Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Average Error: 12.0 → 2.2

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.54363164202423776 \cdot 10^{-164} \lor \neg \left(z \le 1.03870846195001497 \cdot 10^{-26}\right):\\ \;\;\;\;x \cdot \frac{y}{t - z} + x \cdot \left(-\frac{z}{t - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + x \cdot \left(-z\right)}{t - z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r558157 = x;
        double r558158 = y;
        double r558159 = z;
        double r558160 = r558158 - r558159;
        double r558161 = r558157 * r558160;
        double r558162 = t;
        double r558163 = r558162 - r558159;
        double r558164 = r558161 / r558163;
        return r558164;
}

↓

double f(double x, double y, double z, double t) {
        double r558165 = z;
        double r558166 = -2.5436316420242378e-164;
        bool r558167 = r558165 <= r558166;
        double r558168 = 1.038708461950015e-26;
        bool r558169 = r558165 <= r558168;
        double r558170 = !r558169;
        bool r558171 = r558167 || r558170;
        double r558172 = x;
        double r558173 = y;
        double r558174 = t;
        double r558175 = r558174 - r558165;
        double r558176 = r558173 / r558175;
        double r558177 = r558172 * r558176;
        double r558178 = r558165 / r558175;
        double r558179 = -r558178;
        double r558180 = r558172 * r558179;
        double r558181 = r558177 + r558180;
        double r558182 = r558172 * r558173;
        double r558183 = -r558165;
        double r558184 = r558172 * r558183;
        double r558185 = r558182 + r558184;
        double r558186 = r558185 / r558175;
        double r558187 = r558171 ? r558181 : r558186;
        return r558187;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	12.0
Target	2.2
Herbie	2.2

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

Derivation

1. Split input into 2 regimes

2. if z < -2.5436316420242378e-164 or 1.038708461950015e-26 < z

   1. Initial program 15.1

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

   3. Applied *-un-lft-identity 15.1

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

   4. Applied times-frac 0.6

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

   5. Simplified 0.6

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

   7. Applied div-sub 0.6

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

   9. Applied sub-neg 0.6

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

   10. Applied distribute-lft-in 0.6

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

   if -2.5436316420242378e-164 < z < 1.038708461950015e-26

   1. Initial program 5.5

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

   3. Applied sub-neg 5.5

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

   4. Applied distribute-lft-in 5.5

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

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.54363164202423776 \cdot 10^{-164} \lor \neg \left(z \le 1.03870846195001497 \cdot 10^{-26}\right):\\ \;\;\;\;x \cdot \frac{y}{t - z} + x \cdot \left(-\frac{z}{t - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + x \cdot \left(-z\right)}{t - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

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

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