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

Average Error: 22.7 → 0.2

Time: 6.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -360301218.9043175 \lor \neg \left(y \le 149564260.861771911\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r4132 = 1.0;
        double r4133 = x;
        double r4134 = r4132 - r4133;
        double r4135 = y;
        double r4136 = r4134 * r4135;
        double r4137 = r4135 + r4132;
        double r4138 = r4136 / r4137;
        double r4139 = r4132 - r4138;
        return r4139;
}

↓

double f(double x, double y) {
        double r4140 = y;
        double r4141 = -360301218.9043175;
        bool r4142 = r4140 <= r4141;
        double r4143 = 149564260.8617719;
        bool r4144 = r4140 <= r4143;
        double r4145 = !r4144;
        bool r4146 = r4142 || r4145;
        double r4147 = 1.0;
        double r4148 = 1.0;
        double r4149 = r4148 / r4140;
        double r4150 = x;
        double r4151 = r4150 / r4140;
        double r4152 = r4149 - r4151;
        double r4153 = r4147 * r4152;
        double r4154 = r4153 + r4150;
        double r4155 = r4140 + r4147;
        double r4156 = r4148 / r4155;
        double r4157 = r4147 - r4150;
        double r4158 = r4157 * r4140;
        double r4159 = r4156 * r4158;
        double r4160 = r4147 - r4159;
        double r4161 = r4146 ? r4154 : r4160;
        return r4161;
}

Error

Bits error versus x

Bits error versus y

Target

Original	22.7
Target	0.2
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -360301218.9043175 or 149564260.8617719 < y

   1. Initial program 46.1

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

   2. Taylor expanded around inf 0.2

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

   3. Simplified 0.2

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

   if -360301218.9043175 < y < 149564260.8617719

   1. Initial program 0.2

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
   2. Using strategy rm

   3. Applied associate-/l* 0.2

      \[\leadsto 1 - \color{blue}{\frac{1 - x}{\frac{y + 1}{y}}}\]
   4. Using strategy rm

   5. Applied div-inv 0.2

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

   6. Applied *-un-lft-identity 0.2

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

   7. Applied times-frac 0.2

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

   8. Simplified 0.2

      \[\leadsto 1 - \frac{1}{y + 1} \cdot \color{blue}{\left(\left(1 - x\right) \cdot y\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -360301218.9043175 \lor \neg \left(y \le 149564260.861771911\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))