Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.6 → 2.9

Time: 5.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.014343242050049761157872931103442569929 \cdot 10^{-140} \lor \neg \left(x \le 4.682548860145129920579692546790894336778 \cdot 10^{-88}\right):\\ \;\;\;\;\frac{1}{z} \cdot \frac{x}{\frac{y - t}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r498535 = x;
        double r498536 = 2.0;
        double r498537 = r498535 * r498536;
        double r498538 = y;
        double r498539 = z;
        double r498540 = r498538 * r498539;
        double r498541 = t;
        double r498542 = r498541 * r498539;
        double r498543 = r498540 - r498542;
        double r498544 = r498537 / r498543;
        return r498544;
}

↓

double f(double x, double y, double z, double t) {
        double r498545 = x;
        double r498546 = -2.0143432420500498e-140;
        bool r498547 = r498545 <= r498546;
        double r498548 = 4.68254886014513e-88;
        bool r498549 = r498545 <= r498548;
        double r498550 = !r498549;
        bool r498551 = r498547 || r498550;
        double r498552 = 1.0;
        double r498553 = z;
        double r498554 = r498552 / r498553;
        double r498555 = y;
        double r498556 = t;
        double r498557 = r498555 - r498556;
        double r498558 = 2.0;
        double r498559 = r498557 / r498558;
        double r498560 = r498545 / r498559;
        double r498561 = r498554 * r498560;
        double r498562 = r498545 / r498553;
        double r498563 = r498562 / r498559;
        double r498564 = r498551 ? r498561 : r498563;
        return r498564;
}

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	2.9

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

Derivation

1. Split input into 2 regimes

2. if x < -2.0143432420500498e-140 or 4.68254886014513e-88 < x

   1. Initial program 8.3

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

   2. Simplified 7.4

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

   4. Applied *-un-lft-identity 7.4

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

   5. Applied times-frac 7.4

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

   6. Applied *-un-lft-identity 7.4

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

   7. Applied times-frac 3.3

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

   8. Simplified 3.3

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

   if -2.0143432420500498e-140 < x < 4.68254886014513e-88

   1. Initial program 3.4

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

   2. Simplified 2.2

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

   4. Applied *-un-lft-identity 2.2

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

   5. Applied times-frac 2.2

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

   6. Applied associate-/r* 2.2

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

   7. Simplified 2.2

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.014343242050049761157872931103442569929 \cdot 10^{-140} \lor \neg \left(x \le 4.682548860145129920579692546790894336778 \cdot 10^{-88}\right):\\ \;\;\;\;\frac{1}{z} \cdot \frac{x}{\frac{y - t}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))))