Average Error: 6.7 → 3.4
Time: 5.4s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot 2 \le 1.20445290796802163 \cdot 10^{-28}:\\ \;\;\;\;\frac{2 \cdot x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\frac{1}{y - t}}{\sqrt[3]{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{y - t}}{z}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot 2 \le 1.20445290796802163 \cdot 10^{-28}:\\
\;\;\;\;\frac{2 \cdot x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\frac{1}{y - t}}{\sqrt[3]{z}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x \cdot 2}{y - t}}{z}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r712334 = x;
        double r712335 = 2.0;
        double r712336 = r712334 * r712335;
        double r712337 = y;
        double r712338 = z;
        double r712339 = r712337 * r712338;
        double r712340 = t;
        double r712341 = r712340 * r712338;
        double r712342 = r712339 - r712341;
        double r712343 = r712336 / r712342;
        return r712343;
}
↓
double f(double x, double y, double z, double t) {
        double r712344 = x;
        double r712345 = 2.0;
        double r712346 = r712344 * r712345;
        double r712347 = 1.2044529079680216e-28;
        bool r712348 = r712346 <= r712347;
        double r712349 = r712345 * r712344;
        double r712350 = z;
        double r712351 = cbrt(r712350);
        double r712352 = r712351 * r712351;
        double r712353 = r712349 / r712352;
        double r712354 = 1.0;
        double r712355 = y;
        double r712356 = t;
        double r712357 = r712355 - r712356;
        double r712358 = r712354 / r712357;
        double r712359 = r712358 / r712351;
        double r712360 = r712353 * r712359;
        double r712361 = r712346 / r712357;
        double r712362 = r712361 / r712350;
        double r712363 = r712348 ? r712360 : r712362;
        return r712363;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.7
Target2.0
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* x 2.0) < 1.2044529079680216e-28

    1. Initial program 5.7

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified4.5

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}}\]
    3. Using strategy rm

    4. Applied *-commutative4.5

      \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(y - t\right) \cdot z}}\]
    5. Using strategy rm

    6. Applied div-inv4.7

      \[\leadsto \color{blue}{\left(x \cdot 2\right) \cdot \frac{1}{\left(y - t\right) \cdot z}}\]
    7. Using strategy rm

    8. Applied associate-/r*4.5

      \[\leadsto \left(x \cdot 2\right) \cdot \color{blue}{\frac{\frac{1}{y - t}}{z}}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt5.1

      \[\leadsto \left(x \cdot 2\right) \cdot \frac{\frac{1}{y - t}}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

    11. Applied *-un-lft-identity5.1

      \[\leadsto \left(x \cdot 2\right) \cdot \frac{\frac{1}{\color{blue}{1 \cdot \left(y - t\right)}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

    12. Applied *-un-lft-identity5.1

      \[\leadsto \left(x \cdot 2\right) \cdot \frac{\frac{\color{blue}{1 \cdot 1}}{1 \cdot \left(y - t\right)}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

    13. Applied times-frac5.1

      \[\leadsto \left(x \cdot 2\right) \cdot \frac{\color{blue}{\frac{1}{1} \cdot \frac{1}{y - t}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

    14. Applied times-frac5.1

      \[\leadsto \left(x \cdot 2\right) \cdot \color{blue}{\left(\frac{\frac{1}{1}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\frac{1}{y - t}}{\sqrt[3]{z}}\right)}\]

    15. Applied associate-*r*3.7

      \[\leadsto \color{blue}{\left(\left(x \cdot 2\right) \cdot \frac{\frac{1}{1}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\frac{1}{y - t}}{\sqrt[3]{z}}}\]

    16. Simplified3.7

      \[\leadsto \color{blue}{\frac{2 \cdot x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{\frac{1}{y - t}}{\sqrt[3]{z}}\]

    if 1.2044529079680216e-28 < (* x 2.0)

    1. Initial program 9.8

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified9.0

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}}\]
    3. Using strategy rm

    4. Applied *-commutative9.0

      \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(y - t\right) \cdot z}}\]
    5. Using strategy rm

    6. Applied associate-/r*2.5

      \[\leadsto \color{blue}{\frac{\frac{x \cdot 2}{y - t}}{z}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \le 1.20445290796802163 \cdot 10^{-28}:\\ \;\;\;\;\frac{2 \cdot x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\frac{1}{y - t}}{\sqrt[3]{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{y - t}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2) (if (< (/ (* x 2) (- (* y z) (* t z))) 1.0450278273301259e-269) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2)))

  (/ (* x 2) (- (* y z) (* t z))))