Average Error: 7.0 → 5.6
Time: 1.5m
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\frac{\frac{x}{\frac{y - t}{2}}}{z}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\frac{\frac{x}{\frac{y - t}{2}}}{z}
double f(double x, double y, double z, double t) {
        double r380871 = x;
        double r380872 = 2.0;
        double r380873 = r380871 * r380872;
        double r380874 = y;
        double r380875 = z;
        double r380876 = r380874 * r380875;
        double r380877 = t;
        double r380878 = r380877 * r380875;
        double r380879 = r380876 - r380878;
        double r380880 = r380873 / r380879;
        return r380880;
}
↓
double f(double x, double y, double z, double t) {
        double r380881 = x;
        double r380882 = y;
        double r380883 = t;
        double r380884 = r380882 - r380883;
        double r380885 = 2.0;
        double r380886 = r380884 / r380885;
        double r380887 = r380881 / r380886;
        double r380888 = z;
        double r380889 = r380887 / r380888;
        return r380889;
}

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

Original7.0
Target2.2
Herbie5.6
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061113708240820439530037456 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.045027827330126029709547581125571222799 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -6.948586446622268e+96

    1. Initial program 14.7

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

    2. Simplified11.1

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

    4. Applied *-un-lft-identity11.1

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

    5. Applied times-frac11.1

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

    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 -6.948586446622268e+96 < z < 1.3634763454561944e-116

    1. Initial program 3.2

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

    2. Simplified3.2

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

    4. Applied div-inv3.5

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

    5. Simplified3.4

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

    if 1.3634763454561944e-116 < z

    1. Initial program 7.7

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

    2. Simplified6.3

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

    4. Applied *-un-lft-identity6.3

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

    5. Applied times-frac6.3

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

    6. Applied *-un-lft-identity6.3

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

    7. Applied times-frac2.7

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

    8. Simplified2.7

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

  4. Final simplification5.6

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

Reproduce

herbie shell --seed 2019303 
(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))))