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

Average Error: 10.1 → 0.2

Time: 2.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.26172555651194866 \cdot 10^{69} \lor \neg \left(z \le 1.04217124031497941 \cdot 10^{38}\right):\\ \;\;\;\;\left(\frac{x}{\frac{z}{y}} + 1 \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r669086 = x;
        double r669087 = y;
        double r669088 = z;
        double r669089 = r669087 - r669088;
        double r669090 = 1.0;
        double r669091 = r669089 + r669090;
        double r669092 = r669086 * r669091;
        double r669093 = r669092 / r669088;
        return r669093;
}

↓

double f(double x, double y, double z) {
        double r669094 = z;
        double r669095 = -3.2617255565119487e+69;
        bool r669096 = r669094 <= r669095;
        double r669097 = 1.0421712403149794e+38;
        bool r669098 = r669094 <= r669097;
        double r669099 = !r669098;
        bool r669100 = r669096 || r669099;
        double r669101 = x;
        double r669102 = y;
        double r669103 = r669094 / r669102;
        double r669104 = r669101 / r669103;
        double r669105 = 1.0;
        double r669106 = r669101 / r669094;
        double r669107 = r669105 * r669106;
        double r669108 = r669104 + r669107;
        double r669109 = r669108 - r669101;
        double r669110 = r669101 * r669102;
        double r669111 = r669110 / r669094;
        double r669112 = r669111 + r669107;
        double r669113 = r669112 - r669101;
        double r669114 = r669100 ? r669109 : r669113;
        return r669114;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.1
Target	0.5
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -3.2617255565119487e+69 or 1.0421712403149794e+38 < z

   1. Initial program 19.5

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

   2. Taylor expanded around 0 6.1

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

   4. Applied associate-/l* 0.1

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

   if -3.2617255565119487e+69 < z < 1.0421712403149794e+38

   1. Initial program 0.6

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

   2. Taylor expanded around 0 0.4

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.26172555651194866 \cdot 10^{69} \lor \neg \left(z \le 1.04217124031497941 \cdot 10^{38}\right):\\ \;\;\;\;\left(\frac{x}{\frac{z}{y}} + 1 \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

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