Average Error: 0.1 → 0.1
Time: 10.9s
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 r99757 = x;
        double r99758 = y;
        double r99759 = log(r99758);
        double r99760 = r99757 * r99759;
        double r99761 = z;
        double r99762 = r99760 + r99761;
        double r99763 = t;
        double r99764 = r99762 + r99763;
        double r99765 = a;
        double r99766 = r99764 + r99765;
        double r99767 = b;
        double r99768 = 0.5;
        double r99769 = r99767 - r99768;
        double r99770 = c;
        double r99771 = log(r99770);
        double r99772 = r99769 * r99771;
        double r99773 = r99766 + r99772;
        double r99774 = i;
        double r99775 = r99758 * r99774;
        double r99776 = r99773 + r99775;
        return r99776;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r99777 = i;
        double r99778 = y;
        double r99779 = c;
        double r99780 = log(r99779);
        double r99781 = b;
        double r99782 = 0.5;
        double r99783 = r99781 - r99782;
        double r99784 = r99780 * r99783;
        double r99785 = x;
        double r99786 = log(r99778);
        double r99787 = z;
        double r99788 = fma(r99785, r99786, r99787);
        double r99789 = t;
        double r99790 = a;
        double r99791 = r99789 + r99790;
        double r99792 = r99788 + r99791;
        double r99793 = r99784 + r99792;
        double r99794 = fma(r99777, r99778, r99793);
        return r99794;
}

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 2020065 +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)))