Average Error: 6.8 → 3.5
Time: 5.2s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.154512528065834 \cdot 10^{229} \lor \neg \left(z \le 9.1298419035993844 \cdot 10^{-77}\right):\\ \;\;\;\;1 \cdot \frac{\frac{x}{\frac{y - t}{2}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y - t} \cdot \frac{x}{\frac{1}{2}}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.154512528065834 \cdot 10^{229} \lor \neg \left(z \le 9.1298419035993844 \cdot 10^{-77}\right):\\
\;\;\;\;1 \cdot \frac{\frac{x}{\frac{y - t}{2}}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{z}}{y - t} \cdot \frac{x}{\frac{1}{2}}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r552140 = x;
        double r552141 = 2.0;
        double r552142 = r552140 * r552141;
        double r552143 = y;
        double r552144 = z;
        double r552145 = r552143 * r552144;
        double r552146 = t;
        double r552147 = r552146 * r552144;
        double r552148 = r552145 - r552147;
        double r552149 = r552142 / r552148;
        return r552149;
}
↓
double f(double x, double y, double z, double t) {
        double r552150 = z;
        double r552151 = -4.154512528065834e+229;
        bool r552152 = r552150 <= r552151;
        double r552153 = 9.129841903599384e-77;
        bool r552154 = r552150 <= r552153;
        double r552155 = !r552154;
        bool r552156 = r552152 || r552155;
        double r552157 = 1.0;
        double r552158 = x;
        double r552159 = y;
        double r552160 = t;
        double r552161 = r552159 - r552160;
        double r552162 = 2.0;
        double r552163 = r552161 / r552162;
        double r552164 = r552158 / r552163;
        double r552165 = r552164 / r552150;
        double r552166 = r552157 * r552165;
        double r552167 = r552157 / r552150;
        double r552168 = r552167 / r552161;
        double r552169 = r552157 / r552162;
        double r552170 = r552158 / r552169;
        double r552171 = r552168 * r552170;
        double r552172 = r552156 ? r552166 : r552171;
        return r552172;
}

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

Original6.8
Target2.1
Herbie3.5
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.154512528065834e+229 or 9.129841903599384e-77 < z

    1. Initial program 10.0

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

    2. Simplified7.9

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

    4. Applied *-un-lft-identity7.9

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

    5. Applied times-frac7.9

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

    6. Applied *-un-lft-identity7.9

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

    7. Applied times-frac2.5

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

    8. Simplified2.5

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

    10. Applied *-un-lft-identity2.5

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

    11. Applied *-un-lft-identity2.5

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

    12. Applied times-frac2.5

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

    13. Applied associate-*l*2.5

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

    14. Simplified2.4

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

    if -4.154512528065834e+229 < z < 9.129841903599384e-77

    1. Initial program 4.6

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

    2. Simplified4.2

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

    4. Applied *-un-lft-identity4.2

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

    5. Applied times-frac4.2

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

    6. Applied *-un-lft-identity4.2

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

    7. Applied times-frac7.7

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

    8. Simplified7.7

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

    10. Applied div-inv7.7

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

    11. Applied *-un-lft-identity7.7

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

    12. Applied times-frac7.7

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

    13. Applied associate-*r*4.3

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

    14. Simplified4.2

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

  4. Final simplification3.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.154512528065834 \cdot 10^{229} \lor \neg \left(z \le 9.1298419035993844 \cdot 10^{-77}\right):\\ \;\;\;\;1 \cdot \frac{\frac{x}{\frac{y - t}{2}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y - t} \cdot \frac{x}{\frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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))))