Average Error: 2.7 → 0.2
Time: 2.9s
Precision: 64
\[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
↓
\[x + \frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{\frac{e^{z}}{y}} \cdot \sqrt[3]{\frac{e^{z}}{y}}\right) \cdot \sqrt[3]{\frac{e^{z}}{y}}, 1.12837916709551256, -x\right)}\]
x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}
↓
x + \frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{\frac{e^{z}}{y}} \cdot \sqrt[3]{\frac{e^{z}}{y}}\right) \cdot \sqrt[3]{\frac{e^{z}}{y}}, 1.12837916709551256, -x\right)}
double f(double x, double y, double z) {
        double r375563 = x;
        double r375564 = y;
        double r375565 = 1.1283791670955126;
        double r375566 = z;
        double r375567 = exp(r375566);
        double r375568 = r375565 * r375567;
        double r375569 = r375563 * r375564;
        double r375570 = r375568 - r375569;
        double r375571 = r375564 / r375570;
        double r375572 = r375563 + r375571;
        return r375572;
}
↓
double f(double x, double y, double z) {
        double r375573 = x;
        double r375574 = 1.0;
        double r375575 = z;
        double r375576 = exp(r375575);
        double r375577 = y;
        double r375578 = r375576 / r375577;
        double r375579 = cbrt(r375578);
        double r375580 = r375579 * r375579;
        double r375581 = r375580 * r375579;
        double r375582 = 1.1283791670955126;
        double r375583 = -r375573;
        double r375584 = fma(r375581, r375582, r375583);
        double r375585 = r375574 / r375584;
        double r375586 = r375573 + r375585;
        return r375586;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original2.7
Target0.0
Herbie0.2
\[x + \frac{1}{\frac{1.12837916709551256}{y} \cdot e^{z} - x}\]

Derivation

  1. Initial program 2.7

    \[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
  2. Using strategy rm

  3. Applied clear-num2.7

    \[\leadsto x + \color{blue}{\frac{1}{\frac{1.12837916709551256 \cdot e^{z} - x \cdot y}{y}}}\]

  4. Simplified0.0

    \[\leadsto x + \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{e^{z}}{y}, 1.12837916709551256, -x\right)}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.2

    \[\leadsto x + \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{\frac{e^{z}}{y}} \cdot \sqrt[3]{\frac{e^{z}}{y}}\right) \cdot \sqrt[3]{\frac{e^{z}}{y}}}, 1.12837916709551256, -x\right)}\]

  7. Final simplification0.2

    \[\leadsto x + \frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{\frac{e^{z}}{y}} \cdot \sqrt[3]{\frac{e^{z}}{y}}\right) \cdot \sqrt[3]{\frac{e^{z}}{y}}, 1.12837916709551256, -x\right)}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))