Average Error: 2.2 → 0.4
Time: 5.7s
Precision: binary64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot \left(\left(z \cdot z\right) \cdot 0.5 + \left(z + b\right)\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot \left(\left(z \cdot z\right) \cdot 0.5 + \left(z + b\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 (exp (- (* y (- (log z) t)) (* a (+ (* (* z z) 0.5) (+ z b)))))))
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 * (log(z) - t)) - (a * (((z * z) * 0.5) + (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 2.2

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

  2. Taylor expanded around 0 0.4

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(-\left(a \cdot b + \left(a \cdot z + 0.5 \cdot \left(a \cdot {z}^{2}\right)\right)\right)\right)}}\]

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2020224 
(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))))))