Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, D

Average Error: 21.0 → 0.2

Time: 20.7s

Precision: 64

Math

\[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -3480617740589270.5:\\ \;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 4005471382996.025:\\ \;\;\;\;\left(1.0 - \frac{\left(1.0 - x\right) \cdot y}{\mathsf{fma}\left(y, y \cdot y, \left(1.0 \cdot 1.0\right) \cdot 1.0\right)} \cdot \left(y \cdot y\right)\right) - \left(1.0 \cdot 1.0 - 1.0 \cdot y\right) \cdot \frac{\left(1.0 - x\right) \cdot y}{\mathsf{fma}\left(y, y \cdot y, \left(1.0 \cdot 1.0\right) \cdot 1.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r29876961 = 1.0;
        double r29876962 = x;
        double r29876963 = r29876961 - r29876962;
        double r29876964 = y;
        double r29876965 = r29876963 * r29876964;
        double r29876966 = r29876964 + r29876961;
        double r29876967 = r29876965 / r29876966;
        double r29876968 = r29876961 - r29876967;
        return r29876968;
}

↓

double f(double x, double y) {
        double r29876969 = y;
        double r29876970 = -3480617740589270.5;
        bool r29876971 = r29876969 <= r29876970;
        double r29876972 = 1.0;
        double r29876973 = 1.0;
        double r29876974 = r29876973 / r29876969;
        double r29876975 = x;
        double r29876976 = r29876975 / r29876969;
        double r29876977 = r29876974 - r29876976;
        double r29876978 = fma(r29876972, r29876977, r29876975);
        double r29876979 = 4005471382996.025;
        bool r29876980 = r29876969 <= r29876979;
        double r29876981 = r29876972 - r29876975;
        double r29876982 = r29876981 * r29876969;
        double r29876983 = r29876969 * r29876969;
        double r29876984 = r29876972 * r29876972;
        double r29876985 = r29876984 * r29876972;
        double r29876986 = fma(r29876969, r29876983, r29876985);
        double r29876987 = r29876982 / r29876986;
        double r29876988 = r29876987 * r29876983;
        double r29876989 = r29876972 - r29876988;
        double r29876990 = r29876972 * r29876969;
        double r29876991 = r29876984 - r29876990;
        double r29876992 = r29876991 * r29876987;
        double r29876993 = r29876989 - r29876992;
        double r29876994 = r29876980 ? r29876993 : r29876978;
        double r29876995 = r29876971 ? r29876978 : r29876994;
        return r29876995;
}

Error

Bits error versus x

Bits error versus y

Target

Original	21.0
Target	0.2
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y \lt -3693.8482788297247:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891:\\ \;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -3480617740589270.5 or 4005471382996.025 < y

   1. Initial program 43.9

      \[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]

   2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(x + 1.0 \cdot \frac{1}{y}\right) - 1.0 \cdot \frac{x}{y}}\]

   3. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)}\]

   if -3480617740589270.5 < y < 4005471382996.025

   1. Initial program 0.4

      \[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
   2. Using strategy rm

   3. Applied flip3-+ 0.5

      \[\leadsto 1.0 - \frac{\left(1.0 - x\right) \cdot y}{\color{blue}{\frac{{y}^{3} + {1.0}^{3}}{y \cdot y + \left(1.0 \cdot 1.0 - y \cdot 1.0\right)}}}\]

   4. Applied associate-/r/ 0.5

      \[\leadsto 1.0 - \color{blue}{\frac{\left(1.0 - x\right) \cdot y}{{y}^{3} + {1.0}^{3}} \cdot \left(y \cdot y + \left(1.0 \cdot 1.0 - y \cdot 1.0\right)\right)}\]

   5. Simplified 0.5

      \[\leadsto 1.0 - \color{blue}{\frac{y \cdot \left(1.0 - x\right)}{\mathsf{fma}\left(y, y \cdot y, \left(1.0 \cdot 1.0\right) \cdot 1.0\right)}} \cdot \left(y \cdot y + \left(1.0 \cdot 1.0 - y \cdot 1.0\right)\right)\]
   6. Using strategy rm

   7. Applied distribute-rgt-in 0.5

      \[\leadsto 1.0 - \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{y \cdot \left(1.0 - x\right)}{\mathsf{fma}\left(y, y \cdot y, \left(1.0 \cdot 1.0\right) \cdot 1.0\right)} + \left(1.0 \cdot 1.0 - y \cdot 1.0\right) \cdot \frac{y \cdot \left(1.0 - x\right)}{\mathsf{fma}\left(y, y \cdot y, \left(1.0 \cdot 1.0\right) \cdot 1.0\right)}\right)}\]

   8. Applied associate--r+ 0.3

      \[\leadsto \color{blue}{\left(1.0 - \left(y \cdot y\right) \cdot \frac{y \cdot \left(1.0 - x\right)}{\mathsf{fma}\left(y, y \cdot y, \left(1.0 \cdot 1.0\right) \cdot 1.0\right)}\right) - \left(1.0 \cdot 1.0 - y \cdot 1.0\right) \cdot \frac{y \cdot \left(1.0 - x\right)}{\mathsf{fma}\left(y, y \cdot y, \left(1.0 \cdot 1.0\right) \cdot 1.0\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3480617740589270.5:\\ \;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 4005471382996.025:\\ \;\;\;\;\left(1.0 - \frac{\left(1.0 - x\right) \cdot y}{\mathsf{fma}\left(y, y \cdot y, \left(1.0 \cdot 1.0\right) \cdot 1.0\right)} \cdot \left(y \cdot y\right)\right) - \left(1.0 \cdot 1.0 - 1.0 \cdot y\right) \cdot \frac{\left(1.0 - x\right) \cdot y}{\mathsf{fma}\left(y, y \cdot y, \left(1.0 \cdot 1.0\right) \cdot 1.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.0, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))