsto \frac{1}{2 + \log \color{blue}{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}} \]

      7. flip-+ -0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot 2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right) \cdot \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}{2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}}} \]

   7. Applied egg-rr 40.9%

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

      1. distribute-frac-neg 40.9%

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

      2. distribute-frac-neg 40.9%

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

      3. sqr-neg 40.9%

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

      4. clear-num 40.9%

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

      5. frac-2neg 40.9%

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

      6. frac-times 40.9%

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

      7. *-un-lft-identity 40.9%

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

      8. add-sqr-sqrt 40.9%

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

      9. sqrt-unprod 40.9%

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

      10. sqr-neg 40.9%

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

      11. sqrt-unprod -0.0%

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

      12. add-sqr-sqrt 40.8%

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

   9. Applied egg-rr 40.8%

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

   10. Taylor expanded in x around inf 52.0%

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

   if -9.99999975e-6 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 47.2%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.0% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999747378752e-6) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999747378752e-6f) {
		tmp = -s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999747378752e-6)) then
        tmp = -s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999747378752e-6))
		tmp = Float32(Float32(-s) / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999747378752e-6))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999975e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 52.1%

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

      1. mul-1-neg 52.1%

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

      2. unsub-neg 52.1%

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

   5. Simplified 52.1%

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

   6. Taylor expanded in x around inf 47.1%

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

      1. associate-*r/ 47.1%

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

      2. neg-mul-1 47.1%

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

   8. Simplified 47.1%

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

   if -9.99999975e-6 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 47.2%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.9% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999747378752e-6) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999747378752e-6f) {
		tmp = s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999747378752e-6)) then
        tmp = s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999747378752e-6))
		tmp = Float32(s / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999747378752e-6))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999975e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 52.1%

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

      1. mul-1-neg 52.1%

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

      2. unsub-neg 52.1%

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

   5. Simplified 52.1%

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

      1. sub-neg 52.1%

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

      2. neg-mul-1 52.1%

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

      3. add-log-exp 98.7%

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

      4. log-pow 98.7%

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

      5. metadata-eval 98.7%

         \[\leadsto \frac{1}{2 + \log \left({\left(e^{\frac{x}{s}}\right)}^{\color{blue}{\left(3 \cdot -0.3333333333333333\right)}}\right)} \]

      6. pow-pow 98.7%

         \[\leadsto \frac{1}{2 + \log \color{blue}{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}} \]

      7. flip-+ -0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot 2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right) \cdot \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}{2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}}} \]

   7. Applied egg-rr 40.9%

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

      1. distribute-frac-neg 40.9%

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

      2. distribute-frac-neg 40.9%

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

      3. sqr-neg 40.9%

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

      4. clear-num 40.9%

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

      5. frac-2neg 40.9%

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

      6. frac-times 40.9%

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

      7. *-un-lft-identity 40.9%

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

      8. add-sqr-sqrt 40.9%

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

      9. sqrt-unprod 40.9%

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

      10. sqr-neg 40.9%

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

      11. sqrt-unprod -0.0%

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

      12. add-sqr-sqrt 40.8%

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

   9. Applied egg-rr 40.8%

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

   10. Taylor expanded in x around inf 47.0%

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

   if -9.99999975e-6 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 47.2%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.2% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 0.5)

C

float code(float x, float s) {
	return 0.5f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0
end function

Julia

function code(x, s)
	return Float32(0.5)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5);
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0 36.4%

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

4. Final simplification 36.4%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))