Average Error: 0.1 → 0.1
Time: 16.2s
Precision: 64
\[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
↓
\[\mathsf{fma}\left(y, i, \log c \cdot \left(b - 0.5\right) + \left(a + \left(\mathsf{fma}\left(x, \log y, z\right) + t\right)\right)\right)\]
\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(y, i, \log c \cdot \left(b - 0.5\right) + \left(a + \left(\mathsf{fma}\left(x, \log y, z\right) + t\right)\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r98638 = x;
        double r98639 = y;
        double r98640 = log(r98639);
        double r98641 = r98638 * r98640;
        double r98642 = z;
        double r98643 = r98641 + r98642;
        double r98644 = t;
        double r98645 = r98643 + r98644;
        double r98646 = a;
        double r98647 = r98645 + r98646;
        double r98648 = b;
        double r98649 = 0.5;
        double r98650 = r98648 - r98649;
        double r98651 = c;
        double r98652 = log(r98651);
        double r98653 = r98650 * r98652;
        double r98654 = r98647 + r98653;
        double r98655 = i;
        double r98656 = r98639 * r98655;
        double r98657 = r98654 + r98656;
        return r98657;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r98658 = y;
        double r98659 = i;
        double r98660 = c;
        double r98661 = log(r98660);
        double r98662 = b;
        double r98663 = 0.5;
        double r98664 = r98662 - r98663;
        double r98665 = r98661 * r98664;
        double r98666 = a;
        double r98667 = x;
        double r98668 = log(r98658);
        double r98669 = z;
        double r98670 = fma(r98667, r98668, r98669);
        double r98671 = t;
        double r98672 = r98670 + r98671;
        double r98673 = r98666 + r98672;
        double r98674 = r98665 + r98673;
        double r98675 = fma(r98658, r98659, r98674);
        return r98675;
}

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

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

  4. Applied fma-udef0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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)))