Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 7.3 → 5.4

Time: 20.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r402321 = x;
        double r402322 = 2.0;
        double r402323 = r402321 * r402322;
        double r402324 = y;
        double r402325 = z;
        double r402326 = r402324 * r402325;
        double r402327 = t;
        double r402328 = r402327 * r402325;
        double r402329 = r402326 - r402328;
        double r402330 = r402323 / r402329;
        return r402330;
}

↓

double f(double x, double y, double z, double t) {
        double r402331 = x;
        double r402332 = y;
        double r402333 = t;
        double r402334 = r402332 - r402333;
        double r402335 = 2.0;
        double r402336 = r402334 / r402335;
        double r402337 = r402331 / r402336;
        double r402338 = z;
        double r402339 = r402337 / r402338;
        return r402339;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.3
Target	2.2
Herbie	5.4

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061113708240820439530037456 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.045027827330126029709547581125571222799 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -2.1961928853843444e-14

   1. Initial program 11.3

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

   2. Simplified 9.1

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

   4. Applied *-un-lft-identity 9.1

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

   5. Applied times-frac 9.1

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

   6. Applied associate-/r* 1.6

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

   7. Simplified 1.6

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

   if -2.1961928853843444e-14 < z < 1.6092045793144405e-239

   1. Initial program 3.1

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

   2. Simplified 3.1

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

   if 1.6092045793144405e-239 < z

   1. Initial program 7.3

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

   2. Simplified 6.0

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

   4. Applied *-un-lft-identity 6.0

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

   5. Applied times-frac 6.0

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

   6. Applied *-un-lft-identity 6.0

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

   7. Applied times-frac 4.1

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

   8. Simplified 4.1

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

   10. Applied *-un-lft-identity 4.1

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

   11. Applied *-un-lft-identity 4.1

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

   12. Applied times-frac 4.1

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

   13. Applied associate-*l* 4.1

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

   14. Simplified 4.0

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

4. Final simplification 5.4

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

Reproduce

herbie shell --seed 2019297 
(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))) 1.045027827330126e-269) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2)))

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