Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 9.7s

Alternatives: 10

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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{x}{s}}\\ \frac{1}{1 + {\left(e^{{t_0}^{2}}\right)}^{\left(-t_0\right)}} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (cbrt (/ x s))))
   (/ 1.0 (+ 1.0 (pow (exp (pow t_0 2.0)) (- t_0))))))

C

float code(float x, float s) {
	float t_0 = cbrtf((x / s));
	return 1.0f / (1.0f + powf(expf(powf(t_0, 2.0f)), -t_0));
}

Julia

function code(x, s)
	t_0 = cbrt(Float32(x / s))
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (exp((t_0 ^ Float32(2.0))) ^ Float32(-t_0))))
end

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

   3. add-sqr-sqrt 48.0%

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

   4. sqrt-unprod 60.9%

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

   5. sqr-neg 60.9%

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

   6. sqrt-unprod 14.8%

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

   7. add-sqr-sqrt 29.7%

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

   8. add-cube-cbrt 29.7%

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

   9. exp-prod 29.7%

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

   10. pow-flip 29.7%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{1}{1 + {t_0}^{-0.6666666666666666} \cdot {\left(\frac{1}{t_0}\right)}^{0.3333333333333333}} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (/
    1.0
    (+
     1.0
     (* (pow t_0 -0.6666666666666666) (pow (/ 1.0 t_0) 0.3333333333333333))))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	return 1.0f / (1.0f + (powf(t_0, -0.6666666666666666f) * powf((1.0f / t_0), 0.3333333333333333f)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((x / s))
    code = 1.0e0 / (1.0e0 + ((t_0 ** (-0.6666666666666666e0)) * ((1.0e0 / t_0) ** 0.3333333333333333e0)))
end function

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32((t_0 ^ Float32(-0.6666666666666666)) * (Float32(Float32(1.0) / t_0) ^ Float32(0.3333333333333333)))))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((x / s));
	tmp = single(1.0) / (single(1.0) + ((t_0 ^ single(-0.6666666666666666)) * ((single(1.0) / t_0) ^ single(0.3333333333333333))));
end

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

   3. add-sqr-sqrt 48.0%

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

   4. sqrt-unprod 60.9%

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

   5. sqr-neg 60.9%

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

   6. sqrt-unprod 14.8%

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

   7. add-sqr-sqrt 29.7%

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

   8. add-sqr-sqrt 29.7%

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

   9. pow2 29.7%

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

   10. metadata-eval 29.7%

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

   11. pow-flip 29.7%

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

   12. add-sqr-sqrt 14.8%

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

   13. sqrt-unprod 60.8%

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

   14. sqr-neg 60.8%

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

   15. sqrt-unprod 48.0%

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

   16. add-sqr-sqrt 99.7%

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

   17. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

   1. add-cbrt-cube 99.7%

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

   2. pow1/3 99.8%

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

   3. add-sqr-sqrt 99.8%

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

   4. pow1 99.8%

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

   5. pow1/2 99.8%

      \[\leadsto \frac{1}{1 + {\left({\left({\left(e^{\frac{x}{s}}\right)}^{1} \cdot \color{blue}{{\left(e^{\frac{x}{s}}\right)}^{0.5}}\right)}^{0.3333333333333333}\right)}^{-2}} \]

   6. pow-prod-up 99.8%

      \[\leadsto \frac{1}{1 + {\left({\color{blue}{\left({\left(e^{\frac{x}{s}}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{-2}} \]

   7. metadata-eval 99.8%

      \[\leadsto \frac{1}{1 + {\left({\left({\left(e^{\frac{x}{s}}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{-2}} \]

5. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + {\color{blue}{\left({\left({\left(e^{\frac{x}{s}}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{-2}} \]
6. Step-by-step derivation

   1. pow-pow 99.8%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{1.5}\right)}^{\left(0.3333333333333333 \cdot -2\right)}}} \]

   2. metadata-eval 99.8%

      \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{\color{blue}{\left(1 + 0.5\right)}}\right)}^{\left(0.3333333333333333 \cdot -2\right)}} \]

   3. metadata-eval 99.8%

      \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{\left(1 + \color{blue}{1.5 \cdot 0.3333333333333333}\right)}\right)}^{\left(0.3333333333333333 \cdot -2\right)}} \]

   4. pow-prod-up 99.8%

      \[\leadsto \frac{1}{1 + {\color{blue}{\left({\left(e^{\frac{x}{s}}\right)}^{1} \cdot {\left(e^{\frac{x}{s}}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right)}}^{\left(0.3333333333333333 \cdot -2\right)}} \]

   5. pow1 99.8%

      \[\leadsto \frac{1}{1 + {\left(\color{blue}{e^{\frac{x}{s}}} \cdot {\left(e^{\frac{x}{s}}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right)}^{\left(0.3333333333333333 \cdot -2\right)}} \]

   6. pow-pow 99.8%

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{x}{s}} \cdot \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{1.5}\right)}^{0.3333333333333333}}\right)}^{\left(0.3333333333333333 \cdot -2\right)}} \]

   7. unpow-prod-down 99.8%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{\frac{x}{s}}\right)}^{\left(0.3333333333333333 \cdot -2\right)} \cdot {\left({\left({\left(e^{\frac{x}{s}}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}^{\left(0.3333333333333333 \cdot -2\right)}}} \]

   8. metadata-eval 99.8%

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{x}{s}}\right)}^{\color{blue}{-0.6666666666666666}} \cdot {\left({\left({\left(e^{\frac{x}{s}}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}^{\left(0.3333333333333333 \cdot -2\right)}} \]

   9. pow-pow 99.8%

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{x}{s}}\right)}^{-0.6666666666666666} \cdot {\color{blue}{\left({\left(e^{\frac{x}{s}}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right)}}^{\left(0.3333333333333333 \cdot -2\right)}} \]

   10. metadata-eval 99.8%

       \[\leadsto \frac{1}{1 + {\left(e^{\frac{x}{s}}\right)}^{-0.6666666666666666} \cdot {\left({\left(e^{\frac{x}{s}}\right)}^{\color{blue}{0.5}}\right)}^{\left(0.3333333333333333 \cdot -2\right)}} \]

   11. pow1/2 99.8%

       \[\leadsto \frac{1}{1 + {\left(e^{\frac{x}{s}}\right)}^{-0.6666666666666666} \cdot {\color{blue}{\left(\sqrt{e^{\frac{x}{s}}}\right)}}^{\left(0.3333333333333333 \cdot -2\right)}} \]

   12. metadata-eval 99.8%

       \[\leadsto \frac{1}{1 + {\left(e^{\frac{x}{s}}\right)}^{-0.6666666666666666} \cdot {\left(\sqrt{e^{\frac{x}{s}}}\right)}^{\color{blue}{-0.6666666666666666}}} \]

7. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{\frac{x}{s}}\right)}^{-0.6666666666666666} \cdot {\left(\sqrt{e^{\frac{x}{s}}}\right)}^{-0.6666666666666666}}} \]

8. Taylor expanded in x around inf 99.8%

   \[\leadsto \frac{1}{1 + {\left(e^{\frac{x}{s}}\right)}^{-0.6666666666666666} \cdot \color{blue}{{\left(\frac{1}{e^{\frac{x}{s}}}\right)}^{0.3333333333333333}}} \]

9. Final simplification 99.8%

   \[\leadsto \frac{1}{1 + {\left(e^{\frac{x}{s}}\right)}^{-0.6666666666666666} \cdot {\left(\frac{1}{e^{\frac{x}{s}}}\right)}^{0.3333333333333333}} \]

Alternative 3: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + (exp((-2.0e0)) ** ((x / s) * 0.5e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

   3. add-sqr-sqrt 48.0%

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

   4. sqrt-unprod 60.9%

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

   5. sqr-neg 60.9%

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

   6. sqrt-unprod 14.8%

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

   7. add-sqr-sqrt 29.7%

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

   8. add-cube-cbrt 29.7%

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

   9. exp-prod 29.7%

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

   10. pow-flip 29.7%

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

3. Applied egg-rr 99.8%

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

   1. pow-neg 99.8%

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

   2. inv-pow 99.8%

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

   3. pow-exp 99.8%

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

   4. unpow2 99.8%

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

   5. add-cube-cbrt 99.8%

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

   6. pow-exp 99.8%

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

   7. *-commutative 99.8%

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

   8. pow-exp 99.8%

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

   9. add-sqr-sqrt 99.7%

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

   10. sqrt-unprod 99.5%

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

   11. pow-prod-down 99.5%

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

   12. prod-exp 99.5%

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

   13. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

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

   1. sqrt-pow1 99.8%

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

   2. add-log-exp 99.8%

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

   3. div-inv 99.8%

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

   4. metadata-eval 99.8%

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

   5. metadata-eval 99.8%

      \[\leadsto \frac{1}{1 + {\left(e^{-2}\right)}^{\log \left(e^{\frac{x}{s} \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}\right)}} \]

   6. pow-exp 99.7%

      \[\leadsto \frac{1}{1 + {\left(e^{-2}\right)}^{\log \color{blue}{\left({\left(e^{\frac{x}{s}}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right)}}} \]

   7. log-pow 99.8%

      \[\leadsto \frac{1}{1 + {\left(e^{-2}\right)}^{\color{blue}{\left(\left(1.5 \cdot 0.3333333333333333\right) \cdot \log \left(e^{\frac{x}{s}}\right)\right)}}} \]

   8. metadata-eval 99.8%

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

   9. add-log-exp 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 5: 49.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.1%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 63.0%

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

      1. mul-1-neg 63.0%

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

      2. unsub-neg 63.0%

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

   4. Simplified 63.0%

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

4. Final simplification 51.5%

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

Alternative 6: 47.4% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 55.5%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 39.2%

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

      1. mul-1-neg 39.2%

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

      2. unsub-neg 39.2%

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

   4. Simplified 39.2%

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

   5. Taylor expanded in x around inf 39.2%

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

      1. associate-*r/ 39.2%

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

      2. neg-mul-1 39.2%

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

   7. Simplified 39.2%

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

4. Final simplification 49.4%

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

Alternative 7: 47.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.0000000233721948e-7) (/ 1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.0000000233721948e-7f) {
		tmp = 1.0f / (x / s);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-2.0000000233721948e-7)) then
        tmp = 1.0e0 / (x / s)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0000000233721948e-7))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.0000000233721948e-7))
		tmp = single(1.0) / (x / s);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.00000002e-7

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 47.7%

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

      1. mul-1-neg 47.7%

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

      2. unsub-neg 47.7%

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

   4. Simplified 47.7%

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

   5. Taylor expanded in x around inf 44.2%

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

      1. associate-*r/ 44.2%

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

      2. neg-mul-1 44.2%

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

   7. Simplified 44.2%

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

      1. remove-double-neg 44.2%

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

      2. frac-2neg 44.2%

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

      3. clear-num 47.3%

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

      4. inv-pow 47.3%

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

      5. add-sqr-sqrt 47.3%

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

      6. sqrt-unprod 57.0%

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

      7. sqr-neg 57.0%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt 47.0%

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

   9. Applied egg-rr 47.0%

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

       1. unpow-1 47.0%

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

   11. Simplified 47.0%

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

   if -2.00000002e-7 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 50.0%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8: 45.8% accurate, 17.6× 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 99.8%

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

   2. Taylor expanded in x around 0 47.7%

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

      1. mul-1-neg 47.7%

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

      2. unsub-neg 47.7%

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

   4. Simplified 47.7%

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

   5. Taylor expanded in x around inf 44.2%

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

      1. associate-*r/ 44.2%

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

      2. neg-mul-1 44.2%

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

   7. Simplified 44.2%

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

   if -2.00000002e-7 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 50.0%

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

4. Final simplification 48.2%

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

Alternative 9: 45.7% accurate, 21.1× 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(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 99.8%

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

   2. Taylor expanded in x around 0 47.7%

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

      1. mul-1-neg 47.7%

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

      2. unsub-neg 47.7%

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

   4. Simplified 47.7%

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

   5. Taylor expanded in x around inf 44.2%

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

      1. associate-*r/ 44.2%

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

      2. neg-mul-1 44.2%

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

   7. Simplified 44.2%

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

      1. remove-double-neg 44.2%

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

      2. frac-2neg 44.2%

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

      3. clear-num 47.3%

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

      4. expm1-log1p-u 47.3%

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

      5. expm1-udef 94.1%

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

      6. clear-num 94.1%

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

      7. add-sqr-sqrt 94.1%

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

      8. sqrt-unprod 94.1%

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

      9. sqr-neg 94.1%

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

      10. sqrt-unprod -0.0%

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

      11. add-sqr-sqrt 93.8%

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

   9. Applied egg-rr 93.8%

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

       1. expm1-def 43.8%

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

       2. expm1-log1p 43.9%

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

   11. Simplified 43.9%

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

   if -2.00000002e-7 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 50.0%

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

4. Final simplification 48.1%

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

Alternative 10: 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. Taylor expanded in x around 0 37.1%

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

3. Final simplification 37.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))