Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A

Average Error: 0.1 → 0.1

Time: 58.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r13545059 = x;
        double r13545060 = y;
        double r13545061 = r13545059 + r13545060;
        double r13545062 = z;
        double r13545063 = r13545061 + r13545062;
        double r13545064 = t;
        double r13545065 = log(r13545064);
        double r13545066 = r13545062 * r13545065;
        double r13545067 = r13545063 - r13545066;
        double r13545068 = a;
        double r13545069 = 0.5;
        double r13545070 = r13545068 - r13545069;
        double r13545071 = b;
        double r13545072 = r13545070 * r13545071;
        double r13545073 = r13545067 + r13545072;
        return r13545073;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r13545074 = z;
        double r13545075 = 1.0;
        double r13545076 = t;
        double r13545077 = log(r13545076);
        double r13545078 = r13545075 - r13545077;
        double r13545079 = b;
        double r13545080 = a;
        double r13545081 = 0.5;
        double r13545082 = r13545080 - r13545081;
        double r13545083 = x;
        double r13545084 = y;
        double r13545085 = r13545083 + r13545084;
        double r13545086 = fma(r13545079, r13545082, r13545085);
        double r13545087 = fma(r13545074, r13545078, r13545086);
        return r13545087;
}

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

Original	0.1
Target	0.4
Herbie	0.1

\[\left(\left(x + y\right) + \frac{\left(1.0 - {\left(\log t\right)}^{2.0}\right) \cdot z}{1.0 + \log t}\right) + \left(a - 0.5\right) \cdot b\]

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. Simplified 0.1

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

3. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t)))) (* (- a 0.5) b))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))