Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.5 → 2.9

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.5647687804893846 \cdot 10^{-4}:\\ \;\;\;\;1 \cdot \frac{\frac{1}{y - t}}{\frac{\frac{z}{x}}{2}}\\ \mathbf{elif}\;z \le 1.4378345516742572 \cdot 10^{-160}:\\ \;\;\;\;1 \cdot \frac{x}{z \cdot \frac{y - t}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r481764 = x;
        double r481765 = 2.0;
        double r481766 = r481764 * r481765;
        double r481767 = y;
        double r481768 = z;
        double r481769 = r481767 * r481768;
        double r481770 = t;
        double r481771 = r481770 * r481768;
        double r481772 = r481769 - r481771;
        double r481773 = r481766 / r481772;
        return r481773;
}

↓

double f(double x, double y, double z, double t) {
        double r481774 = z;
        double r481775 = -0.00035647687804893846;
        bool r481776 = r481774 <= r481775;
        double r481777 = 1.0;
        double r481778 = y;
        double r481779 = t;
        double r481780 = r481778 - r481779;
        double r481781 = r481777 / r481780;
        double r481782 = x;
        double r481783 = r481774 / r481782;
        double r481784 = 2.0;
        double r481785 = r481783 / r481784;
        double r481786 = r481781 / r481785;
        double r481787 = r481777 * r481786;
        double r481788 = 1.4378345516742572e-160;
        bool r481789 = r481774 <= r481788;
        double r481790 = r481780 / r481784;
        double r481791 = r481774 * r481790;
        double r481792 = r481782 / r481791;
        double r481793 = r481777 * r481792;
        double r481794 = r481782 / r481774;
        double r481795 = r481794 / r481790;
        double r481796 = r481789 ? r481793 : r481795;
        double r481797 = r481776 ? r481787 : r481796;
        return r481797;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.5
Target	2.2
Herbie	2.9

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

Derivation

1. Split input into 3 regimes

2. if z < -0.00035647687804893846

   1. Initial program 9.9

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

   2. Simplified 8.3

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

   4. Applied *-un-lft-identity 8.3

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

   5. Applied times-frac 8.2

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

   6. Applied *-un-lft-identity 8.2

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

   7. Applied times-frac 2.3

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

   8. Simplified 2.3

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

   10. Applied *-un-lft-identity 2.3

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

   11. Applied associate-*l* 2.3

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

   12. Simplified 2.2

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

   14. Applied div-inv 2.2

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

   15. Applied *-un-lft-identity 2.2

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

   16. Applied times-frac 2.3

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

   17. Applied associate-/l* 2.3

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

   18. Simplified 2.3

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

   if -0.00035647687804893846 < z < 1.4378345516742572e-160

   1. Initial program 3.1

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

   2. Simplified 3.1

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

   4. Applied *-un-lft-identity 3.1

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

   5. Applied times-frac 3.1

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

   6. Applied *-un-lft-identity 3.1

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

   7. Applied times-frac 10.8

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

   8. Simplified 10.8

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

   10. Applied *-un-lft-identity 10.8

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

   11. Applied associate-*l* 10.8

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

   12. Simplified 10.7

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

   14. Applied div-inv 10.8

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

   15. Applied associate-/l* 3.2

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

   16. Simplified 3.1

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

   if 1.4378345516742572e-160 < z

   1. Initial program 7.0

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

   2. Simplified 5.8

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

   4. Applied *-un-lft-identity 5.8

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

   5. Applied times-frac 5.7

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

   6. Applied associate-/r* 3.3

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

   7. Simplified 3.3

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.5647687804893846 \cdot 10^{-4}:\\ \;\;\;\;1 \cdot \frac{\frac{1}{y - t}}{\frac{\frac{z}{x}}{2}}\\ \mathbf{elif}\;z \le 1.4378345516742572 \cdot 10^{-160}:\\ \;\;\;\;1 \cdot \frac{x}{z \cdot \frac{y - t}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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))))