Average Error: 29.1 → 29.2
Time: 10.0s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]
\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}
↓
\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r64375 = x;
        double r64376 = y;
        double r64377 = r64375 * r64376;
        double r64378 = z;
        double r64379 = r64377 + r64378;
        double r64380 = r64379 * r64376;
        double r64381 = 27464.7644705;
        double r64382 = r64380 + r64381;
        double r64383 = r64382 * r64376;
        double r64384 = 230661.510616;
        double r64385 = r64383 + r64384;
        double r64386 = r64385 * r64376;
        double r64387 = t;
        double r64388 = r64386 + r64387;
        double r64389 = a;
        double r64390 = r64376 + r64389;
        double r64391 = r64390 * r64376;
        double r64392 = b;
        double r64393 = r64391 + r64392;
        double r64394 = r64393 * r64376;
        double r64395 = c;
        double r64396 = r64394 + r64395;
        double r64397 = r64396 * r64376;
        double r64398 = i;
        double r64399 = r64397 + r64398;
        double r64400 = r64388 / r64399;
        return r64400;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r64401 = x;
        double r64402 = y;
        double r64403 = r64401 * r64402;
        double r64404 = z;
        double r64405 = r64403 + r64404;
        double r64406 = r64405 * r64402;
        double r64407 = 27464.7644705;
        double r64408 = r64406 + r64407;
        double r64409 = r64408 * r64402;
        double r64410 = 230661.510616;
        double r64411 = r64409 + r64410;
        double r64412 = r64411 * r64402;
        double r64413 = t;
        double r64414 = r64412 + r64413;
        double r64415 = 1.0;
        double r64416 = a;
        double r64417 = r64402 + r64416;
        double r64418 = b;
        double r64419 = fma(r64417, r64402, r64418);
        double r64420 = c;
        double r64421 = fma(r64419, r64402, r64420);
        double r64422 = i;
        double r64423 = fma(r64421, r64402, r64422);
        double r64424 = r64423 * r64415;
        double r64425 = r64415 / r64424;
        double r64426 = r64414 * r64425;
        return r64426;
}

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

Bits error versus c

Bits error versus i

Derivation

  1. Initial program 29.1

    \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
  2. Using strategy rm

  3. Applied div-inv29.2

    \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t\right) \cdot \frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}}\]

  4. Simplified29.2

    \[\leadsto \left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t\right) \cdot \color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}}\]

  5. Final simplification29.2

    \[\leadsto \left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))