Logistic function

Percentage Accurate: 99.0% → 99.3%

Time: 1.3min

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

  1. x = -6.225873227348942e-27
  2. s = 1.8661978417986744e-34

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: 99.0% 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: 99.3% 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 99.1%

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

   1. distribute-frac-neg 99.1%

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

   2. exp-neg 99.0%

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

3. Applied egg-rr 99.0%

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

   1. inv-pow 99.0%

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

   2. metadata-eval 99.0%

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

   3. sqrt-pow2 98.9%

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

   4. add-exp-log 98.8%

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

   5. log-rec 98.9%

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

   6. log1p-udef 99.1%

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

   7. sqrt-pow2 99.2%

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

   8. metadata-eval 99.2%

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

   9. inv-pow 99.2%

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

   10. rec-exp 99.3%

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

5. Applied egg-rr 99.3%

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

   1. distribute-neg-frac 99.3%

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 2: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. distribute-frac-neg 99.1%

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

   2. exp-neg 99.0%

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

3. Applied egg-rr 99.0%

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

   1. inv-pow 99.1%

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

   2. rec-exp 99.1%

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

5. Applied egg-rr 99.1%

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

6. Final simplification 99.1%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(1 + \mathsf{expm1}\left(\frac{-x}{s}\right)\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (+ 1.0 (expm1 (/ (- x) s))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + (1.0f + expm1f((-x / s))));
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + expm1(Float32(Float32(-x) / s)))))
end

Derivation

1. Initial program 99.1%

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

   1. distribute-frac-neg 99.1%

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

   2. exp-neg 99.0%

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

3. Applied egg-rr 99.0%

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

   1. inv-pow 99.0%

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

   2. metadata-eval 99.0%

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

   3. sqrt-pow2 98.9%

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

   4. expm1-log1p-u 98.8%

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

   5. expm1-udef 98.8%

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

   6. log1p-udef 98.8%

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

   7. sqrt-pow2 99.0%

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

   8. metadata-eval 99.0%

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

   9. inv-pow 99.0%

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

   10. add-exp-log 99.0%

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

   11. rec-exp 99.1%

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

5. Applied egg-rr 99.1%

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

   1. associate--l+ 99.1%

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

   2. expm1-def 99.1%

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

   3. distribute-neg-frac 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

2. Final simplification 99.1%

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

Alternative 5: 80.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t_0 \leq 50:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \left(\frac{1}{s} \cdot \left(x \cdot \frac{x}{s}\right)\right) - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot \left(s \cdot 2\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -1.0)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 50.0)
       (/ 1.0 (+ 2.0 (- (* 0.5 (* (/ 1.0 s) (* x (/ x s)))) (/ x s))))
       (/ (/ (* s (* s 2.0)) x) x)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 50.0f) {
		tmp = 1.0f / (2.0f + ((0.5f * ((1.0f / s) * (x * (x / s)))) - (x / s)));
	} else {
		tmp = ((s * (s * 2.0f)) / x) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-1.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 50.0e0) then
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * ((1.0e0 / s) * (x * (x / s)))) - (x / s)))
    else
        tmp = ((s * (s * 2.0e0)) / x) / x
    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(-1.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(Float32(Float32(1.0) / s) * Float32(x * Float32(x / s)))) - Float32(x / s))));
	else
		tmp = Float32(Float32(Float32(s * Float32(s * Float32(2.0))) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(50.0))
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * ((single(1.0) / s) * (x * (x / s)))) - (x / s)));
	else
		tmp = ((s * (s * single(2.0))) / x) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 46.6%

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

   if -1 < (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 42.1%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. mul-1-neg 42.1%

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

      2. unsub-neg 42.1%

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

      3. unpow2 42.1%

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

      4. unpow2 42.1%

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

      5. times-frac 86.6%

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

   4. Simplified 86.6%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. clear-num 86.6%

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

      2. frac-times 86.6%

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

      3. *-un-lft-identity 86.6%

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

   6. Applied egg-rr 86.6%

      \[\leadsto \frac{1}{2 + \left(0.5 \cdot \color{blue}{\frac{x}{\frac{s}{x} \cdot s}} - \frac{x}{s}\right)} \]
   7. Step-by-step derivation

      1. clear-num 86.6%

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

      2. inv-pow 86.6%

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

      3. *-commutative 86.6%

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

   8. Applied egg-rr 86.6%

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

      1. unpow-1 86.6%

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

      2. associate-/l* 86.6%

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

   10. Simplified 86.6%

       \[\leadsto \frac{1}{2 + \left(0.5 \cdot \color{blue}{\frac{1}{\frac{s}{\frac{x}{\frac{s}{x}}}}} - \frac{x}{s}\right)} \]
   11. Step-by-step derivation

       1. associate-/r/ 86.6%

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

       2. div-inv 86.6%

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

       3. clear-num 86.6%

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

   12. Applied egg-rr 86.6%

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

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

   1. Initial program 97.7%

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

   2. Taylor expanded in x around 0 6.6%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. mul-1-neg 6.6%

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

      2. unsub-neg 6.6%

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

      3. unpow2 6.6%

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

      4. unpow2 6.6%

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

      5. times-frac 7.3%

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

   4. Simplified 7.3%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. clear-num 7.3%

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

      2. frac-times 7.3%

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

      3. *-un-lft-identity 7.3%

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

   6. Applied egg-rr 7.3%

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

   7. Taylor expanded in x around inf 6.6%

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

      1. associate-*r/ 6.6%

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

      2. unpow2 6.6%

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

      3. associate-/r* 58.1%

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

      4. unpow2 58.1%

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

      5. associate-*r* 58.1%

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

   9. Simplified 58.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \left(\frac{1}{s} \cdot \left(x \cdot \frac{x}{s}\right)\right) - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot \left(s \cdot 2\right)}{x}}{x}\\ \end{array} \]

Alternative 6: 80.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t_0 \leq 50:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{1}{\frac{s}{x} \cdot \frac{s}{x}} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot \left(s \cdot 2\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -1.0)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 50.0)
       (/ 1.0 (+ 2.0 (- (* 0.5 (/ 1.0 (* (/ s x) (/ s x)))) (/ x s))))
       (/ (/ (* s (* s 2.0)) x) x)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 50.0f) {
		tmp = 1.0f / (2.0f + ((0.5f * (1.0f / ((s / x) * (s / x)))) - (x / s)));
	} else {
		tmp = ((s * (s * 2.0f)) / x) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-1.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 50.0e0) then
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * (1.0e0 / ((s / x) * (s / x)))) - (x / s)))
    else
        tmp = ((s * (s * 2.0e0)) / x) / x
    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(-1.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(Float32(1.0) / Float32(Float32(s / x) * Float32(s / x)))) - Float32(x / s))));
	else
		tmp = Float32(Float32(Float32(s * Float32(s * Float32(2.0))) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(50.0))
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * (single(1.0) / ((s / x) * (s / x)))) - (x / s)));
	else
		tmp = ((s * (s * single(2.0))) / x) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 46.6%

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

   if -1 < (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 42.1%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. mul-1-neg 42.1%

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

      2. unsub-neg 42.1%

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

      3. unpow2 42.1%

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

      4. unpow2 42.1%

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

      5. times-frac 86.6%

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

   4. Simplified 86.6%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. clear-num 86.6%

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

      2. clear-num 86.6%

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

      3. frac-times 86.6%

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

      4. metadata-eval 86.6%

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

   6. Applied egg-rr 86.6%

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

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

   1. Initial program 97.7%

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

   2. Taylor expanded in x around 0 6.6%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. mul-1-neg 6.6%

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

      2. unsub-neg 6.6%

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

      3. unpow2 6.6%

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

      4. unpow2 6.6%

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

      5. times-frac 7.3%

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

   4. Simplified 7.3%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. clear-num 7.3%

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

      2. frac-times 7.3%

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

      3. *-un-lft-identity 7.3%

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

   6. Applied egg-rr 7.3%

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

   7. Taylor expanded in x around inf 6.6%

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

      1. associate-*r/ 6.6%

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

      2. unpow2 6.6%

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

      3. associate-/r* 58.1%

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

      4. unpow2 58.1%

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

      5. associate-*r* 58.1%

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

   9. Simplified 58.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{1}{\frac{s}{x} \cdot \frac{s}{x}} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot \left(s \cdot 2\right)}{x}}{x}\\ \end{array} \]

Alternative 7: 80.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t_0 \leq 50:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot \left(s \cdot 2\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -1.0)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 50.0)
       (/ 1.0 (+ 2.0 (- (* 0.5 (* (/ x s) (/ x s))) (/ x s))))
       (/ (/ (* s (* s 2.0)) x) x)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 50.0f) {
		tmp = 1.0f / (2.0f + ((0.5f * ((x / s) * (x / s))) - (x / s)));
	} else {
		tmp = ((s * (s * 2.0f)) / x) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-1.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 50.0e0) then
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * ((x / s) * (x / s))) - (x / s)))
    else
        tmp = ((s * (s * 2.0e0)) / x) / x
    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(-1.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(Float32(x / s) * Float32(x / s))) - Float32(x / s))));
	else
		tmp = Float32(Float32(Float32(s * Float32(s * Float32(2.0))) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(50.0))
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * ((x / s) * (x / s))) - (x / s)));
	else
		tmp = ((s * (s * single(2.0))) / x) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 46.6%

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

   if -1 < (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 42.1%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. mul-1-neg 42.1%

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

      2. unsub-neg 42.1%

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

      3. unpow2 42.1%

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

      4. unpow2 42.1%

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

      5. times-frac 86.6%

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

   4. Simplified 86.6%

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

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

   1. Initial program 97.7%

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

   2. Taylor expanded in x around 0 6.6%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. mul-1-neg 6.6%

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

      2. unsub-neg 6.6%

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

      3. unpow2 6.6%

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

      4. unpow2 6.6%

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

      5. times-frac 7.3%

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

   4. Simplified 7.3%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. clear-num 7.3%

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

      2. frac-times 7.3%

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

      3. *-un-lft-identity 7.3%

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

   6. Applied egg-rr 7.3%

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

   7. Taylor expanded in x around inf 6.6%

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

      1. associate-*r/ 6.6%

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

      2. unpow2 6.6%

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

      3. associate-/r* 58.1%

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

      4. unpow2 58.1%

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

      5. associate-*r* 58.1%

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

   9. Simplified 58.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot \left(s \cdot 2\right)}{x}}{x}\\ \end{array} \]

Alternative 8: 80.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t_0 \leq 50:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x}{s \cdot \frac{s}{x}} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot \left(s \cdot 2\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -1.0)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 50.0)
       (/ 1.0 (+ 2.0 (- (* 0.5 (/ x (* s (/ s x)))) (/ x s))))
       (/ (/ (* s (* s 2.0)) x) x)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 50.0f) {
		tmp = 1.0f / (2.0f + ((0.5f * (x / (s * (s / x)))) - (x / s)));
	} else {
		tmp = ((s * (s * 2.0f)) / x) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-1.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 50.0e0) then
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * (x / (s * (s / x)))) - (x / s)))
    else
        tmp = ((s * (s * 2.0e0)) / x) / x
    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(-1.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(x / Float32(s * Float32(s / x)))) - Float32(x / s))));
	else
		tmp = Float32(Float32(Float32(s * Float32(s * Float32(2.0))) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(50.0))
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * (x / (s * (s / x)))) - (x / s)));
	else
		tmp = ((s * (s * single(2.0))) / x) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 46.6%

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

   if -1 < (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 42.1%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. mul-1-neg 42.1%

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

      2. unsub-neg 42.1%

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

      3. unpow2 42.1%

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

      4. unpow2 42.1%

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

      5. times-frac 86.6%

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

   4. Simplified 86.6%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. clear-num 86.6%

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

      2. frac-times 86.6%

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

      3. *-un-lft-identity 86.6%

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

   6. Applied egg-rr 86.6%

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

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

   1. Initial program 97.7%

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

   2. Taylor expanded in x around 0 6.6%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. mul-1-neg 6.6%

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

      2. unsub-neg 6.6%

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

      3. unpow2 6.6%

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

      4. unpow2 6.6%

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

      5. times-frac 7.3%

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

   4. Simplified 7.3%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. clear-num 7.3%

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

      2. frac-times 7.3%

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

      3. *-un-lft-identity 7.3%

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

   6. Applied egg-rr 7.3%

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

   7. Taylor expanded in x around inf 6.6%

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

      1. associate-*r/ 6.6%

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

      2. unpow2 6.6%

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

      3. associate-/r* 58.1%

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

      4. unpow2 58.1%

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

      5. associate-*r* 58.1%

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

   9. Simplified 58.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x}{s \cdot \frac{s}{x}} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot \left(s \cdot 2\right)}{x}}{x}\\ \end{array} \]

Alternative 9: 80.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -3:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{s \cdot s}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -3.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 2.0) (+ 0.5 (* (/ x s) 0.25)) (* 2.0 (/ (/ (* s s) x) x))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -3.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 2.0f * (((s * s) / x) / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-3.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 2.0e0 * (((s * s) / x) / x)
    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(-3.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(Float32(s * s) / x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-3.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(2.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(2.0) * (((s * s) / x) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -3

   1. Initial program 99.8%

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

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 48.8%

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

   5. Taylor expanded in x around inf 48.7%

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

   if -3 < (/.f32 (neg.f32 x) s) < 2

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 91.4%

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

      1. *-commutative 91.4%

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

   4. Simplified 91.4%

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

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

   1. Initial program 97.4%

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

   2. Taylor expanded in x around 0 5.3%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. mul-1-neg 5.3%

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

      2. unsub-neg 5.3%

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

      3. unpow2 5.3%

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

      4. unpow2 5.3%

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

      5. times-frac 13.0%

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

   4. Simplified 13.0%

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

   5. Taylor expanded in x around inf 5.3%

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

      1. unpow2 5.3%

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

      2. associate-/r* 38.4%

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

      3. unpow2 38.4%

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

   7. Simplified 38.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -3:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{s \cdot s}{x}}{x}\\ \end{array} \]

Alternative 10: 80.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -3:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot \left(s \cdot 2\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -3.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 2.0) (+ 0.5 (* (/ x s) 0.25)) (/ (/ (* s (* s 2.0)) x) x)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -3.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = ((s * (s * 2.0f)) / x) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-3.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = ((s * (s * 2.0e0)) / x) / x
    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(-3.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(Float32(s * Float32(s * Float32(2.0))) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-3.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(2.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = ((s * (s * single(2.0))) / x) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -3

   1. Initial program 99.8%

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

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 48.8%

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

   5. Taylor expanded in x around inf 48.7%

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

   if -3 < (/.f32 (neg.f32 x) s) < 2

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 91.4%

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

      1. *-commutative 91.4%

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

   4. Simplified 91.4%

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

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

   1. Initial program 97.4%

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

   2. Taylor expanded in x around 0 5.3%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. mul-1-neg 5.3%

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

      2. unsub-neg 5.3%

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

      3. unpow2 5.3%

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

      4. unpow2 5.3%

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

      5. times-frac 13.0%

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

   4. Simplified 13.0%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. clear-num 13.0%

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

      2. frac-times 13.0%

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

      3. *-un-lft-identity 13.0%

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

   6. Applied egg-rr 13.0%

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

   7. Taylor expanded in x around inf 5.3%

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

      1. associate-*r/ 5.3%

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

      2. unpow2 5.3%

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

      3. associate-/r* 38.4%

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

      4. unpow2 38.4%

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

      5. associate-*r* 38.4%

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

   9. Simplified 38.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -3:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot \left(s \cdot 2\right)}{x}}{x}\\ \end{array} \]

Alternative 11: 80.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -3:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot \left(s \cdot 2\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -3.0)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 2.0) (+ 0.5 (* (/ x s) 0.25)) (/ (/ (* s (* s 2.0)) x) x)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -3.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = ((s * (s * 2.0f)) / x) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-3.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = ((s * (s * 2.0e0)) / x) / x
    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(-3.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(Float32(s * Float32(s * Float32(2.0))) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-3.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(2.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = ((s * (s * single(2.0))) / x) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -3

   1. Initial program 99.8%

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

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 48.8%

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

   if -3 < (/.f32 (neg.f32 x) s) < 2

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 91.4%

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

      1. *-commutative 91.4%

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

   4. Simplified 91.4%

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

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

   1. Initial program 97.4%

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

   2. Taylor expanded in x around 0 5.3%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. mul-1-neg 5.3%

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

      2. unsub-neg 5.3%

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

      3. unpow2 5.3%

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

      4. unpow2 5.3%

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

      5. times-frac 13.0%

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

   4. Simplified 13.0%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. clear-num 13.0%

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

      2. frac-times 13.0%

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

      3. *-un-lft-identity 13.0%

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

   6. Applied egg-rr 13.0%

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

   7. Taylor expanded in x around inf 5.3%

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

      1. associate-*r/ 5.3%

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

      2. unpow2 5.3%

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

      3. associate-/r* 38.4%

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

      4. unpow2 38.4%

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

      5. associate-*r* 38.4%

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

   9. Simplified 38.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -3:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot \left(s \cdot 2\right)}{x}}{x}\\ \end{array} \]

Alternative 12: 74.9% accurate, 8.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -10.0) (- 1.0 (/ s x)) (+ 0.5 (* (/ x s) 0.25))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -10.0f) {
		tmp = 1.0f - (s / x);
	} else {
		tmp = 0.5f + ((x / s) * 0.25f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-10.0e0)) then
        tmp = 1.0e0 - (s / x)
    else
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    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(Float32(1.0) - Float32(s / x));
	else
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-10.0))
		tmp = single(1.0) - (s / x);
	else
		tmp = single(0.5) + ((x / s) * single(0.25));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 49.8%

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

   5. Taylor expanded in x around inf 49.7%

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

      1. +-commutative 49.7%

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

      2. mul-1-neg 49.7%

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

      3. unsub-neg 49.7%

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

   7. Simplified 49.7%

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

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

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0 75.4%

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

      1. *-commutative 75.4%

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

   4. Simplified 75.4%

      \[\leadsto \color{blue}{0.5 + \frac{x}{s} \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

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

Alternative 13: 75.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -3:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -3.0) (/ 1.0 (+ 1.0 (/ s x))) (+ 0.5 (* (/ x s) 0.25))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -3.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else {
		tmp = 0.5f + ((x / s) * 0.25f);
	}
	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) <= (-3.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-3.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	else
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-3.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	else
		tmp = single(0.5) + ((x / s) * single(0.25));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -3

   1. Initial program 99.8%

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

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 48.8%

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

   5. Taylor expanded in x around inf 48.7%

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

   if -3 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0 75.8%

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

      1. *-commutative 75.8%

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

   4. Simplified 75.8%

      \[\leadsto \color{blue}{0.5 + \frac{x}{s} \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -3:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \end{array} \]

Alternative 14: 62.3% 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 99.1%

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

2. Taylor expanded in x around 0 60.0%

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

3. Final simplification 60.0%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023278 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))