Average Error: 0.1 → 0.1
Time: 7.6s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \]
↓
\[\mathsf{fma}\left(b, a - 0.5, y\right) + \mathsf{fma}\left(z, 1 - \log t, x\right) \]
\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, y\right) + \mathsf{fma}\left(z, 1 - \log t, x\right)

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))
↓
(FPCore (x y z t a b)
 :precision binary64
 (+ (fma b (- a 0.5) y) (fma z (- 1.0 (log t)) x)))
double code(double x, double y, double z, double t, double a, double b) {
	return (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return fma(b, (a - 0.5), y) + fma(z, (1.0 - log(t)), x);
}

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
Target0.4
Herbie0.1
\[\left(\left(x + y\right) + \frac{\left(1 - {\log t}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b \]

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

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

  3. Taylor expanded in z around 0 0.1

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

  4. Simplified0.1

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

  5. Applied +-commutative_binary640.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2022077 
(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) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t)))) (* (- a 0.5) b))

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