Average Error: 2.0 → 0.5
Time: 6.7s
Precision: binary64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot \left(b + \left(z + \left(z \cdot z\right) \cdot \left(z \cdot 0.3333333333333333 + 0.5\right)\right)\right)}\]
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(b + \left(z + \left(z \cdot z\right) \cdot \left(z \cdot 0.3333333333333333 + 0.5\right)\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 (+ b (+ z (* (* z z) (+ (* z 0.3333333333333333) 0.5)))))))))
double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (x * ((double) exp(((double) (((double) (y * ((double) (((double) log(z)) - t)))) + ((double) (a * ((double) (((double) log(((double) (1.0 - z)))) - b))))))))));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (x * ((double) exp(((double) (((double) (y * ((double) (((double) log(z)) - t)))) - ((double) (a * ((double) (b + ((double) (z + ((double) (((double) (z * z)) * ((double) (((double) (z * 0.3333333333333333)) + 0.5))))))))))))))));
}

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 0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(-\left(z + \left(0.3333333333333333 \cdot {z}^{3} + 0.5 \cdot {z}^{2}\right)\right)\right)} - b\right)}\]

  3. Simplified0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\left(-z\right) - \left(z \cdot z\right) \cdot \left(z \cdot 0.3333333333333333 + 0.5\right)\right)} - b\right)}\]

  4. Final simplification0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) - a \cdot \left(b + \left(z + \left(z \cdot z\right) \cdot \left(z \cdot 0.3333333333333333 + 0.5\right)\right)\right)}\]

Reproduce

herbie shell --seed 2020219 
(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))))))