Average Error: 1.9 → 1.9
Time: 26.4s
Precision: binary64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
↓
\[\frac{x \cdot e^{\mathsf{fma}\left(\log z, y, \mathsf{fma}\left(\log a, t + -1, -b\right)\right)}}{y} \]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{x \cdot e^{\mathsf{fma}\left(\log z, y, \mathsf{fma}\left(\log a, t + -1, -b\right)\right)}}{y}

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))
↓
(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (fma (log z) y (fma (log a) (+ t -1.0) (- b))))) y))
double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp(((y * log(z)) + ((t - 1.0) * log(a))) - b)) / y;
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp(fma(log(z), y, fma(log(a), (t + -1.0), -b)))) / y;
}

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

Target

Original1.9
Target11.0
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array} \]

Derivation

  1. Initial program 1.9

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

  2. Applied add-exp-log_binary641.9

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

  3. Simplified1.9

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

  4. Final simplification1.9

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

Reproduce

herbie shell --seed 2022117 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))