Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Average Error: 10.3 → 0.4

Time: 11.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.253272645269980309068047732029063302043 \cdot 10^{-46} \lor \neg \left(x \le 7.454484818730052574552465194726905505901 \cdot 10^{96}\right):\\ \;\;\;\;\left(1 \cdot \frac{x}{z} - x\right) + \frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}} + \left(1 \cdot \frac{x}{z} - x\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r559256 = x;
        double r559257 = y;
        double r559258 = z;
        double r559259 = r559257 - r559258;
        double r559260 = 1.0;
        double r559261 = r559259 + r559260;
        double r559262 = r559256 * r559261;
        double r559263 = r559262 / r559258;
        return r559263;
}

↓

double f(double x, double y, double z) {
        double r559264 = x;
        double r559265 = -1.2532726452699803e-46;
        bool r559266 = r559264 <= r559265;
        double r559267 = 7.454484818730053e+96;
        bool r559268 = r559264 <= r559267;
        double r559269 = !r559268;
        bool r559270 = r559266 || r559269;
        double r559271 = 1.0;
        double r559272 = z;
        double r559273 = r559264 / r559272;
        double r559274 = r559271 * r559273;
        double r559275 = r559274 - r559264;
        double r559276 = y;
        double r559277 = r559272 / r559276;
        double r559278 = r559264 / r559277;
        double r559279 = r559275 + r559278;
        double r559280 = 1.0;
        double r559281 = r559276 * r559264;
        double r559282 = r559272 / r559281;
        double r559283 = r559280 / r559282;
        double r559284 = r559283 + r559275;
        double r559285 = r559270 ? r559279 : r559284;
        return r559285;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.3
Target	0.5
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -1.2532726452699803e-46 or 7.454484818730053e+96 < x

   1. Initial program 27.0

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 0.3

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

   5. Applied add-cube-cbrt 1.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{1 \cdot z} \cdot \left(\left(y - z\right) + 1\right)\]

   6. Applied times-frac 1.4

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \frac{\sqrt[3]{x}}{z}\right)} \cdot \left(\left(y - z\right) + 1\right)\]

   7. Applied associate-*l* 1.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \left(\left(y - z\right) + 1\right)\right)}\]

   8. Taylor expanded around 0 9.0

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

   9. Simplified 0.2

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

   if -1.2532726452699803e-46 < x < 7.454484818730053e+96

   1. Initial program 1.2

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

   2. Simplified 14.2

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 14.2

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

   5. Applied add-cube-cbrt 15.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{1 \cdot z} \cdot \left(\left(y - z\right) + 1\right)\]

   6. Applied times-frac 15.2

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \frac{\sqrt[3]{x}}{z}\right)} \cdot \left(\left(y - z\right) + 1\right)\]

   7. Applied associate-*l* 8.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \left(\left(y - z\right) + 1\right)\right)}\]

   8. Taylor expanded around 0 0.5

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

   9. Simplified 4.7

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}} + \left(1 \cdot \frac{x}{z} - x\right)}\]
   10. Using strategy rm

   11. Applied clear-num 4.7

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

   12. Simplified 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.253272645269980309068047732029063302043 \cdot 10^{-46} \lor \neg \left(x \le 7.454484818730052574552465194726905505901 \cdot 10^{96}\right):\\ \;\;\;\;\left(1 \cdot \frac{x}{z} - x\right) + \frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}} + \left(1 \cdot \frac{x}{z} - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

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