Average Error: 2.2 → 0.5
Time: 8.4s
Precision: binary64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(z \cdot \left(-1 - z \cdot 0.5\right) - b\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(z \cdot \left(-1 - z \cdot 0.5\right) - b\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 (- -1.0 (* z 0.5))) 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 * (-1.0 - (z * 0.5))) - 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 0.5

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

  3. Simplified0.5

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

  4. Taylor expanded around 0 0.5

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

  5. Simplified0.5

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

  7. Applied distribute-lft-out--_binary640.5

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

  8. Simplified0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{\left(z \cdot \left(-1 - z \cdot 0.5\right) - b\right)}}\]

  9. Final simplification0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(z \cdot \left(-1 - z \cdot 0.5\right) - b\right)}\]

Reproduce

herbie shell --seed 2020273 
(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))))))