Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I

Average Error: 3.8 → 3.1

Time: 32.8s

Precision: 64

Math

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

↓

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r289056 = x;
        double r289057 = y;
        double r289058 = 2.0;
        double r289059 = z;
        double r289060 = t;
        double r289061 = a;
        double r289062 = r289060 + r289061;
        double r289063 = sqrt(r289062);
        double r289064 = r289059 * r289063;
        double r289065 = r289064 / r289060;
        double r289066 = b;
        double r289067 = c;
        double r289068 = r289066 - r289067;
        double r289069 = 5.0;
        double r289070 = 6.0;
        double r289071 = r289069 / r289070;
        double r289072 = r289061 + r289071;
        double r289073 = 3.0;
        double r289074 = r289060 * r289073;
        double r289075 = r289058 / r289074;
        double r289076 = r289072 - r289075;
        double r289077 = r289068 * r289076;
        double r289078 = r289065 - r289077;
        double r289079 = r289058 * r289078;
        double r289080 = exp(r289079);
        double r289081 = r289057 * r289080;
        double r289082 = r289056 + r289081;
        double r289083 = r289056 / r289082;
        return r289083;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r289084 = x;
        double r289085 = y;
        double r289086 = 2.0;
        double r289087 = z;
        double r289088 = t;
        double r289089 = a;
        double r289090 = r289088 + r289089;
        double r289091 = sqrt(r289090);
        double r289092 = r289088 / r289091;
        double r289093 = r289087 / r289092;
        double r289094 = b;
        double r289095 = c;
        double r289096 = r289094 - r289095;
        double r289097 = 5.0;
        double r289098 = 6.0;
        double r289099 = r289097 / r289098;
        double r289100 = r289089 + r289099;
        double r289101 = 3.0;
        double r289102 = r289088 * r289101;
        double r289103 = r289086 / r289102;
        double r289104 = r289100 - r289103;
        double r289105 = r289096 * r289104;
        double r289106 = r289093 - r289105;
        double r289107 = r289086 * r289106;
        double r289108 = exp(r289107);
        double r289109 = r289085 * r289108;
        double r289110 = r289084 + r289109;
        double r289111 = r289084 / r289110;
        return r289111;
}

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

Bits error versus c

Target

Original	3.8
Target	3.2
Herbie	3.1

\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581057561884576920117070548 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333333703407674875052180141211 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547088010424937268931048836 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

1. Initial program 3.8

   \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
2. Using strategy rm

3. Applied associate-/l* 3.1

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{\frac{t}{\sqrt{t + a}}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

4. Final simplification 3.1

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :herbie-target
  (if (< t -2.1183266448915811e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.83333333333333337 c)) (* a b))))))) (if (< t 5.19658877065154709e-123) (/ x (+ x (* y (exp (* 2 (/ (- (* (* z (sqrt (+ t a))) (* (* 3 t) (- a (/ 5 6)))) (* (- (* (+ (/ 5 6) a) (* 3 t)) 2) (* (- a (/ 5 6)) (* (- b c) t)))) (* (* (* t t) 3) (- a (/ 5 6))))))))) (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3))))))))))))

  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))