Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.4 → 1.9

Time: 10.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{x \cdot \left(y + z\right)}{z}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -9.6027108909063706 \cdot 10^{-114}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \le 1.80510424301144413 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r517578 = x;
        double r517579 = y;
        double r517580 = z;
        double r517581 = r517579 + r517580;
        double r517582 = r517578 * r517581;
        double r517583 = r517582 / r517580;
        return r517583;
}

↓

double f(double x, double y, double z) {
        double r517584 = z;
        double r517585 = -9.60271089090637e-114;
        bool r517586 = r517584 <= r517585;
        double r517587 = x;
        double r517588 = y;
        double r517589 = r517588 / r517584;
        double r517590 = r517587 * r517589;
        double r517591 = r517587 + r517590;
        double r517592 = 1.805104243011444e-23;
        bool r517593 = r517584 <= r517592;
        double r517594 = 1.0;
        double r517595 = r517594 / r517584;
        double r517596 = r517587 * r517588;
        double r517597 = r517595 * r517596;
        double r517598 = r517597 + r517587;
        double r517599 = r517584 / r517588;
        double r517600 = r517587 / r517599;
        double r517601 = r517600 + r517587;
        double r517602 = r517593 ? r517598 : r517601;
        double r517603 = r517586 ? r517591 : r517602;
        return r517603;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.4
Target	3.0
Herbie	1.9

\[\frac{x}{\frac{z}{y + z}}\]

Derivation

1. Split input into 3 regimes

2. if z < -9.60271089090637e-114

   1. Initial program 13.4

      \[\frac{x \cdot \left(y + z\right)}{z}\]

   2. Simplified 2.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, y, x\right)}\]
   3. Using strategy rm

   4. Applied fma-udef 2.9

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y + x}\]

   5. Simplified 4.6

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} + x\]
   6. Using strategy rm

   7. Applied associate-/l* 0.6

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} + x\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 0.6

      \[\leadsto \frac{x}{\frac{z}{\color{blue}{1 \cdot y}}} + x\]

   10. Applied *-un-lft-identity 0.6

       \[\leadsto \frac{x}{\frac{\color{blue}{1 \cdot z}}{1 \cdot y}} + x\]

   11. Applied times-frac 0.6

       \[\leadsto \frac{x}{\color{blue}{\frac{1}{1} \cdot \frac{z}{y}}} + x\]

   12. Applied *-un-lft-identity 0.6

       \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{1}{1} \cdot \frac{z}{y}} + x\]

   13. Applied times-frac 0.6

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{1}} \cdot \frac{x}{\frac{z}{y}}} + x\]

   14. Simplified 0.6

       \[\leadsto \color{blue}{1} \cdot \frac{x}{\frac{z}{y}} + x\]

   15. Simplified 0.8

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

   if -9.60271089090637e-114 < z < 1.805104243011444e-23

   1. Initial program 8.3

      \[\frac{x \cdot \left(y + z\right)}{z}\]

   2. Simplified 9.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, y, x\right)}\]
   3. Using strategy rm

   4. Applied fma-udef 9.5

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y + x}\]

   5. Simplified 4.9

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} + x\]
   6. Using strategy rm

   7. Applied associate-/l* 8.7

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} + x\]
   8. Using strategy rm

   9. Applied div-inv 8.7

      \[\leadsto \frac{x}{\color{blue}{z \cdot \frac{1}{y}}} + x\]

   10. Applied *-un-lft-identity 8.7

       \[\leadsto \frac{\color{blue}{1 \cdot x}}{z \cdot \frac{1}{y}} + x\]

   11. Applied times-frac 5.0

       \[\leadsto \color{blue}{\frac{1}{z} \cdot \frac{x}{\frac{1}{y}}} + x\]

   12. Simplified 5.0

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

   if 1.805104243011444e-23 < z

   1. Initial program 15.4

      \[\frac{x \cdot \left(y + z\right)}{z}\]

   2. Simplified 2.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, y, x\right)}\]
   3. Using strategy rm

   4. Applied fma-udef 2.6

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y + x}\]

   5. Simplified 4.9

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} + x\]
   6. Using strategy rm

   7. Applied associate-/l* 0.1

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.6027108909063706 \cdot 10^{-114}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \le 1.80510424301144413 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))