Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.6 → 2.9

Time: 9.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.89093263392816965 \cdot 10^{114}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 2.9355937162195811 \cdot 10^{56}:\\ \;\;\;\;\frac{1}{\frac{\frac{z \cdot \left(y - t\right)}{2}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x} \cdot \left(\frac{1}{\frac{z}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r543491 = x;
        double r543492 = 2.0;
        double r543493 = r543491 * r543492;
        double r543494 = y;
        double r543495 = z;
        double r543496 = r543494 * r543495;
        double r543497 = t;
        double r543498 = r543497 * r543495;
        double r543499 = r543496 - r543498;
        double r543500 = r543493 / r543499;
        return r543500;
}

↓

double f(double x, double y, double z, double t) {
        double r543501 = z;
        double r543502 = -1.8909326339281697e+114;
        bool r543503 = r543501 <= r543502;
        double r543504 = x;
        double r543505 = r543504 / r543501;
        double r543506 = y;
        double r543507 = t;
        double r543508 = r543506 - r543507;
        double r543509 = 2.0;
        double r543510 = r543508 / r543509;
        double r543511 = r543505 / r543510;
        double r543512 = 2.935593716219581e+56;
        bool r543513 = r543501 <= r543512;
        double r543514 = 1.0;
        double r543515 = r543501 * r543508;
        double r543516 = r543515 / r543509;
        double r543517 = r543516 / r543504;
        double r543518 = r543514 / r543517;
        double r543519 = cbrt(r543504);
        double r543520 = r543501 / r543519;
        double r543521 = r543514 / r543520;
        double r543522 = r543519 / r543510;
        double r543523 = r543521 * r543522;
        double r543524 = r543519 * r543523;
        double r543525 = r543513 ? r543518 : r543524;
        double r543526 = r543503 ? r543511 : r543525;
        return r543526;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.6
Target	2.0
Herbie	2.9

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

Derivation

1. Split input into 3 regimes

2. if z < -1.8909326339281697e+114

   1. Initial program 13.4

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

   2. Simplified 9.7

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

   4. Applied *-un-lft-identity 9.7

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

   5. Applied times-frac 9.7

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

   6. Applied associate-/r* 2.1

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

   7. Simplified 2.1

      \[\leadsto \frac{\color{blue}{\frac{x}{z}}}{\frac{y - t}{2}}\]

   if -1.8909326339281697e+114 < z < 2.935593716219581e+56

   1. Initial program 2.6

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

   2. Simplified 2.6

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

   4. Applied clear-num 3.1

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

   if 2.935593716219581e+56 < z

   1. Initial program 12.0

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

   2. Simplified 9.8

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

   4. Applied *-un-lft-identity 9.8

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

   5. Applied times-frac 9.8

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

   6. Applied add-cube-cbrt 10.2

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

   7. Applied times-frac 1.8

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

   8. Simplified 1.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{x}}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\]
   9. Using strategy rm

   10. Applied div-inv 1.8

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

   11. Applied associate-*l* 2.9

       \[\leadsto \color{blue}{\sqrt[3]{x} \cdot \left(\frac{1}{\frac{z}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.89093263392816965 \cdot 10^{114}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 2.9355937162195811 \cdot 10^{56}:\\ \;\;\;\;\frac{1}{\frac{\frac{z \cdot \left(y - t\right)}{2}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x} \cdot \left(\frac{1}{\frac{z}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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))))