Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B

Average Error: 15.3 → 0.7

Time: 9.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.548240730790440881436662006949490760264 \cdot 10^{219}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.442433262831919710603838797359517016977 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le -0.0 \lor \neg \left(\frac{y}{z} \le 2.381732747802579320943017921487277387689 \cdot 10^{244}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r384642 = x;
        double r384643 = y;
        double r384644 = z;
        double r384645 = r384643 / r384644;
        double r384646 = t;
        double r384647 = r384645 * r384646;
        double r384648 = r384647 / r384646;
        double r384649 = r384642 * r384648;
        return r384649;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r384650 = y;
        double r384651 = z;
        double r384652 = r384650 / r384651;
        double r384653 = -1.548240730790441e+219;
        bool r384654 = r384652 <= r384653;
        double r384655 = x;
        double r384656 = r384655 / r384651;
        double r384657 = r384650 * r384656;
        double r384658 = -1.4424332628319197e-140;
        bool r384659 = r384652 <= r384658;
        double r384660 = r384652 * r384655;
        double r384661 = -0.0;
        bool r384662 = r384652 <= r384661;
        double r384663 = 2.3817327478025793e+244;
        bool r384664 = r384652 <= r384663;
        double r384665 = !r384664;
        bool r384666 = r384662 || r384665;
        double r384667 = r384651 / r384650;
        double r384668 = r384655 / r384667;
        double r384669 = r384666 ? r384657 : r384668;
        double r384670 = r384659 ? r384660 : r384669;
        double r384671 = r384654 ? r384657 : r384670;
        return r384671;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	15.3
Target	1.6
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.206722051230450047215521150762600712224 \cdot 10^{245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.90752223693390632993316700759382836344 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.658954423153415216825328199697215652986 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.008718050240713347941382056648619307142 \cdot 10^{217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -1.548240730790441e+219 or -1.4424332628319197e-140 < (/ y z) < -0.0 or 2.3817327478025793e+244 < (/ y z)

   1. Initial program 25.6

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

   2. Simplified 1.0

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

   4. Applied *-un-lft-identity 1.0

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

   5. Applied times-frac 16.9

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

   6. Simplified 16.9

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

   8. Applied pow1 16.9

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

   9. Applied pow1 16.9

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

   10. Applied pow-prod-down 16.9

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

   11. Simplified 0.9

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

   if -1.548240730790441e+219 < (/ y z) < -1.4424332628319197e-140

   1. Initial program 8.5

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

   2. Simplified 10.5

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

   4. Applied *-un-lft-identity 10.5

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

   5. Applied times-frac 0.3

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

   6. Simplified 0.3

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

   if -0.0 < (/ y z) < 2.3817327478025793e+244

   1. Initial program 10.9

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

   2. Simplified 7.4

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

   4. Applied associate-/l* 0.7

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.548240730790440881436662006949490760264 \cdot 10^{219}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.442433262831919710603838797359517016977 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le -0.0 \lor \neg \left(\frac{y}{z} \le 2.381732747802579320943017921487277387689 \cdot 10^{244}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

  (* x (/ (* (/ y z) t) t)))