Average Error: 4.0 → 1.7
Time: 40.5s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\frac{t}{\sqrt[3]{z}}} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)}, x\right)}\]
\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 r23843208 = x;
        double r23843209 = y;
        double r23843210 = 2.0;
        double r23843211 = z;
        double r23843212 = t;
        double r23843213 = a;
        double r23843214 = r23843212 + r23843213;
        double r23843215 = sqrt(r23843214);
        double r23843216 = r23843211 * r23843215;
        double r23843217 = r23843216 / r23843212;
        double r23843218 = b;
        double r23843219 = c;
        double r23843220 = r23843218 - r23843219;
        double r23843221 = 5.0;
        double r23843222 = 6.0;
        double r23843223 = r23843221 / r23843222;
        double r23843224 = r23843213 + r23843223;
        double r23843225 = 3.0;
        double r23843226 = r23843212 * r23843225;
        double r23843227 = r23843210 / r23843226;
        double r23843228 = r23843224 - r23843227;
        double r23843229 = r23843220 * r23843228;
        double r23843230 = r23843217 - r23843229;
        double r23843231 = r23843210 * r23843230;
        double r23843232 = exp(r23843231);
        double r23843233 = r23843209 * r23843232;
        double r23843234 = r23843208 + r23843233;
        double r23843235 = r23843208 / r23843234;
        return r23843235;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r23843236 = x;
        double r23843237 = y;
        double r23843238 = 2.0;
        double r23843239 = c;
        double r23843240 = b;
        double r23843241 = r23843239 - r23843240;
        double r23843242 = 5.0;
        double r23843243 = 6.0;
        double r23843244 = r23843242 / r23843243;
        double r23843245 = t;
        double r23843246 = r23843238 / r23843245;
        double r23843247 = 3.0;
        double r23843248 = r23843246 / r23843247;
        double r23843249 = a;
        double r23843250 = r23843248 - r23843249;
        double r23843251 = r23843244 - r23843250;
        double r23843252 = r23843249 + r23843245;
        double r23843253 = sqrt(r23843252);
        double r23843254 = z;
        double r23843255 = cbrt(r23843254);
        double r23843256 = r23843245 / r23843255;
        double r23843257 = r23843253 / r23843256;
        double r23843258 = r23843255 * r23843255;
        double r23843259 = r23843257 * r23843258;
        double r23843260 = fma(r23843241, r23843251, r23843259);
        double r23843261 = r23843238 * r23843260;
        double r23843262 = exp(r23843261);
        double r23843263 = fma(r23843237, r23843262, r23843236);
        double r23843264 = r23843236 / r23843263;
        return r23843264;
}

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)))))))))))