Average Error: 0.1 → 0.1
Time: 13.2s
Precision: 64
\[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
↓
\[\mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right) + \left(\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right)\right)\]
\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i
↓
\mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right) + \left(\mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r82570 = x;
        double r82571 = y;
        double r82572 = log(r82571);
        double r82573 = r82570 * r82572;
        double r82574 = z;
        double r82575 = r82573 + r82574;
        double r82576 = t;
        double r82577 = r82575 + r82576;
        double r82578 = a;
        double r82579 = r82577 + r82578;
        double r82580 = b;
        double r82581 = 0.5;
        double r82582 = r82580 - r82581;
        double r82583 = c;
        double r82584 = log(r82583);
        double r82585 = r82582 * r82584;
        double r82586 = r82579 + r82585;
        double r82587 = i;
        double r82588 = r82571 * r82587;
        double r82589 = r82586 + r82588;
        return r82589;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r82590 = i;
        double r82591 = y;
        double r82592 = c;
        double r82593 = log(r82592);
        double r82594 = b;
        double r82595 = 0.5;
        double r82596 = r82594 - r82595;
        double r82597 = r82593 * r82596;
        double r82598 = x;
        double r82599 = log(r82591);
        double r82600 = z;
        double r82601 = fma(r82598, r82599, r82600);
        double r82602 = t;
        double r82603 = a;
        double r82604 = r82602 + r82603;
        double r82605 = r82601 + r82604;
        double r82606 = r82597 + r82605;
        double r82607 = fma(r82590, r82591, r82606);
        return r82607;
}

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 0.1

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, \mathsf{fma}\left(x, \log y, z\right) + \left(t + a\right)\right)\right)}\]
  3. Using strategy rm

  4. Applied fma-udef0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))