Average Error: 0.6 → 0.6
Time: 4.1s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\sqrt[3]{{\left(\log \left(e^{x} + 1\right)\right)}^{2} \cdot \log \left(e^{x} + 1\right)} - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\sqrt[3]{{\left(\log \left(e^{x} + 1\right)\right)}^{2} \cdot \log \left(e^{x} + 1\right)} - x \cdot y
double code(double x, double y) {
	return (log((1.0 + exp(x))) - (x * y));
}
↓
double code(double x, double y) {
	return (cbrt((pow(log((exp(x) + 1.0)), 2.0) * log((exp(x) + 1.0)))) - (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

Original0.6
Target0.1
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.6

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

  3. Applied add-cbrt-cube0.6

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

  4. Simplified0.6

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

  6. Applied add-cube-cbrt1.7

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

  7. Applied unpow-prod-down1.8

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

  8. Simplified1.4

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

  9. Simplified0.6

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

  10. Final simplification0.6

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

Reproduce

herbie shell --seed 2020091 +o rules:numerics
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y))))

  (- (log (+ 1 (exp x))) (* x y)))