Average Error: 6.6 → 2.3
Time: 4.5s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.694063912547288218817191837339691890919 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 3167.128584297215184051310643553733825684:\\ \;\;\;\;\frac{x}{z \cdot \frac{y - t}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y - t}{2}}}{z}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -6.694063912547288218817191837339691890919 \cdot 10^{-98}:\\
\;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\

\mathbf{elif}\;z \le 3167.128584297215184051310643553733825684:\\
\;\;\;\;\frac{x}{z \cdot \frac{y - t}{2}}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r523505 = x;
        double r523506 = 2.0;
        double r523507 = r523505 * r523506;
        double r523508 = y;
        double r523509 = z;
        double r523510 = r523508 * r523509;
        double r523511 = t;
        double r523512 = r523511 * r523509;
        double r523513 = r523510 - r523512;
        double r523514 = r523507 / r523513;
        return r523514;
}
↓
double f(double x, double y, double z, double t) {
        double r523515 = z;
        double r523516 = -6.694063912547288e-98;
        bool r523517 = r523515 <= r523516;
        double r523518 = x;
        double r523519 = r523518 / r523515;
        double r523520 = y;
        double r523521 = t;
        double r523522 = r523520 - r523521;
        double r523523 = 2.0;
        double r523524 = r523522 / r523523;
        double r523525 = r523519 / r523524;
        double r523526 = 3167.128584297215;
        bool r523527 = r523515 <= r523526;
        double r523528 = r523515 * r523524;
        double r523529 = r523518 / r523528;
        double r523530 = r523518 / r523524;
        double r523531 = r523530 / r523515;
        double r523532 = r523527 ? r523529 : r523531;
        double r523533 = r523517 ? r523525 : r523532;
        return r523533;
}

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

Original6.6
Target2.0
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061113708240820439530037456 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.045027827330126029709547581125571222799 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -6.694063912547288e-98

    1. Initial program 8.1

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

    2. Simplified6.9

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

    4. Applied *-un-lft-identity6.9

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

    5. Applied times-frac6.9

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

    6. Applied associate-/r*2.3

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

    7. Simplified2.3

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

    if -6.694063912547288e-98 < z < 3167.128584297215

    1. Initial program 2.3

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

    2. Simplified2.3

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

    4. Applied *-un-lft-identity2.3

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

    5. Applied times-frac2.3

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

    6. Applied *-un-lft-identity2.3

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

    7. Applied times-frac10.2

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

    8. Simplified10.2

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

    10. Applied associate-*l/10.1

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

    11. Simplified10.1

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

    13. Applied div-inv10.2

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

    14. Applied associate-/l*2.4

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

    15. Simplified2.3

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

    if 3167.128584297215 < z

    1. Initial program 10.3

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

    2. Simplified8.4

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

    4. Applied *-un-lft-identity8.4

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

    5. Applied times-frac8.4

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

    6. Applied *-un-lft-identity8.4

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

    7. Applied times-frac2.2

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

    8. Simplified2.2

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

    10. Applied associate-*l/2.2

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

    11. Simplified2.2

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

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.694063912547288218817191837339691890919 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 3167.128584297215184051310643553733825684:\\ \;\;\;\;\frac{x}{z \cdot \frac{y - t}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y - t}{2}}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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))))