Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 7.8s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. neg-mul-1 99.8%

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

   2. add-sqr-sqrt 99.8%

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

   3. times-frac 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

6. Simplified 99.8%

   \[\leadsto \frac{1}{1 + e^{\color{blue}{\frac{\frac{-1}{\sqrt{s}} \cdot x}{\sqrt{s}}}}} \]
7. Add Preprocessing

Alternative 2: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{-{s}^{2}}{s}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.999999936531045e-19)
   (/ (/ (- (pow s 2.0)) s) x)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.999999936531045e-19f) {
		tmp = (-powf(s, 2.0f) / s) / x;
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.999999936531045e-19)) then
        tmp = (-(s ** 2.0e0) / s) / x
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.999999936531045e-19))
		tmp = Float32(Float32(Float32(-(s ^ Float32(2.0))) / s) / x);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.999999936531045e-19))
		tmp = (-(s ^ single(2.0)) / s) / x;
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999994e-19

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 58.7%

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

      1. neg-mul-1 58.7%

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

      2. unsub-neg 58.7%

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

   5. Simplified 58.7%

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

   6. Taylor expanded in x around inf 48.9%

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

      1. associate-*r/ 48.9%

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

      2. neg-mul-1 48.9%

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

   8. Simplified 48.9%

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

      1. neg-sub0 48.9%

         \[\leadsto \frac{\color{blue}{0 - s}}{x} \]

      2. flip-- 69.3%

         \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}}}{x} \]

      3. metadata-eval 69.3%

         \[\leadsto \frac{\frac{\color{blue}{0} - s \cdot s}{0 + s}}{x} \]

      4. pow2 69.3%

         \[\leadsto \frac{\frac{0 - \color{blue}{{s}^{2}}}{0 + s}}{x} \]

      5. add-sqr-sqrt 69.3%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}}}{x} \]

      6. sqrt-unprod 15.1%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s \cdot s}}}}{x} \]

      7. sqr-neg 15.1%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}}}{x} \]

      8. sqrt-unprod -0.0%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}{x} \]

      9. add-sqr-sqrt 68.1%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\left(-s\right)}}}{x} \]

      10. sub-neg 68.1%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{0 - s}}}{x} \]

      11. neg-sub0 68.1%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{-s}}}{x} \]

      12. add-sqr-sqrt -0.0%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}{x} \]

      13. sqrt-unprod 15.1%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}}{x} \]

      14. sqr-neg 15.1%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\sqrt{\color{blue}{s \cdot s}}}}{x} \]

      15. sqrt-unprod 69.3%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}{x} \]

      16. add-sqr-sqrt 69.3%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{s}}}{x} \]

   10. Applied egg-rr 69.3%

       \[\leadsto \frac{\color{blue}{\frac{0 - {s}^{2}}{s}}}{x} \]
   11. Step-by-step derivation

       1. sub0-neg 69.3%

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

   12. Simplified 69.3%

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

   if -1.99999994e-19 < x

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 93.6%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 4: 77.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.999999809593135 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot 2 - x\right) \cdot \frac{1}{s \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.999999809593135e-22)
   (/ 1.0 (* x (* (- (* s 2.0) x) (/ 1.0 (* s x)))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.999999809593135e-22f) {
		tmp = 1.0f / (x * (((s * 2.0f) - x) * (1.0f / (s * x))));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.999999809593135e-22)) then
        tmp = 1.0e0 / (x * (((s * 2.0e0) - x) * (1.0e0 / (s * x))))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.999999809593135e-22))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(Float32(s * Float32(2.0)) - x) * Float32(Float32(1.0) / Float32(s * x)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.999999809593135e-22))
		tmp = single(1.0) / (x * (((s * single(2.0)) - x) * (single(1.0) / (s * x))));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.9999998e-22

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 58.8%

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

      1. neg-mul-1 58.8%

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

      2. unsub-neg 58.8%

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

   5. Simplified 58.8%

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

   6. Taylor expanded in x around inf 58.6%

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

      1. sub-neg 58.6%

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

      2. associate-*r/ 58.6%

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

      3. metadata-eval 58.6%

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

      4. distribute-neg-frac 58.6%

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

      5. metadata-eval 58.6%

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

   8. Simplified 58.6%

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

      1. frac-add 62.4%

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

      2. div-inv 65.8%

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

      3. fma-define 65.8%

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

   10. Applied egg-rr 65.8%

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

       1. *-commutative 65.8%

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

       2. neg-mul-1 65.8%

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

       3. fmm-def 65.8%

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

       4. *-commutative 65.8%

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

   12. Simplified 65.8%

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

   if -5.9999998e-22 < x

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 94.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.999999809593135 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot 2 - x\right) \cdot \frac{1}{s \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + s \cdot 2}{s \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999996399175e-23)
   (/ 1.0 (* x (/ (+ x (* s 2.0)) (* s x))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999996399175e-23f) {
		tmp = 1.0f / (x * ((x + (s * 2.0f)) / (s * x)));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.9999999996399175e-23)) then
        tmp = 1.0e0 / (x * ((x + (s * 2.0e0)) / (s * x)))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999996399175e-23))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(x + Float32(s * Float32(2.0))) / Float32(s * x))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.9999999996399175e-23))
		tmp = single(1.0) / (x * ((x + (s * single(2.0))) / (s * x)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2e-23

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 59.0%

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

      1. neg-mul-1 59.0%

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

      2. unsub-neg 59.0%

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

   5. Simplified 59.0%

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

   6. Taylor expanded in x around inf 58.8%

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

      1. sub-neg 58.8%

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

      2. associate-*r/ 58.8%

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

      3. metadata-eval 58.8%

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

      4. distribute-neg-frac 58.8%

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

      5. metadata-eval 58.8%

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

   8. Simplified 58.8%

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

      1. +-commutative 58.8%

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

      2. frac-2neg 58.8%

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

      3. metadata-eval 58.8%

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

      4. frac-add 63.1%

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

      5. *-un-lft-identity 63.1%

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

      6. add-sqr-sqrt -0.0%

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

      7. sqrt-unprod 52.6%

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

      8. sqr-neg 52.6%

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

      9. sqrt-unprod 52.6%

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

      10. add-sqr-sqrt 52.6%

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

      11. add-sqr-sqrt -0.0%

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

      12. sqrt-unprod 76.2%

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

      13. sqr-neg 76.2%

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

      14. sqrt-unprod 61.8%

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

      15. add-sqr-sqrt 61.8%

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

   10. Applied egg-rr 61.8%

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

   if -2e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 95.7%

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

Alternative 6: 71.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x \cdot \frac{-1}{s}}\\ \mathbf{elif}\;x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999873689376e-5)
   (/ 1.0 (* x (/ -1.0 s)))
   (if (<= x 5.000000156871975e-23) (+ 0.5 (/ (* x 0.25) s)) 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999873689376e-5f) {
		tmp = 1.0f / (x * (-1.0f / s));
	} else if (x <= 5.000000156871975e-23f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999873689376e-5)) then
        tmp = 1.0e0 / (x * ((-1.0e0) / s))
    else if (x <= 5.000000156871975e-23) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999873689376e-5))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(-1.0) / s)));
	elseif (x <= Float32(5.000000156871975e-23))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999873689376e-5))
		tmp = single(1.0) / (x * (single(-1.0) / s));
	elseif (x <= single(5.000000156871975e-23))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -4.99999987e-5

   1. Initial program 100.0%

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

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

   6. Taylor expanded in x around inf 63.1%

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

      1. sub-neg 63.1%

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

      2. associate-*r/ 63.1%

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

      3. metadata-eval 63.1%

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

      4. distribute-neg-frac 63.1%

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

      5. metadata-eval 63.1%

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

   8. Simplified 63.1%

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

   9. Taylor expanded in x around inf 63.1%

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

   if -4.99999987e-5 < x < 5.00000016e-23

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 74.2%

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

      1. associate-*r/ 74.2%

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

   5. Simplified 74.2%

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

   if 5.00000016e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 96.2%

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

   6. Taylor expanded in x around inf 91.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x \cdot \frac{-1}{s}}\\ \mathbf{elif}\;x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999873689376e-5)
   (/ 1.0 (/ x s))
   (if (<= x 5.000000156871975e-23) (+ 0.5 (/ (* x 0.25) s)) 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999873689376e-5f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 5.000000156871975e-23f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999873689376e-5)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 5.000000156871975e-23) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999873689376e-5))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(5.000000156871975e-23))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999873689376e-5))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(5.000000156871975e-23))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -4.99999987e-5

   1. Initial program 100.0%

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

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

   6. Taylor expanded in x around inf 57.5%

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

      1. associate-*r/ 57.5%

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

      2. neg-mul-1 57.5%

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

   8. Simplified 57.5%

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

      1. clear-num 63.1%

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

      2. inv-pow 63.1%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 74.3%

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

      5. sqr-neg 74.3%

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

      6. sqrt-unprod 63.1%

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

      7. add-sqr-sqrt 63.1%

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

   10. Applied egg-rr 63.1%

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

       1. unpow-1 63.1%

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

   12. Simplified 63.1%

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

   if -4.99999987e-5 < x < 5.00000016e-23

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 74.2%

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

      1. associate-*r/ 74.2%

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

   5. Simplified 74.2%

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

   if 5.00000016e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 96.2%

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

   6. Taylor expanded in x around inf 91.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000179695649 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.000000179695649e-37)
   (/ 1.0 (- 2.0 (/ x s)))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.000000179695649e-37f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.000000179695649e-37)) then
        tmp = 1.0e0 / (2.0e0 - (x / s))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.000000179695649e-37))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.000000179695649e-37))
		tmp = single(1.0) / (single(2.0) - (x / s));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000018e-37

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 63.5%

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

      1. neg-mul-1 63.5%

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

      2. unsub-neg 63.5%

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

   5. Simplified 63.5%

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

   if -5.00000018e-37 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 95.5%

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

Alternative 9: 71.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 5.000000156871975e-23) (/ 1.0 (- 2.0 (/ x s))) 1.0))

C

float code(float x, float s) {
	float tmp;
	if (x <= 5.000000156871975e-23f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 5.000000156871975e-23) then
        tmp = 1.0e0 / (2.0e0 - (x / s))
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(5.000000156871975e-23))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(5.000000156871975e-23))
		tmp = single(1.0) / (single(2.0) - (x / s));
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5.00000016e-23

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 67.7%

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

      1. neg-mul-1 67.7%

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

      2. unsub-neg 67.7%

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

   5. Simplified 67.7%

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

   if 5.00000016e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 96.2%

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

   6. Taylor expanded in x around inf 91.3%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 70.1% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999873689376e-5)
   (/ 1.0 (/ x s))
   (if (<= x 5.000000156871975e-23) 0.5 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999873689376e-5f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 5.000000156871975e-23f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999873689376e-5)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 5.000000156871975e-23) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999873689376e-5))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(5.000000156871975e-23))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999873689376e-5))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(5.000000156871975e-23))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -4.99999987e-5

   1. Initial program 100.0%

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

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

   6. Taylor expanded in x around inf 57.5%

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

      1. associate-*r/ 57.5%

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

      2. neg-mul-1 57.5%

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

   8. Simplified 57.5%

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

      1. clear-num 63.1%

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

      2. inv-pow 63.1%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 74.3%

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

      5. sqr-neg 74.3%

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

      6. sqrt-unprod 63.1%

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

      7. add-sqr-sqrt 63.1%

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

   10. Applied egg-rr 63.1%

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

       1. unpow-1 63.1%

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

   12. Simplified 63.1%

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

   if -4.99999987e-5 < x < 5.00000016e-23

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 70.4%

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

   if 5.00000016e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 96.2%

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

   6. Taylor expanded in x around inf 91.3%

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

Alternative 11: 68.7% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999873689376e-5)
   (* s (/ 1.0 x))
   (if (<= x 5.000000156871975e-23) 0.5 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999873689376e-5f) {
		tmp = s * (1.0f / x);
	} else if (x <= 5.000000156871975e-23f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999873689376e-5)) then
        tmp = s * (1.0e0 / x)
    else if (x <= 5.000000156871975e-23) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999873689376e-5))
		tmp = Float32(s * Float32(Float32(1.0) / x));
	elseif (x <= Float32(5.000000156871975e-23))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999873689376e-5))
		tmp = s * (single(1.0) / x);
	elseif (x <= single(5.000000156871975e-23))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -4.99999987e-5

   1. Initial program 100.0%

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

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

   6. Taylor expanded in x around inf 57.5%

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

      1. associate-*r/ 57.5%

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

      2. neg-mul-1 57.5%

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

   8. Simplified 57.5%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 70.7%

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

      3. sqr-neg 70.7%

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

      4. sqrt-unprod 57.5%

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

      5. add-sqr-sqrt 57.5%

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

      6. div-inv 57.5%

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

   10. Applied egg-rr 57.5%

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

   if -4.99999987e-5 < x < 5.00000016e-23

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 70.4%

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

   if 5.00000016e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 96.2%

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

   6. Taylor expanded in x around inf 91.3%

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

Alternative 12: 68.7% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{elif}\;x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999873689376e-5)
   (/ s x)
   (if (<= x 5.000000156871975e-23) 0.5 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999873689376e-5f) {
		tmp = s / x;
	} else if (x <= 5.000000156871975e-23f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999873689376e-5)) then
        tmp = s / x
    else if (x <= 5.000000156871975e-23) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999873689376e-5))
		tmp = Float32(s / x);
	elseif (x <= Float32(5.000000156871975e-23))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999873689376e-5))
		tmp = s / x;
	elseif (x <= single(5.000000156871975e-23))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -4.99999987e-5

   1. Initial program 100.0%

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

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

   6. Taylor expanded in x around inf 57.5%

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

      1. associate-*r/ 57.5%

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

      2. neg-mul-1 57.5%

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

   8. Simplified 57.5%

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

      1. neg-sub0 57.5%

         \[\leadsto \frac{\color{blue}{0 - s}}{x} \]

      2. sub-neg 57.5%

         \[\leadsto \frac{\color{blue}{0 + \left(-s\right)}}{x} \]

      3. add-sqr-sqrt -0.0%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}{x} \]

      4. sqrt-unprod 70.7%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}{x} \]

      5. sqr-neg 70.7%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{s \cdot s}}}{x} \]

      6. sqrt-unprod 57.5%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}}{x} \]

      7. add-sqr-sqrt 57.5%

         \[\leadsto \frac{0 + \color{blue}{s}}{x} \]

   10. Applied egg-rr 57.5%

       \[\leadsto \frac{\color{blue}{0 + s}}{x} \]
   11. Step-by-step derivation

       1. +-lft-identity 57.5%

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

   12. Simplified 57.5%

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

   if -4.99999987e-5 < x < 5.00000016e-23

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 70.4%

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

   if 5.00000016e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 96.2%

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

   6. Taylor expanded in x around inf 91.3%

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

Alternative 13: 57.6% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (if (<= x 5.000000156871975e-23) 0.5 1.0))

C

float code(float x, float s) {
	float tmp;
	if (x <= 5.000000156871975e-23f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 5.000000156871975e-23) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(5.000000156871975e-23))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(5.000000156871975e-23))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5.00000016e-23

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 37.7%

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

   if 5.00000016e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 96.2%

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

   6. Taylor expanded in x around inf 91.3%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 34.8% 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 36.1%

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

Reproduce

herbie shell --seed 2024180 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))