Average Error: 0.5 → 0.6
Time: 3.8s
Precision: binary64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\sqrt[3]{e^{\sqrt[3]{\log \log \left(1 + e^{x}\right)} \cdot \left(3 \cdot \left({\left(\sqrt[3]{\sqrt[3]{\log \log \left(1 + e^{x}\right)}}\right)}^{4} \cdot {\left(\sqrt[3]{\sqrt[3]{\log \log \left(1 + e^{x}\right)}}\right)}^{2}\right)\right)}} - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\sqrt[3]{e^{\sqrt[3]{\log \log \left(1 + e^{x}\right)} \cdot \left(3 \cdot \left({\left(\sqrt[3]{\sqrt[3]{\log \log \left(1 + e^{x}\right)}}\right)}^{4} \cdot {\left(\sqrt[3]{\sqrt[3]{\log \log \left(1 + e^{x}\right)}}\right)}^{2}\right)\right)}} - x \cdot y
(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))
↓
(FPCore (x y)
 :precision binary64
 (-
  (cbrt
   (exp
    (*
     (cbrt (log (log (+ 1.0 (exp x)))))
     (*
      3.0
      (*
       (pow (cbrt (cbrt (log (log (+ 1.0 (exp x)))))) 4.0)
       (pow (cbrt (cbrt (log (log (+ 1.0 (exp x)))))) 2.0))))))
  (* x y)))
double code(double x, double y) {
	return log(1.0 + exp(x)) - (x * y);
}
↓
double code(double x, double y) {
	return cbrt(exp(cbrt(log(log(1.0 + exp(x)))) * (3.0 * (pow(cbrt(cbrt(log(log(1.0 + exp(x))))), 4.0) * pow(cbrt(cbrt(log(log(1.0 + exp(x))))), 2.0))))) - (x * y);
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.5
Target0.1
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

    \[\log \left(1 + e^{x}\right) - x \cdot y\]
  2. Using strategy rm

  3. Applied add-cbrt-cube_binary64_28670.5

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

  4. Simplified0.5

    \[\leadsto \sqrt[3]{\color{blue}{{\log \left(1 + e^{x}\right)}^{3}}} - x \cdot y\]
  5. Using strategy rm

  6. Applied pow-to-exp_binary64_29001.1

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

  7. Simplified1.1

    \[\leadsto \sqrt[3]{e^{\color{blue}{3 \cdot \log \log \left(1 + e^{x}\right)}}} - x \cdot y\]
  8. Using strategy rm

  9. Applied add-cube-cbrt_binary64_28661.1

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

  10. Applied associate-*r*_binary64_27711.1

    \[\leadsto \sqrt[3]{e^{\color{blue}{\left(3 \cdot \left(\sqrt[3]{\log \log \left(1 + e^{x}\right)} \cdot \sqrt[3]{\log \log \left(1 + e^{x}\right)}\right)\right) \cdot \sqrt[3]{\log \log \left(1 + e^{x}\right)}}}} - x \cdot y\]
  11. Using strategy rm

  12. Applied add-cube-cbrt_binary64_28661.1

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

  13. Applied add-cube-cbrt_binary64_28660.6

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

  14. Applied swap-sqr_binary64_27980.6

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

  15. Simplified0.6

    \[\leadsto \sqrt[3]{e^{\left(3 \cdot \left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\log \log \left(1 + e^{x}\right)}}\right)}^{4}} \cdot \left(\sqrt[3]{\sqrt[3]{\log \log \left(1 + e^{x}\right)}} \cdot \sqrt[3]{\sqrt[3]{\log \log \left(1 + e^{x}\right)}}\right)\right)\right) \cdot \sqrt[3]{\log \log \left(1 + e^{x}\right)}}} - x \cdot y\]

  16. Simplified0.6

    \[\leadsto \sqrt[3]{e^{\left(3 \cdot \left({\left(\sqrt[3]{\sqrt[3]{\log \log \left(1 + e^{x}\right)}}\right)}^{4} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt[3]{\log \log \left(1 + e^{x}\right)}}\right)}^{2}}\right)\right) \cdot \sqrt[3]{\log \log \left(1 + e^{x}\right)}}} - x \cdot y\]

  17. Final simplification0.6

    \[\leadsto \sqrt[3]{e^{\sqrt[3]{\log \log \left(1 + e^{x}\right)} \cdot \left(3 \cdot \left({\left(\sqrt[3]{\sqrt[3]{\log \log \left(1 + e^{x}\right)}}\right)}^{4} \cdot {\left(\sqrt[3]{\sqrt[3]{\log \log \left(1 + e^{x}\right)}}\right)}^{2}\right)\right)}} - x \cdot y\]

Reproduce

herbie shell --seed 2020288 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1.0 (exp x))) (* x y)) (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y))))

  (- (log (+ 1.0 (exp x))) (* x y)))