Average Error: 4.0 → 1.7
Time: 40.0s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\frac{t}{\sqrt[3]{z}}} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)}, x\right)}\]
\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}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\frac{t}{\sqrt[3]{z}}} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r21770220 = x;
        double r21770221 = y;
        double r21770222 = 2.0;
        double r21770223 = z;
        double r21770224 = t;
        double r21770225 = a;
        double r21770226 = r21770224 + r21770225;
        double r21770227 = sqrt(r21770226);
        double r21770228 = r21770223 * r21770227;
        double r21770229 = r21770228 / r21770224;
        double r21770230 = b;
        double r21770231 = c;
        double r21770232 = r21770230 - r21770231;
        double r21770233 = 5.0;
        double r21770234 = 6.0;
        double r21770235 = r21770233 / r21770234;
        double r21770236 = r21770225 + r21770235;
        double r21770237 = 3.0;
        double r21770238 = r21770224 * r21770237;
        double r21770239 = r21770222 / r21770238;
        double r21770240 = r21770236 - r21770239;
        double r21770241 = r21770232 * r21770240;
        double r21770242 = r21770229 - r21770241;
        double r21770243 = r21770222 * r21770242;
        double r21770244 = exp(r21770243);
        double r21770245 = r21770221 * r21770244;
        double r21770246 = r21770220 + r21770245;
        double r21770247 = r21770220 / r21770246;
        return r21770247;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r21770248 = x;
        double r21770249 = y;
        double r21770250 = 2.0;
        double r21770251 = c;
        double r21770252 = b;
        double r21770253 = r21770251 - r21770252;
        double r21770254 = 5.0;
        double r21770255 = 6.0;
        double r21770256 = r21770254 / r21770255;
        double r21770257 = t;
        double r21770258 = r21770250 / r21770257;
        double r21770259 = 3.0;
        double r21770260 = r21770258 / r21770259;
        double r21770261 = a;
        double r21770262 = r21770260 - r21770261;
        double r21770263 = r21770256 - r21770262;
        double r21770264 = r21770261 + r21770257;
        double r21770265 = sqrt(r21770264);
        double r21770266 = z;
        double r21770267 = cbrt(r21770266);
        double r21770268 = r21770257 / r21770267;
        double r21770269 = r21770265 / r21770268;
        double r21770270 = r21770267 * r21770267;
        double r21770271 = r21770269 * r21770270;
        double r21770272 = fma(r21770253, r21770263, r21770271);
        double r21770273 = r21770250 * r21770272;
        double r21770274 = exp(r21770273);
        double r21770275 = fma(r21770249, r21770274, r21770248);
        double r21770276 = r21770248 / r21770275;
        return r21770276;
}

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

Original4.0
Target3.0
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581057561884576920117070548 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333333703407674875052180141211 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547088010424937268931048836 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

  1. Initial program 4.0

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

  2. Simplified1.7

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

  4. Applied add-cube-cbrt1.7

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

  5. Applied *-un-lft-identity1.7

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

  6. Applied times-frac1.7

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

  7. Applied *-un-lft-identity1.7

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

  8. Applied sqrt-prod1.7

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

  9. Applied times-frac1.7

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

  10. Simplified1.7

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

  11. Final simplification1.7

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2.0 (/ (- (* (* z (sqrt (+ t a))) (* (* 3.0 t) (- a (/ 5.0 6.0)))) (* (- (* (+ (/ 5.0 6.0) a) (* 3.0 t)) 2.0) (* (- a (/ 5.0 6.0)) (* (- b c) t)))) (* (* (* t t) 3.0) (- a (/ 5.0 6.0))))))))) (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))))

  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))