Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 11.9s

Alternatives: 13

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

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32((Float32(exp(1)) ^ Float32(Float32(-0.5) / Float32(s / x))) / exp(Float32(Float32(x / s) * Float32(0.5))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + ((single(2.71828182845904523536) ^ (single(-0.5) / (s / x))) / exp(((x / s) * single(0.5)))));
end

Derivation

1. Initial program 99.7%

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

   1. distribute-frac-neg 99.7%

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

   2. exp-neg 99.7%

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

   3. add-sqr-sqrt 48.4%

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

   4. sqrt-unprod 61.1%

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

   5. sqr-neg 61.1%

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

   6. sqrt-unprod 15.0%

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

   7. add-sqr-sqrt 29.8%

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

   8. add-sqr-sqrt 29.8%

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

   9. associate-/r* 29.7%

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

   10. add-sqr-sqrt 15.0%

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

   11. sqrt-unprod 28.0%

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

   12. sqr-neg 28.0%

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

   13. sqrt-unprod 13.0%

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

   14. add-sqr-sqrt 26.0%

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

3. Applied egg-rr 99.6%

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

   1. frac-2neg 99.6%

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

   2. div-inv 99.6%

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

   3. metadata-eval 99.6%

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

   4. metadata-eval 99.6%

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

   5. metadata-eval 99.6%

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

   6. distribute-neg-frac 99.6%

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

   7. frac-2neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. metadata-eval 99.6%

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

   3. unpow1/2 99.6%

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

   4. exp-prod 99.7%

      \[\leadsto \frac{1}{1 + \frac{\frac{1}{\color{blue}{e^{\frac{x}{s} \cdot 0.5}}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   5. *-commutative 99.7%

      \[\leadsto \frac{1}{1 + \frac{\frac{1}{e^{\color{blue}{0.5 \cdot \frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   6. exp-neg 99.7%

      \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{-0.5 \cdot \frac{x}{s}}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   7. distribute-lft-neg-in 99.7%

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

   8. metadata-eval 99.7%

      \[\leadsto \frac{1}{1 + \frac{e^{\color{blue}{-0.5} \cdot \frac{x}{s}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   9. associate-*r/ 99.7%

      \[\leadsto \frac{1}{1 + \frac{e^{\color{blue}{\frac{-0.5 \cdot x}{s}}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   10. *-commutative 99.7%

       \[\leadsto \frac{1}{1 + \frac{e^{\frac{\color{blue}{x \cdot -0.5}}{s}}}{\sqrt{e^{\frac{x}{s}}}}} \]

7. Simplified 99.7%

   \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{\frac{x \cdot -0.5}{s}}}}{\sqrt{e^{\frac{x}{s}}}}} \]
8. Step-by-step derivation

   1. pow1/2 99.7%

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

   2. pow-exp 99.7%

      \[\leadsto \frac{1}{1 + \frac{e^{\frac{x \cdot -0.5}{s}}}{\color{blue}{e^{\frac{x}{s} \cdot 0.5}}}} \]

9. Applied egg-rr 99.7%

   \[\leadsto \frac{1}{1 + \frac{e^{\frac{x \cdot -0.5}{s}}}{\color{blue}{e^{\frac{x}{s} \cdot 0.5}}}} \]
10. Step-by-step derivation

    1. *-un-lft-identity 99.7%

       \[\leadsto \frac{1}{1 + \frac{e^{\color{blue}{1 \cdot \frac{x \cdot -0.5}{s}}}}{e^{\frac{x}{s} \cdot 0.5}}} \]

    2. exp-prod 99.7%

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

    3. e-exp-1 99.7%

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

    4. *-commutative 99.7%

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

    5. associate-/l* 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (1.0f + (powf(expf(-0.5f), (x / s)) / expf(((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((-0.5e0)) ** (x / s)) / exp(((x / s) * 0.5e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. distribute-frac-neg 99.7%

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

   2. exp-neg 99.7%

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

   3. add-sqr-sqrt 48.4%

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

   4. sqrt-unprod 61.1%

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

   5. sqr-neg 61.1%

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

   6. sqrt-unprod 15.0%

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

   7. add-sqr-sqrt 29.8%

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

   8. add-sqr-sqrt 29.8%

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

   9. associate-/r* 29.7%

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

   10. add-sqr-sqrt 15.0%

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

   11. sqrt-unprod 28.0%

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

   12. sqr-neg 28.0%

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

   13. sqrt-unprod 13.0%

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

   14. add-sqr-sqrt 26.0%

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

3. Applied egg-rr 99.6%

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

   1. frac-2neg 99.6%

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

   2. div-inv 99.6%

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

   3. metadata-eval 99.6%

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

   4. metadata-eval 99.6%

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

   5. metadata-eval 99.6%

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

   6. distribute-neg-frac 99.6%

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

   7. frac-2neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. metadata-eval 99.6%

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

   3. unpow1/2 99.6%

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

   4. exp-prod 99.7%

      \[\leadsto \frac{1}{1 + \frac{\frac{1}{\color{blue}{e^{\frac{x}{s} \cdot 0.5}}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   5. *-commutative 99.7%

      \[\leadsto \frac{1}{1 + \frac{\frac{1}{e^{\color{blue}{0.5 \cdot \frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   6. exp-neg 99.7%

      \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{-0.5 \cdot \frac{x}{s}}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   7. distribute-lft-neg-in 99.7%

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

   8. metadata-eval 99.7%

      \[\leadsto \frac{1}{1 + \frac{e^{\color{blue}{-0.5} \cdot \frac{x}{s}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   9. associate-*r/ 99.7%

      \[\leadsto \frac{1}{1 + \frac{e^{\color{blue}{\frac{-0.5 \cdot x}{s}}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   10. *-commutative 99.7%

       \[\leadsto \frac{1}{1 + \frac{e^{\frac{\color{blue}{x \cdot -0.5}}{s}}}{\sqrt{e^{\frac{x}{s}}}}} \]

7. Simplified 99.7%

   \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{\frac{x \cdot -0.5}{s}}}}{\sqrt{e^{\frac{x}{s}}}}} \]
8. Step-by-step derivation

   1. pow1/2 99.7%

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

   2. pow-exp 99.7%

      \[\leadsto \frac{1}{1 + \frac{e^{\frac{x \cdot -0.5}{s}}}{\color{blue}{e^{\frac{x}{s} \cdot 0.5}}}} \]

9. Applied egg-rr 99.7%

   \[\leadsto \frac{1}{1 + \frac{e^{\frac{x \cdot -0.5}{s}}}{\color{blue}{e^{\frac{x}{s} \cdot 0.5}}}} \]

10. Taylor expanded in x around inf 99.7%

    \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{-0.5 \cdot \frac{x}{s}}}}{e^{\frac{x}{s} \cdot 0.5}}} \]
11. Step-by-step derivation

    1. exp-prod 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \frac{e^{\frac{-0.5 \cdot x}{s}}}{e^{\frac{x}{s} \cdot 0.5}}} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (1.0f + (expf(((-0.5f * x) / s)) / expf(((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((((-0.5e0) * x) / s)) / exp(((x / s) * 0.5e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. distribute-frac-neg 99.7%

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

   2. exp-neg 99.7%

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

   3. add-sqr-sqrt 48.4%

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

   4. sqrt-unprod 61.1%

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

   5. sqr-neg 61.1%

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

   6. sqrt-unprod 15.0%

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

   7. add-sqr-sqrt 29.8%

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

   8. add-sqr-sqrt 29.8%

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

   9. associate-/r* 29.7%

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

   10. add-sqr-sqrt 15.0%

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

   11. sqrt-unprod 28.0%

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

   12. sqr-neg 28.0%

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

   13. sqrt-unprod 13.0%

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

   14. add-sqr-sqrt 26.0%

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

3. Applied egg-rr 99.6%

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

   1. frac-2neg 99.6%

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

   2. div-inv 99.6%

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

   3. metadata-eval 99.6%

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

   4. metadata-eval 99.6%

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

   5. metadata-eval 99.6%

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

   6. distribute-neg-frac 99.6%

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

   7. frac-2neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. metadata-eval 99.6%

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

   3. unpow1/2 99.6%

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

   4. exp-prod 99.7%

      \[\leadsto \frac{1}{1 + \frac{\frac{1}{\color{blue}{e^{\frac{x}{s} \cdot 0.5}}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   5. *-commutative 99.7%

      \[\leadsto \frac{1}{1 + \frac{\frac{1}{e^{\color{blue}{0.5 \cdot \frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   6. exp-neg 99.7%

      \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{-0.5 \cdot \frac{x}{s}}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   7. distribute-lft-neg-in 99.7%

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

   8. metadata-eval 99.7%

      \[\leadsto \frac{1}{1 + \frac{e^{\color{blue}{-0.5} \cdot \frac{x}{s}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   9. associate-*r/ 99.7%

      \[\leadsto \frac{1}{1 + \frac{e^{\color{blue}{\frac{-0.5 \cdot x}{s}}}}{\sqrt{e^{\frac{x}{s}}}}} \]

   10. *-commutative 99.7%

       \[\leadsto \frac{1}{1 + \frac{e^{\frac{\color{blue}{x \cdot -0.5}}{s}}}{\sqrt{e^{\frac{x}{s}}}}} \]

7. Simplified 99.7%

   \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{\frac{x \cdot -0.5}{s}}}}{\sqrt{e^{\frac{x}{s}}}}} \]
8. Step-by-step derivation

   1. pow1/2 99.7%

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

   2. pow-exp 99.7%

      \[\leadsto \frac{1}{1 + \frac{e^{\frac{x \cdot -0.5}{s}}}{\color{blue}{e^{\frac{x}{s} \cdot 0.5}}}} \]

9. Applied egg-rr 99.7%

   \[\leadsto \frac{1}{1 + \frac{e^{\frac{x \cdot -0.5}{s}}}{\color{blue}{e^{\frac{x}{s} \cdot 0.5}}}} \]

10. Final simplification 99.7%

    \[\leadsto \frac{1}{1 + \frac{e^{\frac{-0.5 \cdot x}{s}}}{e^{\frac{x}{s} \cdot 0.5}}} \]

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 99.7%

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

2. Final simplification 99.7%

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

Alternative 5: 48.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -100:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -100.0)
     0.5
     (if (<= t_0 1.0)
       (+ 0.5 (/ (* x 0.25) s))
       (if (<= t_0 INFINITY)
         (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (/ x s)))
         (/ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -100.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else if (t_0 <= ((float) INFINITY)) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / (x / s));
	} else {
		tmp = 1.0f / (x / s);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-100.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	elseif (t_0 <= Float32(Inf))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-100.0))
		tmp = single(0.5);
	elseif (t_0 <= single(1.0))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	elseif (t_0 <= single(Inf))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / (x / s));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

2. if (/.f32 (neg.f32 x) s) < -100

   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 -100 < (/.f32 (neg.f32 x) s) < 1

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 95.7%

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

      1. associate-*r/ 95.7%

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

   4. Simplified 95.7%

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

   if 1 < (/.f32 (neg.f32 x) s) < +inf.0

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 44.9%

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

      1. mul-1-neg 44.9%

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

      2. unsub-neg 44.9%

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

   4. Simplified 44.9%

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

      1. sub-neg 44.9%

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

      2. flip-+ 36.8%

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

      3. metadata-eval 36.8%

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

      4. distribute-neg-frac 36.8%

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

      5. distribute-neg-frac 36.8%

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

      6. distribute-neg-frac 36.8%

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

   6. Applied egg-rr 36.8%

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

   7. Taylor expanded in x around inf 36.7%

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

   if +inf.0 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 46.3%

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

      1. mul-1-neg 46.3%

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

      2. unsub-neg 46.3%

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

   4. Simplified 46.3%

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

      1. sub-neg 46.3%

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

      2. flip-+ 42.6%

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

      3. metadata-eval 42.6%

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

      4. distribute-neg-frac 42.6%

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

      5. distribute-neg-frac 42.6%

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

      6. distribute-neg-frac 42.6%

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

   6. Applied egg-rr 42.6%

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

      1. distribute-frac-neg 42.6%

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

      2. distribute-frac-neg 42.6%

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

      3. sqr-neg 42.6%

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

      4. clear-num 42.6%

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

      5. frac-2neg 42.6%

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

      6. frac-times 44.0%

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

      7. *-un-lft-identity 44.0%

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

      8. add-sqr-sqrt 28.8%

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

      9. sqrt-unprod 45.4%

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

      10. sqr-neg 45.4%

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

      11. sqrt-unprod 16.3%

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

      12. add-sqr-sqrt 44.9%

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

   8. Applied egg-rr 44.9%

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

   9. Taylor expanded in x around inf 21.7%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -100:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{elif}\;\frac{-x}{s} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]

Alternative 6: 50.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 0.5f;
	} else if (t_0 <= ((float) INFINITY)) {
		tmp = 1.0f / ((4.0f - (x * ((x / s) / s))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / (x / s);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(Inf))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x * Float32(Float32(x / s) / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -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) < +inf.0

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 66.7%

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

      1. mul-1-neg 66.7%

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

      2. unsub-neg 66.7%

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

   4. Simplified 66.7%

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

      1. sub-neg 66.7%

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

      2. flip-+ 62.2%

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

      3. metadata-eval 62.2%

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

      4. distribute-neg-frac 62.2%

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

      5. distribute-neg-frac 62.2%

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

      6. distribute-neg-frac 62.2%

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

   6. Applied egg-rr 62.2%

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

      1. distribute-frac-neg 62.2%

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

      2. distribute-frac-neg 62.2%

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

      3. sqr-neg 62.2%

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

      4. clear-num 62.2%

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

      5. frac-2neg 62.2%

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

      6. frac-times 64.3%

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

      7. *-un-lft-identity 64.3%

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

      8. add-sqr-sqrt 43.1%

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

      9. sqrt-unprod 64.8%

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

      10. sqr-neg 64.8%

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

      11. sqrt-unprod 21.3%

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

      12. add-sqr-sqrt 64.1%

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

   8. Applied egg-rr 64.1%

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

      1. div-inv 66.7%

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

      2. *-commutative 66.7%

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

      3. associate-/r* 66.7%

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

      4. clear-num 66.7%

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

      5. add-sqr-sqrt -0.0%

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

      6. sqrt-unprod 61.5%

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

      7. sqr-neg 61.5%

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

      8. sqrt-unprod 67.0%

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

      9. add-sqr-sqrt 67.0%

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

   10. Applied egg-rr 67.0%

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

   if +inf.0 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 46.3%

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

      1. mul-1-neg 46.3%

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

      2. unsub-neg 46.3%

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

   4. Simplified 46.3%

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

      1. sub-neg 46.3%

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

      2. flip-+ 42.6%

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

      3. metadata-eval 42.6%

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

      4. distribute-neg-frac 42.6%

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

      5. distribute-neg-frac 42.6%

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

      6. distribute-neg-frac 42.6%

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

   6. Applied egg-rr 42.6%

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

      1. distribute-frac-neg 42.6%

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

      2. distribute-frac-neg 42.6%

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

      3. sqr-neg 42.6%

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

      4. clear-num 42.6%

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

      5. frac-2neg 42.6%

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

      6. frac-times 44.0%

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

      7. *-un-lft-identity 44.0%

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

      8. add-sqr-sqrt 28.8%

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

      9. sqrt-unprod 45.4%

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

      10. sqr-neg 45.4%

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

      11. sqrt-unprod 16.3%

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

      12. add-sqr-sqrt 44.9%

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

   8. Applied egg-rr 44.9%

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

   9. Taylor expanded in x around inf 21.7%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]

Alternative 7: 49.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -100:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -100.0)
     0.5
     (if (<= t_0 1.0) (+ 0.5 (/ (* x 0.25) s)) (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -100.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} 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 <= (-100.0e0)) then
        tmp = 0.5e0
    else if (t_0 <= 1.0e0) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    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(-100.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	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(-100.0))
		tmp = single(0.5);
	elseif (t_0 <= single(1.0))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -100

   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 -100 < (/.f32 (neg.f32 x) s) < 1

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 95.7%

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

      1. associate-*r/ 95.7%

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

   4. Simplified 95.7%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 44.9%

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

      1. mul-1-neg 44.9%

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

      2. unsub-neg 44.9%

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

   4. Simplified 44.9%

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

   5. Taylor expanded in x around inf 44.9%

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

      1. mul-1-neg 44.9%

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

      2. distribute-frac-neg 44.9%

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

   7. Simplified 44.9%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -100:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 8: 49.3% 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 99.6%

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

   2. Taylor expanded in x around 0 66.7%

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

      1. mul-1-neg 66.7%

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

      2. unsub-neg 66.7%

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

   4. Simplified 66.7%

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

4. Final simplification 53.9%

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

Alternative 9: 47.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 1.0f) {
		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 <= 1.0e0) 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(1.0))
		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(1.0))
		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) < 1

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 55.2%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 44.9%

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

      1. mul-1-neg 44.9%

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

      2. unsub-neg 44.9%

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

   4. Simplified 44.9%

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

   5. Taylor expanded in x around inf 44.9%

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

      1. mul-1-neg 44.9%

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

      2. distribute-frac-neg 44.9%

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

   7. Simplified 44.9%

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

4. Final simplification 51.4%

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

Alternative 10: 47.4% 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 51.2%

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

      1. mul-1-neg 51.2%

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

      2. unsub-neg 51.2%

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

   4. Simplified 51.2%

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

      1. sub-neg 51.2%

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

      2. flip-+ 40.5%

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

      3. metadata-eval 40.5%

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

      4. distribute-neg-frac 40.5%

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

      5. distribute-neg-frac 40.5%

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

      6. distribute-neg-frac 40.5%

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

   6. Applied egg-rr 40.5%

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

      1. distribute-frac-neg 40.5%

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

      2. distribute-frac-neg 40.5%

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

      3. sqr-neg 40.5%

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

      4. clear-num 40.5%

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

      5. frac-2neg 40.5%

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

      6. frac-times 40.5%

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

      7. *-un-lft-identity 40.5%

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

      8. add-sqr-sqrt 40.5%

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

      9. sqrt-unprod 40.5%

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

      10. sqr-neg 40.5%

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

      11. sqrt-unprod -0.0%

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

      12. add-sqr-sqrt 40.1%

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

   8. Applied egg-rr 40.1%

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

   9. Taylor expanded in x around inf 50.8%

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

   if -2.00000002e-7 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 51.4%

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

4. Final simplification 51.2%

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

Alternative 11: 46.1% 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 51.2%

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

      1. mul-1-neg 51.2%

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

      2. unsub-neg 51.2%

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

   4. Simplified 51.2%

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

   5. Taylor expanded in x around inf 43.9%

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

      1. associate-*r/ 43.9%

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

      2. neg-mul-1 43.9%

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

   7. Simplified 43.9%

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

   if -2.00000002e-7 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 51.4%

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

4. Final simplification 49.0%

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

Alternative 12: 46.0% 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 51.2%

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

      1. mul-1-neg 51.2%

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

      2. unsub-neg 51.2%

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

   4. Simplified 51.2%

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

      1. sub-neg 51.2%

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

      2. flip-+ 40.5%

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

      3. metadata-eval 40.5%

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

      4. distribute-neg-frac 40.5%

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

      5. distribute-neg-frac 40.5%

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

      6. distribute-neg-frac 40.5%

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

   6. Applied egg-rr 40.5%

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

      1. distribute-frac-neg 40.5%

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

      2. distribute-frac-neg 40.5%

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

      3. sqr-neg 40.5%

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

      4. clear-num 40.5%

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

      5. frac-2neg 40.5%

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

      6. frac-times 40.5%

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

      7. *-un-lft-identity 40.5%

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

      8. add-sqr-sqrt 40.5%

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

      9. sqrt-unprod 40.5%

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

      10. sqr-neg 40.5%

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

      11. sqrt-unprod -0.0%

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

      12. add-sqr-sqrt 40.1%

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

   8. Applied egg-rr 40.1%

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

   9. Taylor expanded in x around inf 43.4%

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

   if -2.00000002e-7 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 51.4%

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

4. Final simplification 48.9%

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

Alternative 13: 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 99.7%

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

2. Taylor expanded in x around 0 37.4%

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

3. Final simplification 37.4%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))