Average Error: 2.3 → 0.4
Time: 33.4s
Precision: binary64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
↓
\[x \cdot e^{\log z \cdot y - \mathsf{fma}\left(a, z + b, y \cdot t\right)} \]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{\log z \cdot y - \mathsf{fma}\left(a, z + b, y \cdot t\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 (exp (- (* (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 * exp((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 add-cbrt-cube_binary640.3

    \[\leadsto x \cdot \color{blue}{\sqrt[3]{\left(e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)} \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}\right) \cdot e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}}} \]

  4. Simplified0.3

    \[\leadsto x \cdot \sqrt[3]{\color{blue}{{\left(e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)\right)}\right)}^{3}}} \]

  5. Taylor expanded in z around 0 0.6

    \[\leadsto x \cdot \sqrt[3]{{\left(e^{\color{blue}{\log z \cdot y - \left(a \cdot b + \left(a \cdot z + y \cdot t\right)\right)}}\right)}^{3}} \]

  6. Simplified0.4

    \[\leadsto x \cdot \sqrt[3]{{\left(e^{\color{blue}{\log z \cdot y - \mathsf{fma}\left(a, b + z, y \cdot t\right)}}\right)}^{3}} \]

  7. Taylor expanded in z around inf 0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2022068 
(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))))))