Average Error: 0.1 → 0.1
Time: 23.4s
Precision: 64
\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]
↓
\[\mathsf{fma}\left(b, a - 0.5, \mathsf{fma}\left(\log \left(\frac{e}{t}\right), z, y\right)\right) + x\]
\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(\log \left(\frac{e}{t}\right), z, y\right)\right) + x
double f(double x, double y, double z, double t, double a, double b) {
        double r309137 = x;
        double r309138 = y;
        double r309139 = r309137 + r309138;
        double r309140 = z;
        double r309141 = r309139 + r309140;
        double r309142 = t;
        double r309143 = log(r309142);
        double r309144 = r309140 * r309143;
        double r309145 = r309141 - r309144;
        double r309146 = a;
        double r309147 = 0.5;
        double r309148 = r309146 - r309147;
        double r309149 = b;
        double r309150 = r309148 * r309149;
        double r309151 = r309145 + r309150;
        return r309151;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r309152 = b;
        double r309153 = a;
        double r309154 = 0.5;
        double r309155 = r309153 - r309154;
        double r309156 = exp(1.0);
        double r309157 = t;
        double r309158 = r309156 / r309157;
        double r309159 = log(r309158);
        double r309160 = z;
        double r309161 = y;
        double r309162 = fma(r309159, r309160, r309161);
        double r309163 = fma(r309152, r309155, r309162);
        double r309164 = x;
        double r309165 = r309163 + r309164;
        return r309165;
}

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
Target0.3
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

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

  4. Applied add-log-exp0.1

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

  5. Applied diff-log0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019235 +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)))