Average Error: 0.4 → 0.9
Time: 29.8s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[(\left(\sqrt{\log_* (1 + e^{x})}\right) \cdot \left(\sqrt{\log_* (1 + e^{x})}\right) + \left(x \cdot \left(-y\right)\right))_*\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
(\left(\sqrt{\log_* (1 + e^{x})}\right) \cdot \left(\sqrt{\log_* (1 + e^{x})}\right) + \left(x \cdot \left(-y\right)\right))_*
double f(double x, double y) {
        double r5465663 = 1.0;
        double r5465664 = x;
        double r5465665 = exp(r5465664);
        double r5465666 = r5465663 + r5465665;
        double r5465667 = log(r5465666);
        double r5465668 = y;
        double r5465669 = r5465664 * r5465668;
        double r5465670 = r5465667 - r5465669;
        return r5465670;
}
↓
double f(double x, double y) {
        double r5465671 = x;
        double r5465672 = exp(r5465671);
        double r5465673 = log1p(r5465672);
        double r5465674 = sqrt(r5465673);
        double r5465675 = y;
        double r5465676 = -r5465675;
        double r5465677 = r5465671 * r5465676;
        double r5465678 = fma(r5465674, r5465674, r5465677);
        return r5465678;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.4
Target0.1
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.4

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

  2. Simplified0.4

    \[\leadsto \color{blue}{\log_* (1 + e^{x}) - y \cdot x}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.9

    \[\leadsto \color{blue}{\sqrt{\log_* (1 + e^{x})} \cdot \sqrt{\log_* (1 + e^{x})}} - y \cdot x\]

  5. Applied fma-neg0.9

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

  6. Final simplification0.9

    \[\leadsto (\left(\sqrt{\log_* (1 + e^{x})}\right) \cdot \left(\sqrt{\log_* (1 + e^{x})}\right) + \left(x \cdot \left(-y\right)\right))_*\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x y)
  :name "Logistic regression 2"

  :herbie-target
  (if (<= x 0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y))))

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