Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.6 → 3.4

Time: 6.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.15387447039997915 \cdot 10^{140}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 1.85210845097292845 \cdot 10^{55}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1} \cdot \frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{1}{\sqrt{2}}}}{\frac{z}{\frac{\sqrt[3]{x}}{\frac{y - t}{\sqrt{2}}}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r540575 = x;
        double r540576 = 2.0;
        double r540577 = r540575 * r540576;
        double r540578 = y;
        double r540579 = z;
        double r540580 = r540578 * r540579;
        double r540581 = t;
        double r540582 = r540581 * r540579;
        double r540583 = r540580 - r540582;
        double r540584 = r540577 / r540583;
        return r540584;
}

↓

double f(double x, double y, double z, double t) {
        double r540585 = z;
        double r540586 = -4.153874470399979e+140;
        bool r540587 = r540585 <= r540586;
        double r540588 = x;
        double r540589 = r540588 / r540585;
        double r540590 = y;
        double r540591 = t;
        double r540592 = r540590 - r540591;
        double r540593 = 2.0;
        double r540594 = r540592 / r540593;
        double r540595 = r540589 / r540594;
        double r540596 = 1.8521084509729285e+55;
        bool r540597 = r540585 <= r540596;
        double r540598 = r540592 * r540585;
        double r540599 = r540588 / r540598;
        double r540600 = r540599 * r540593;
        double r540601 = 1.0;
        double r540602 = r540601 / r540601;
        double r540603 = cbrt(r540588);
        double r540604 = r540603 * r540603;
        double r540605 = sqrt(r540593);
        double r540606 = r540601 / r540605;
        double r540607 = r540604 / r540606;
        double r540608 = r540592 / r540605;
        double r540609 = r540603 / r540608;
        double r540610 = r540585 / r540609;
        double r540611 = r540607 / r540610;
        double r540612 = r540602 * r540611;
        double r540613 = r540597 ? r540600 : r540612;
        double r540614 = r540587 ? r540595 : r540613;
        return r540614;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.6
Target	2.0
Herbie	3.4

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

Derivation

1. Split input into 3 regimes

2. if z < -4.153874470399979e+140

   1. Initial program 14.1

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

   2. Simplified 10.2

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

   4. Applied *-un-lft-identity 10.2

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

   5. Applied times-frac 10.2

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

   6. Applied associate-/r* 2.1

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

   7. Simplified 2.1

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

   if -4.153874470399979e+140 < z < 1.8521084509729285e+55

   1. Initial program 2.9

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

   2. Simplified 2.8

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

   4. Applied *-un-lft-identity 2.8

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

   5. Applied times-frac 2.8

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

   6. Applied *-un-lft-identity 2.8

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

   7. Applied times-frac 7.2

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

   8. Simplified 7.2

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

   10. Applied associate-/r/ 7.3

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

   11. Applied associate-*r* 7.3

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

   12. Simplified 2.8

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

   if 1.8521084509729285e+55 < z

   1. Initial program 12.1

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

   2. Simplified 9.8

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

   4. Applied *-un-lft-identity 9.8

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

   5. Applied times-frac 9.8

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

   6. Applied *-un-lft-identity 9.8

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

   7. Applied times-frac 2.6

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

   8. Simplified 2.6

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

   10. Applied *-un-lft-identity 2.6

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

   11. Applied *-un-lft-identity 2.6

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

   12. Applied times-frac 2.6

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

   13. Applied associate-*l* 2.6

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

   14. Simplified 2.5

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

   16. Applied add-sqr-sqrt 3.0

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

   17. Applied *-un-lft-identity 3.0

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

   18. Applied times-frac 2.9

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

   19. Applied add-cube-cbrt 3.1

       \[\leadsto \frac{1}{1} \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{1}{\sqrt{2}} \cdot \frac{y - t}{\sqrt{2}}}}{z}\]

   20. Applied times-frac 3.1

       \[\leadsto \frac{1}{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{1}{\sqrt{2}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{\sqrt{2}}}}}{z}\]

   21. Applied associate-/l* 6.1

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

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.15387447039997915 \cdot 10^{140}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 1.85210845097292845 \cdot 10^{55}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1} \cdot \frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{1}{\sqrt{2}}}}{\frac{z}{\frac{\sqrt[3]{x}}{\frac{y - t}{\sqrt{2}}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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))))