Average Error: 0.1 → 0.1
Time: 5.1s
Precision: 64
\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]
↓
\[\mathsf{fma}\left(b, a - 0.5, \mathsf{fma}\left(z, 1 - \log t, x + y\right)\right)\]
\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b
↓
\mathsf{fma}\left(b, a - 0.5, \mathsf{fma}\left(z, 1 - \log t, x + y\right)\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r336265 = x;
        double r336266 = y;
        double r336267 = r336265 + r336266;
        double r336268 = z;
        double r336269 = r336267 + r336268;
        double r336270 = t;
        double r336271 = log(r336270);
        double r336272 = r336268 * r336271;
        double r336273 = r336269 - r336272;
        double r336274 = a;
        double r336275 = 0.5;
        double r336276 = r336274 - r336275;
        double r336277 = b;
        double r336278 = r336276 * r336277;
        double r336279 = r336273 + r336278;
        return r336279;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r336280 = b;
        double r336281 = a;
        double r336282 = 0.5;
        double r336283 = r336281 - r336282;
        double r336284 = z;
        double r336285 = 1.0;
        double r336286 = t;
        double r336287 = log(r336286);
        double r336288 = r336285 - r336287;
        double r336289 = x;
        double r336290 = y;
        double r336291 = r336289 + r336290;
        double r336292 = fma(r336284, r336288, r336291);
        double r336293 = fma(r336280, r336283, r336292);
        return r336293;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original0.1
Target0.4
Herbie0.1
\[\left(\left(x + y\right) + \frac{\left(1 - {\left(\log t\right)}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b\]

Derivation

  1. Initial program 0.1

    \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(b, a - 0.5, \left(\left(x + y\right) + z\right) - z \cdot \log t\right)}\]

  3. Taylor expanded around 0 0.1

    \[\leadsto \mathsf{fma}\left(b, a - 0.5, \color{blue}{\left(x + \left(z + y\right)\right) - \log t \cdot z}\right)\]

  4. Simplified0.1

    \[\leadsto \mathsf{fma}\left(b, a - 0.5, \color{blue}{\mathsf{fma}\left(z, 1 - \log t, x + y\right)}\right)\]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(b, a - 0.5, \mathsf{fma}\left(z, 1 - \log t, x + y\right)\right)\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1 (pow (log t) 2)) z) (+ 1 (log t)))) (* (- a 0.5) b))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))