Average Error: 2.3 → 0.5
Time: 15.8s
Precision: binary64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
↓
\[x \cdot {e}^{\left(\log z \cdot y - \mathsf{fma}\left(a, z + b, y \cdot t\right)\right)} \]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot {e}^{\left(\log z \cdot y - \mathsf{fma}\left(a, z + b, y \cdot t\right)\right)}

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

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

Derivation

  1. Initial program 2.3

    \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]

  2. Simplified0.3

    \[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]

  3. Applied *-un-lft-identity_binary640.3

    \[\leadsto x \cdot e^{\color{blue}{1 \cdot \mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}} \]

  4. Applied exp-prod_binary640.3

    \[\leadsto x \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)\right)}} \]

  5. Simplified0.3

    \[\leadsto x \cdot {\color{blue}{e}}^{\left(\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)\right)} \]

  6. Taylor expanded in z around 0 0.8

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

  7. Simplified0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2022088 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))