Average Error: 2.1 → 1.7
Time: 23.6s
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 r26247362 = x;
        double r26247363 = y;
        double r26247364 = r26247362 / r26247363;
        double r26247365 = z;
        double r26247366 = t;
        double r26247367 = r26247365 - r26247366;
        double r26247368 = r26247364 * r26247367;
        double r26247369 = r26247368 + r26247366;
        return r26247369;
}
↓
double f(double x, double y, double z, double t) {
        double r26247370 = z;
        double r26247371 = t;
        double r26247372 = r26247370 - r26247371;
        double r26247373 = cbrt(r26247372);
        double r26247374 = r26247373 * r26247373;
        double r26247375 = 1.0;
        double r26247376 = x;
        double r26247377 = cbrt(r26247376);
        double r26247378 = r26247377 * r26247377;
        double r26247379 = r26247375 / r26247378;
        double r26247380 = r26247374 / r26247379;
        double r26247381 = y;
        double r26247382 = r26247381 / r26247377;
        double r26247383 = r26247373 / r26247382;
        double r26247384 = fma(r26247380, r26247383, r26247371);
        return r26247384;
}

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))