Average Error: 2.0 → 0.4
Time: 14.4s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{\left(y \cdot \mathsf{fma}\left(\sqrt[3]{\log z} \cdot \sqrt[3]{\log z}, \sqrt[3]{\log z}, -t \cdot 1\right) + y \cdot \mathsf{fma}\left(-t, 1, t\right)\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{\left(y \cdot \mathsf{fma}\left(\sqrt[3]{\log z} \cdot \sqrt[3]{\log z}, \sqrt[3]{\log z}, -t \cdot 1\right) + y \cdot \mathsf{fma}\left(-t, 1, t\right)\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}
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((((y * fma((cbrt(log(z)) * cbrt(log(z))), cbrt(log(z)), -(t * 1.0))) + (y * fma(-t, 1.0, t))) + (a * ((log(1.0) - ((0.5 * (pow(z, 2.0) / pow(1.0, 2.0))) + (1.0 * z))) - b)))));
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 2.0

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

  2. Taylor expanded around 0 0.4

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right)} - b\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.4

    \[\leadsto x \cdot e^{y \cdot \left(\log z - \color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\]

  5. Applied add-cube-cbrt0.4

    \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\left(\sqrt[3]{\log z} \cdot \sqrt[3]{\log z}\right) \cdot \sqrt[3]{\log z}} - \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\]

  6. Applied prod-diff0.4

    \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\log z} \cdot \sqrt[3]{\log z}, \sqrt[3]{\log z}, -\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{t}, \sqrt[3]{t} \cdot \sqrt[3]{t}, \sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right)} + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\]

  7. Applied distribute-lft-in3.3

    \[\leadsto x \cdot e^{\color{blue}{\left(y \cdot \mathsf{fma}\left(\sqrt[3]{\log z} \cdot \sqrt[3]{\log z}, \sqrt[3]{\log z}, -\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) + y \cdot \mathsf{fma}\left(-\sqrt[3]{t}, \sqrt[3]{t} \cdot \sqrt[3]{t}, \sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right)} + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\]

  8. Simplified3.3

    \[\leadsto x \cdot e^{\left(\color{blue}{y \cdot \mathsf{fma}\left(\sqrt[3]{\log z} \cdot \sqrt[3]{\log z}, \sqrt[3]{\log z}, -t \cdot 1\right)} + y \cdot \mathsf{fma}\left(-\sqrt[3]{t}, \sqrt[3]{t} \cdot \sqrt[3]{t}, \sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\]

  9. Simplified0.4

    \[\leadsto x \cdot e^{\left(y \cdot \mathsf{fma}\left(\sqrt[3]{\log z} \cdot \sqrt[3]{\log z}, \sqrt[3]{\log z}, -t \cdot 1\right) + \color{blue}{y \cdot \mathsf{fma}\left(-t, 1, t\right)}\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\]

  10. Final simplification0.4

    \[\leadsto x \cdot e^{\left(y \cdot \mathsf{fma}\left(\sqrt[3]{\log z} \cdot \sqrt[3]{\log z}, \sqrt[3]{\log z}, -t \cdot 1\right) + y \cdot \mathsf{fma}\left(-t, 1, t\right)\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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 z)) b))))))