Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 9.1s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

4. Applied egg-rr 99.8%

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

   1. add-exp-log 99.8%

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

   2. log-rec 99.8%

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

   3. log1p-expm1-u 99.8%

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

   4. log1p-define 99.8%

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

   5. rec-exp 99.8%

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

   6. expm1-log1p-u 99.8%

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

   7. distribute-neg-frac2 99.8%

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

6. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 85.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ t_1 := \frac{x}{s} + 2\\ \mathbf{if}\;t\_0 \leq -0.0020000000949949026:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t\_0 \leq 9.999999884841548 \cdot 10^{+26}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \frac{\frac{1}{s}}{\frac{s}{x}} - 4}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot t\_1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))) (t_1 (+ (/ x s) 2.0)))
   (if (<= t_0 -0.0020000000949949026)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 9.999999884841548e+26)
       (/ -1.0 (/ (- (* x (/ (/ 1.0 s) (/ s x))) 4.0) t_1))
       (/ 1.0 (/ (* x t_1) x))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float t_1 = (x / s) + 2.0f;
	float tmp;
	if (t_0 <= -0.0020000000949949026f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 9.999999884841548e+26f) {
		tmp = -1.0f / (((x * ((1.0f / s) / (s / x))) - 4.0f) / t_1);
	} else {
		tmp = 1.0f / ((x * t_1) / 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) :: t_1
    real(4) :: tmp
    t_0 = x / -s
    t_1 = (x / s) + 2.0e0
    if (t_0 <= (-0.0020000000949949026e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 9.999999884841548e+26) then
        tmp = (-1.0e0) / (((x * ((1.0e0 / s) / (s / x))) - 4.0e0) / t_1)
    else
        tmp = 1.0e0 / ((x * t_1) / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	t_1 = Float32(Float32(x / s) + Float32(2.0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0020000000949949026))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(9.999999884841548e+26))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(x * Float32(Float32(Float32(1.0) / s) / Float32(s / x))) - Float32(4.0)) / t_1));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * t_1) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	t_1 = (x / s) + single(2.0);
	tmp = single(0.0);
	if (t_0 <= single(-0.0020000000949949026))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(9.999999884841548e+26))
		tmp = single(-1.0) / (((x * ((single(1.0) / s) / (s / x))) - single(4.0)) / t_1);
	else
		tmp = single(1.0) / ((x * t_1) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 91.3%

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

      1. +-commutative 91.3%

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

   7. Simplified 91.3%

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

   if -0.00200000009 < (/.f32 (neg.f32 x) s) < 9.99999988e26

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 62.1%

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

      1. mul-1-neg 62.1%

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

      2. unsub-neg 62.1%

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

   5. Simplified 62.1%

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

      1. sub-neg 62.1%

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

      2. flip-+ 75.4%

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

      3. metadata-eval 75.4%

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

      4. distribute-neg-frac2 75.4%

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

      5. distribute-neg-frac2 75.4%

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

      6. distribute-neg-frac2 75.4%

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

   7. Applied egg-rr 75.4%

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

      1. clear-num 75.4%

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

      2. clear-num 75.4%

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

      3. frac-times 75.4%

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

      4. metadata-eval 75.4%

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

      5. add-sqr-sqrt -0.0%

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

      6. sqrt-unprod 75.8%

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

      7. sqr-neg 75.8%

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

      8. sqrt-unprod 75.2%

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

      9. add-sqr-sqrt 75.2%

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

      10. add-sqr-sqrt -0.0%

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

      11. sqrt-unprod 75.9%

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

      12. sqr-neg 75.9%

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

      13. sqrt-unprod 75.4%

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

      14. add-sqr-sqrt 75.4%

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

   9. Applied egg-rr 75.4%

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

       1. associate-/r* 75.4%

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

       2. clear-num 75.4%

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

       3. div-inv 75.4%

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

       4. *-un-lft-identity 75.4%

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

       5. times-frac 79.0%

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

   11. Applied egg-rr 79.0%

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

   if 9.99999988e26 < (/.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 64.8%

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

      1. mul-1-neg 64.8%

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

      2. unsub-neg 64.8%

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

   5. Simplified 64.8%

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

   6. Taylor expanded in x around inf 64.8%

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

      1. associate-*r/ 64.8%

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

      2. metadata-eval 64.8%

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

   8. Simplified 64.8%

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

   9. Taylor expanded in x around 0 64.8%

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

       1. mul-1-neg 64.8%

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

       2. sub-neg 64.8%

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

   11. Simplified 64.8%

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

       1. associate-*r/ 100.0%

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

       2. sub-neg 100.0%

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

       3. distribute-frac-neg2 100.0%

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

       4. add-sqr-sqrt -0.0%

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

       5. sqrt-unprod 100.0%

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

       6. sqr-neg 100.0%

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

       7. sqrt-unprod 100.0%

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

       8. add-sqr-sqrt 100.0%

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

   13. Applied egg-rr 100.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.0020000000949949026:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 9.999999884841548 \cdot 10^{+26}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \frac{\frac{1}{s}}{\frac{s}{x}} - 4}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(\frac{x}{s} + 2\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ t_1 := \frac{x}{s} + 2\\ \mathbf{if}\;t\_0 \leq -0.0020000000949949026:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t\_0 \leq 9.999999884841548 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot t\_1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))) (t_1 (+ (/ x s) 2.0)))
   (if (<= t_0 -0.0020000000949949026)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 9.999999884841548e+26)
       (/ 1.0 (/ (- 4.0 (/ x (* s (/ s x)))) t_1))
       (/ 1.0 (/ (* x t_1) x))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float t_1 = (x / s) + 2.0f;
	float tmp;
	if (t_0 <= -0.0020000000949949026f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 9.999999884841548e+26f) {
		tmp = 1.0f / ((4.0f - (x / (s * (s / x)))) / t_1);
	} else {
		tmp = 1.0f / ((x * t_1) / 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) :: t_1
    real(4) :: tmp
    t_0 = x / -s
    t_1 = (x / s) + 2.0e0
    if (t_0 <= (-0.0020000000949949026e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 9.999999884841548e+26) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / t_1)
    else
        tmp = 1.0e0 / ((x * t_1) / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	t_1 = Float32(Float32(x / s) + Float32(2.0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0020000000949949026))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(9.999999884841548e+26))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s * Float32(s / x)))) / t_1));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * t_1) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	t_1 = (x / s) + single(2.0);
	tmp = single(0.0);
	if (t_0 <= single(-0.0020000000949949026))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(9.999999884841548e+26))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / t_1);
	else
		tmp = single(1.0) / ((x * t_1) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 91.3%

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

      1. +-commutative 91.3%

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

   7. Simplified 91.3%

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

   if -0.00200000009 < (/.f32 (neg.f32 x) s) < 9.99999988e26

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 62.1%

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

      1. mul-1-neg 62.1%

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

      2. unsub-neg 62.1%

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

   5. Simplified 62.1%

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

      1. sub-neg 62.1%

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

      2. flip-+ 75.4%

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

      3. metadata-eval 75.4%

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

      4. distribute-neg-frac2 75.4%

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

      5. distribute-neg-frac2 75.4%

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

      6. distribute-neg-frac2 75.4%

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

   7. Applied egg-rr 75.4%

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

      1. clear-num 75.4%

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

      2. frac-times 78.1%

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

      3. *-un-lft-identity 78.1%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 75.8%

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

      6. sqr-neg 75.8%

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

      7. sqrt-unprod 77.9%

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

      8. add-sqr-sqrt 77.9%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 75.9%

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

      11. sqr-neg 75.9%

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

      12. sqrt-unprod 78.1%

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

      13. add-sqr-sqrt 78.1%

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

   9. Applied egg-rr 78.1%

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

   if 9.99999988e26 < (/.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 64.8%

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

      1. mul-1-neg 64.8%

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

      2. unsub-neg 64.8%

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

   5. Simplified 64.8%

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

   6. Taylor expanded in x around inf 64.8%

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

      1. associate-*r/ 64.8%

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

      2. metadata-eval 64.8%

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

   8. Simplified 64.8%

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

   9. Taylor expanded in x around 0 64.8%

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

       1. mul-1-neg 64.8%

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

       2. sub-neg 64.8%

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

   11. Simplified 64.8%

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

       1. associate-*r/ 100.0%

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

       2. sub-neg 100.0%

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

       3. distribute-frac-neg2 100.0%

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

       4. add-sqr-sqrt -0.0%

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

       5. sqrt-unprod 100.0%

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

       6. sqr-neg 100.0%

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

       7. sqrt-unprod 100.0%

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

       8. add-sqr-sqrt 100.0%

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

   13. Applied egg-rr 100.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.0020000000949949026:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 9.999999884841548 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(\frac{x}{s} + 2\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 0.5:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t\_0 \leq 9.999999884841548 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(\frac{x}{s} + 2\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 0.5)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 9.999999884841548e+26)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (/ x s)))
       (/ 1.0 (/ (* x (+ (/ x s) 2.0)) x))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= 0.5f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 9.999999884841548e+26f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / (x / s));
	} else {
		tmp = 1.0f / ((x * ((x / s) + 2.0f)) / 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 <= 0.5e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 9.999999884841548e+26) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / (x / s))
    else
        tmp = 1.0e0 / ((x * ((x / s) + 2.0e0)) / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.5))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(9.999999884841548e+26))
		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(Float32(x * Float32(Float32(x / s) + Float32(2.0))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(0.5))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(9.999999884841548e+26))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / (x / s));
	else
		tmp = single(1.0) / ((x * ((x / s) + single(2.0))) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 93.2%

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

      1. +-commutative 93.2%

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

   7. Simplified 93.2%

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

   if 0.5 < (/.f32 (neg.f32 x) s) < 9.99999988e26

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0 9.2%

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

      1. mul-1-neg 9.2%

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

      2. unsub-neg 9.2%

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

   5. Simplified 9.2%

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

      1. sub-neg 9.2%

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

      2. flip-+ 42.8%

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

      3. metadata-eval 42.8%

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

      4. distribute-neg-frac2 42.8%

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

      5. distribute-neg-frac2 42.8%

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

      6. distribute-neg-frac2 42.8%

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

   7. Applied egg-rr 42.8%

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

   8. Taylor expanded in x around inf 42.7%

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

   if 9.99999988e26 < (/.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 64.8%

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

      1. mul-1-neg 64.8%

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

      2. unsub-neg 64.8%

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

   5. Simplified 64.8%

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

   6. Taylor expanded in x around inf 64.8%

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

      1. associate-*r/ 64.8%

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

      2. metadata-eval 64.8%

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

   8. Simplified 64.8%

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

   9. Taylor expanded in x around 0 64.8%

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

       1. mul-1-neg 64.8%

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

       2. sub-neg 64.8%

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

   11. Simplified 64.8%

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

       1. associate-*r/ 100.0%

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

       2. sub-neg 100.0%

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

       3. distribute-frac-neg2 100.0%

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

       4. add-sqr-sqrt -0.0%

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

       5. sqrt-unprod 100.0%

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

       6. sqr-neg 100.0%

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

       7. sqrt-unprod 100.0%

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

       8. add-sqr-sqrt 100.0%

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

   13. Applied egg-rr 100.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.5:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 9.999999884841548 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(\frac{x}{s} + 2\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 5.7× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 5.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (/ (* x (+ (/ x s) 2.0)) x))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 5.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / ((x * ((x / s) + 2.0f)) / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 5.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / ((x * ((x / s) + 2.0e0)) / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(5.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(x / s) + Float32(2.0))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(5.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / ((x * ((x / s) + single(2.0))) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 92.8%

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

      1. +-commutative 92.8%

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

   7. Simplified 92.8%

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

   if 5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 37.9%

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

      1. mul-1-neg 37.9%

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

      2. unsub-neg 37.9%

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

   5. Simplified 37.9%

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

   6. Taylor expanded in x around inf 37.9%

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

      1. associate-*r/ 37.9%

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

      2. metadata-eval 37.9%

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

   8. Simplified 37.9%

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

   9. Taylor expanded in x around 0 37.9%

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

       1. mul-1-neg 37.9%

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

       2. sub-neg 37.9%

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

   11. Simplified 37.9%

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

       1. associate-*r/ 58.3%

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

       2. sub-neg 58.3%

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

       3. distribute-frac-neg2 58.3%

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

       4. add-sqr-sqrt -0.0%

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

       5. sqrt-unprod 75.9%

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

       6. sqr-neg 75.9%

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

       7. sqrt-unprod 58.3%

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

       8. add-sqr-sqrt 58.3%

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

   13. Applied egg-rr 58.3%

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

4. Final simplification 81.9%

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

Alternative 7: 74.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.004000000189989805:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -0.004000000189989805)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -0.004000000189989805f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} 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) <= (-0.004000000189989805e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(-0.004000000189989805))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(-0.004000000189989805))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	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) < -0.00400000019

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 91.4%

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

      1. +-commutative 91.4%

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

   7. Simplified 91.4%

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

   if -0.00400000019 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 63.1%

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

      1. mul-1-neg 63.1%

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

      2. unsub-neg 63.1%

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

   5. Simplified 63.1%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.004000000189989805:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.2% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.1%

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

      1. mul-1-neg 5.1%

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

      2. unsub-neg 5.1%

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

   5. Simplified 5.1%

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

   6. Taylor expanded in x around inf 5.1%

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

      1. associate-*r/ 5.1%

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

      2. metadata-eval 5.1%

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0 28.1%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 62.9%

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

      1. mul-1-neg 62.9%

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

      2. unsub-neg 62.9%

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

   5. Simplified 62.9%

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

4. Final simplification 48.6%

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

Alternative 9: 47.7% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 51.3%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 37.7%

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

      1. mul-1-neg 37.7%

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

      2. unsub-neg 37.7%

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

   5. Simplified 37.7%

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

   6. Taylor expanded in x around inf 37.7%

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

      1. associate-*r/ 37.7%

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

      2. metadata-eval 37.7%

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

   8. Simplified 37.7%

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

   9. Taylor expanded in x around inf 37.7%

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

4. Final simplification 46.9%

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

Alternative 10: 47.6% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 51.3%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 37.7%

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

      1. mul-1-neg 37.7%

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

      2. unsub-neg 37.7%

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

   5. Simplified 37.7%

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

   6. Taylor expanded in x around inf 37.7%

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

      1. mul-1-neg 37.7%

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

      2. distribute-frac-neg2 37.7%

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

   8. Simplified 37.7%

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

4. Final simplification 46.9%

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

Alternative 11: 46.1% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;-\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.0000000233721948e-7) (- (/ s x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.0000000233721948e-7f) {
		tmp = -(s / x);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-2.0000000233721948e-7)) then
        tmp = -(s / x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0000000233721948e-7))
		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(-2.0000000233721948e-7))
		tmp = -(s / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.00000002e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 45.4%

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

      1. mul-1-neg 45.4%

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

      2. unsub-neg 45.4%

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

   5. Simplified 45.4%

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

   6. Taylor expanded in x around inf 40.8%

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

      1. associate-*r/ 40.8%

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

      2. neg-mul-1 40.8%

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

   8. Simplified 40.8%

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

   if -2.00000002e-7 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 47.3%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;-\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.0% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 36.9%

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

Reproduce

herbie shell --seed 2024090 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))