Average Error: 3.9 → 2.7
Time: 8.5s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\]
\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(z \cdot \sqrt{t + a}, \frac{1}{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 r64187 = x;
        double r64188 = y;
        double r64189 = 2.0;
        double r64190 = z;
        double r64191 = t;
        double r64192 = a;
        double r64193 = r64191 + r64192;
        double r64194 = sqrt(r64193);
        double r64195 = r64190 * r64194;
        double r64196 = r64195 / r64191;
        double r64197 = b;
        double r64198 = c;
        double r64199 = r64197 - r64198;
        double r64200 = 5.0;
        double r64201 = 6.0;
        double r64202 = r64200 / r64201;
        double r64203 = r64192 + r64202;
        double r64204 = 3.0;
        double r64205 = r64191 * r64204;
        double r64206 = r64189 / r64205;
        double r64207 = r64203 - r64206;
        double r64208 = r64199 * r64207;
        double r64209 = r64196 - r64208;
        double r64210 = r64189 * r64209;
        double r64211 = exp(r64210);
        double r64212 = r64188 * r64211;
        double r64213 = r64187 + r64212;
        double r64214 = r64187 / r64213;
        return r64214;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r64215 = x;
        double r64216 = y;
        double r64217 = 2.0;
        double r64218 = z;
        double r64219 = t;
        double r64220 = a;
        double r64221 = r64219 + r64220;
        double r64222 = sqrt(r64221);
        double r64223 = r64218 * r64222;
        double r64224 = 1.0;
        double r64225 = r64224 / r64219;
        double r64226 = 5.0;
        double r64227 = 6.0;
        double r64228 = r64226 / r64227;
        double r64229 = r64220 + r64228;
        double r64230 = 3.0;
        double r64231 = r64219 * r64230;
        double r64232 = r64217 / r64231;
        double r64233 = r64229 - r64232;
        double r64234 = b;
        double r64235 = c;
        double r64236 = r64234 - r64235;
        double r64237 = r64233 * r64236;
        double r64238 = -r64237;
        double r64239 = fma(r64223, r64225, r64238);
        double r64240 = -r64236;
        double r64241 = r64240 + r64236;
        double r64242 = r64233 * r64241;
        double r64243 = r64239 + r64242;
        double r64244 = r64217 * r64243;
        double r64245 = exp(r64244);
        double r64246 = r64216 * r64245;
        double r64247 = r64215 + r64246;
        double r64248 = r64215 / r64247;
        return r64248;
}

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 3.9

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

  3. Applied div-inv3.9

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(z \cdot \sqrt{t + a}\right) \cdot \frac{1}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

  4. Applied prod-diff23.0

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \mathsf{fma}\left(-\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right), b - c, \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right)\right)}}}\]

  5. Simplified2.7

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \color{blue}{\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)}\right)}}\]

  6. Final simplification2.7

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\]

Reproduce

herbie shell --seed 2020035 +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)))))))))))