Average Error: 7.0 → 2.2
Time: 8.1s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7935255452.70021820068359375:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 1.460013591902958990208612458924325025091 \cdot 10^{-43}:\\ \;\;\;\;\frac{1}{\frac{\frac{z \cdot \left(y - t\right)}{2}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{x}{\frac{y - t}{2}}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7935255452.70021820068359375:\\
\;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\

\mathbf{elif}\;z \le 1.460013591902958990208612458924325025091 \cdot 10^{-43}:\\
\;\;\;\;\frac{1}{\frac{\frac{z \cdot \left(y - t\right)}{2}}{x}}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r566730 = x;
        double r566731 = 2.0;
        double r566732 = r566730 * r566731;
        double r566733 = y;
        double r566734 = z;
        double r566735 = r566733 * r566734;
        double r566736 = t;
        double r566737 = r566736 * r566734;
        double r566738 = r566735 - r566737;
        double r566739 = r566732 / r566738;
        return r566739;
}
↓
double f(double x, double y, double z, double t) {
        double r566740 = z;
        double r566741 = -7935255452.700218;
        bool r566742 = r566740 <= r566741;
        double r566743 = x;
        double r566744 = r566743 / r566740;
        double r566745 = y;
        double r566746 = t;
        double r566747 = r566745 - r566746;
        double r566748 = 2.0;
        double r566749 = r566747 / r566748;
        double r566750 = r566744 / r566749;
        double r566751 = 1.460013591902959e-43;
        bool r566752 = r566740 <= r566751;
        double r566753 = 1.0;
        double r566754 = r566740 * r566747;
        double r566755 = r566754 / r566748;
        double r566756 = r566755 / r566743;
        double r566757 = r566753 / r566756;
        double r566758 = r566753 / r566740;
        double r566759 = r566743 / r566749;
        double r566760 = r566758 * r566759;
        double r566761 = r566752 ? r566757 : r566760;
        double r566762 = r566742 ? r566750 : r566761;
        return r566762;
}

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
Target2.0
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061113708240820439530037456 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.045027827330125861587720199944080049996 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -7935255452.700218

    1. Initial program 11.0

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

    2. Simplified9.0

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

    4. Applied *-un-lft-identity9.0

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

    5. Applied times-frac9.0

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

    6. Applied associate-/r*1.6

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

    7. Simplified1.6

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

    if -7935255452.700218 < z < 1.460013591902959e-43

    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 clear-num2.6

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

    if 1.460013591902959e-43 < z

    1. Initial program 10.1

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

    2. Simplified8.1

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

    4. Applied *-un-lft-identity8.1

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

    5. Applied times-frac8.0

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

    6. Applied *-un-lft-identity8.0

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

    7. Applied times-frac2.2

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

    8. Simplified2.2

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

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7935255452.70021820068359375:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 1.460013591902958990208612458924325025091 \cdot 10^{-43}:\\ \;\;\;\;\frac{1}{\frac{\frac{z \cdot \left(y - t\right)}{2}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{x}{\frac{y - t}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(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))))