Average Error: 2.1 → 1.7
Time: 22.6s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\mathsf{fma}\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}, \frac{\sqrt[3]{z - t}}{\frac{y}{\sqrt[3]{x}}}, t\right)\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\mathsf{fma}\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}, \frac{\sqrt[3]{z - t}}{\frac{y}{\sqrt[3]{x}}}, t\right)
double f(double x, double y, double z, double t) {
        double r21728577 = x;
        double r21728578 = y;
        double r21728579 = r21728577 / r21728578;
        double r21728580 = z;
        double r21728581 = t;
        double r21728582 = r21728580 - r21728581;
        double r21728583 = r21728579 * r21728582;
        double r21728584 = r21728583 + r21728581;
        return r21728584;
}
↓
double f(double x, double y, double z, double t) {
        double r21728585 = z;
        double r21728586 = t;
        double r21728587 = r21728585 - r21728586;
        double r21728588 = cbrt(r21728587);
        double r21728589 = r21728588 * r21728588;
        double r21728590 = 1.0;
        double r21728591 = x;
        double r21728592 = cbrt(r21728591);
        double r21728593 = r21728592 * r21728592;
        double r21728594 = r21728590 / r21728593;
        double r21728595 = r21728589 / r21728594;
        double r21728596 = y;
        double r21728597 = r21728596 / r21728592;
        double r21728598 = r21728588 / r21728597;
        double r21728599 = fma(r21728595, r21728598, r21728586);
        return r21728599;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.1
Target2.2
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;z \lt 2.759456554562692 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Initial program 2.1

    \[\frac{x}{y} \cdot \left(z - t\right) + t\]

  2. Taylor expanded around 0 6.2

    \[\leadsto \color{blue}{\left(\frac{x \cdot z}{y} - \frac{t \cdot x}{y}\right)} + t\]

  3. Simplified2.1

    \[\leadsto \color{blue}{\frac{z - t}{\frac{y}{x}}} + t\]
  4. Using strategy rm

  5. Applied add-cube-cbrt2.6

    \[\leadsto \frac{z - t}{\frac{y}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}} + t\]

  6. Applied *-un-lft-identity2.6

    \[\leadsto \frac{z - t}{\frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}} + t\]

  7. Applied times-frac2.6

    \[\leadsto \frac{z - t}{\color{blue}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{y}{\sqrt[3]{x}}}} + t\]

  8. Applied add-cube-cbrt2.7

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{y}{\sqrt[3]{x}}} + t\]

  9. Applied times-frac1.7

    \[\leadsto \color{blue}{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\sqrt[3]{z - t}}{\frac{y}{\sqrt[3]{x}}}} + t\]

  10. Applied fma-def1.7

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

  11. Final simplification1.7

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"

  :herbie-target
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

  (+ (* (/ x y) (- z t)) t))