Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Average Error: 1.9 → 1.9

Time: 15.5s

Precision: binary64

Cost: 20160

Math

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

↓

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

FPCore

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

C

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(((y * log(z)) + (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

Original	1.9
Target	11.2
Herbie	1.9

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

Alternatives

Alternative 1
Error	2.0
Cost	20224

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

Alternative 2
Error	16.1
Cost	53635

\[\begin{array}{l} \mathbf{if}\;b \leq -1.3978983991038887 \cdot 10^{-256}:\\ \;\;\;\;\frac{x \cdot \left(\sqrt{{z}^{y} \cdot \frac{{a}^{\left(t - 1\right)}}{e^{b}}} \cdot \sqrt{{z}^{y} \cdot \frac{{a}^{\left(t - 1\right)}}{e^{b}}}\right)}{y}\\ \mathbf{elif}\;b \leq 3.7659505387627232 \cdot 10^{-205}:\\ \;\;\;\;\frac{\left(x \cdot {a}^{\left(\frac{t - 1}{2}\right)}\right) \cdot \frac{{a}^{\left(\frac{t - 1}{2}\right)}}{e^{b}}}{y}\\ \mathbf{elif}\;b \leq 473.3663386303636:\\ \;\;\;\;\frac{x \cdot \left(\sqrt{{z}^{y} \cdot \frac{{a}^{\left(t - 1\right)}}{e^{b}}} \cdot \sqrt{{z}^{y} \cdot \frac{{a}^{\left(t - 1\right)}}{e^{b}}}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{{a}^{\left(t - 1\right)}}{e^{b}}} \cdot \left(x \cdot \sqrt{\frac{{a}^{\left(t - 1\right)}}{e^{b}}}\right)}{y}\\ \end{array}\]

Alternative 3
Error	16.1
Cost	53132

\[\begin{array}{l} \mathbf{if}\;b \leq -1.6629075640860505 \cdot 10^{-256} \lor \neg \left(b \leq 7.01325613414813 \cdot 10^{-206}\right) \land b \leq 30.30279161798115:\\ \;\;\;\;\frac{x \cdot \left(\sqrt{{z}^{y} \cdot \frac{{a}^{\left(t - 1\right)}}{e^{b}}} \cdot \sqrt{{z}^{y} \cdot \frac{{a}^{\left(t - 1\right)}}{e^{b}}}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{{a}^{\left(t - 1\right)}}{e^{b}}} \cdot \left(x \cdot \sqrt{\frac{{a}^{\left(t - 1\right)}}{e^{b}}}\right)}{y}\\ \end{array}\]

Alternative 4
Error	18.7
Cost	52672

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

Alternative 5
Error	28.0
Cost	52928

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

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. Simplified 1.9

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

3. Final simplification 1.9

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

Reproduce

herbie shell --seed 2021043 
(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))