Average Error: 3.8 → 1.8
Time: 14.7s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\]
\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(\mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r97189 = x;
        double r97190 = y;
        double r97191 = 2.0;
        double r97192 = z;
        double r97193 = t;
        double r97194 = a;
        double r97195 = r97193 + r97194;
        double r97196 = sqrt(r97195);
        double r97197 = r97192 * r97196;
        double r97198 = r97197 / r97193;
        double r97199 = b;
        double r97200 = c;
        double r97201 = r97199 - r97200;
        double r97202 = 5.0;
        double r97203 = 6.0;
        double r97204 = r97202 / r97203;
        double r97205 = r97194 + r97204;
        double r97206 = 3.0;
        double r97207 = r97193 * r97206;
        double r97208 = r97191 / r97207;
        double r97209 = r97205 - r97208;
        double r97210 = r97201 * r97209;
        double r97211 = r97198 - r97210;
        double r97212 = r97191 * r97211;
        double r97213 = exp(r97212);
        double r97214 = r97190 * r97213;
        double r97215 = r97189 + r97214;
        double r97216 = r97189 / r97215;
        return r97216;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r97217 = x;
        double r97218 = y;
        double r97219 = 2.0;
        double r97220 = z;
        double r97221 = t;
        double r97222 = cbrt(r97221);
        double r97223 = r97222 * r97222;
        double r97224 = r97220 / r97223;
        double r97225 = a;
        double r97226 = r97221 + r97225;
        double r97227 = sqrt(r97226);
        double r97228 = r97227 / r97222;
        double r97229 = 5.0;
        double r97230 = 6.0;
        double r97231 = r97229 / r97230;
        double r97232 = r97225 + r97231;
        double r97233 = 3.0;
        double r97234 = r97221 * r97233;
        double r97235 = r97219 / r97234;
        double r97236 = r97232 - r97235;
        double r97237 = b;
        double r97238 = c;
        double r97239 = r97237 - r97238;
        double r97240 = r97236 * r97239;
        double r97241 = -r97240;
        double r97242 = fma(r97224, r97228, r97241);
        double r97243 = -r97239;
        double r97244 = r97243 + r97239;
        double r97245 = r97236 * r97244;
        double r97246 = r97242 + r97245;
        double r97247 = r97219 * r97246;
        double r97248 = exp(r97247);
        double r97249 = r97218 * r97248;
        double r97250 = r97217 + r97249;
        double r97251 = r97217 / r97250;
        return r97251;
}

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

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 add-cube-cbrt3.8

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

  4. Applied times-frac2.6

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

  5. Applied prod-diff22.0

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

  6. Simplified1.8

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

  7. Final simplification1.8

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))