Average Error: 3.7 → 1.5
Time: 43.6s
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{\sqrt[3]{a + t}}}{\frac{t}{\sqrt[3]{z}}} \cdot \frac{\sqrt{\sqrt[3]{a + t} \cdot \sqrt[3]{a + t}}}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right)}, x\right)}\]
\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}
↓
\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{\sqrt[3]{a + t}}}{\frac{t}{\sqrt[3]{z}}} \cdot \frac{\sqrt{\sqrt[3]{a + t} \cdot \sqrt[3]{a + t}}}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r13511191 = x;
        double r13511192 = y;
        double r13511193 = 2.0;
        double r13511194 = z;
        double r13511195 = t;
        double r13511196 = a;
        double r13511197 = r13511195 + r13511196;
        double r13511198 = sqrt(r13511197);
        double r13511199 = r13511194 * r13511198;
        double r13511200 = r13511199 / r13511195;
        double r13511201 = b;
        double r13511202 = c;
        double r13511203 = r13511201 - r13511202;
        double r13511204 = 5.0;
        double r13511205 = 6.0;
        double r13511206 = r13511204 / r13511205;
        double r13511207 = r13511196 + r13511206;
        double r13511208 = 3.0;
        double r13511209 = r13511195 * r13511208;
        double r13511210 = r13511193 / r13511209;
        double r13511211 = r13511207 - r13511210;
        double r13511212 = r13511203 * r13511211;
        double r13511213 = r13511200 - r13511212;
        double r13511214 = r13511193 * r13511213;
        double r13511215 = exp(r13511214);
        double r13511216 = r13511192 * r13511215;
        double r13511217 = r13511191 + r13511216;
        double r13511218 = r13511191 / r13511217;
        return r13511218;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r13511219 = x;
        double r13511220 = y;
        double r13511221 = 2.0;
        double r13511222 = c;
        double r13511223 = b;
        double r13511224 = r13511222 - r13511223;
        double r13511225 = 5.0;
        double r13511226 = 6.0;
        double r13511227 = r13511225 / r13511226;
        double r13511228 = t;
        double r13511229 = r13511221 / r13511228;
        double r13511230 = 3.0;
        double r13511231 = r13511229 / r13511230;
        double r13511232 = a;
        double r13511233 = r13511231 - r13511232;
        double r13511234 = r13511227 - r13511233;
        double r13511235 = r13511232 + r13511228;
        double r13511236 = cbrt(r13511235);
        double r13511237 = sqrt(r13511236);
        double r13511238 = z;
        double r13511239 = cbrt(r13511238);
        double r13511240 = r13511228 / r13511239;
        double r13511241 = r13511237 / r13511240;
        double r13511242 = r13511236 * r13511236;
        double r13511243 = sqrt(r13511242);
        double r13511244 = 1.0;
        double r13511245 = r13511239 * r13511239;
        double r13511246 = r13511244 / r13511245;
        double r13511247 = r13511243 / r13511246;
        double r13511248 = r13511241 * r13511247;
        double r13511249 = fma(r13511224, r13511234, r13511248);
        double r13511250 = r13511221 * r13511249;
        double r13511251 = exp(r13511250);
        double r13511252 = fma(r13511220, r13511251, r13511219);
        double r13511253 = r13511219 / r13511252;
        return r13511253;
}

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

Original3.7
Target3.0
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) - \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(3.0 \cdot t\right) - 2.0\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \end{array}\]

Derivation

  1. Initial program 3.7

    \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

  2. Simplified1.8

    \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{\frac{t}{z}}\right)}, x\right)}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt1.8

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{\frac{t}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}}\right)}, x\right)}\]

  5. Applied *-un-lft-identity1.8

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{\frac{\color{blue}{1 \cdot t}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\right)}, x\right)}\]

  6. Applied times-frac1.8

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{\color{blue}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{t}{\sqrt[3]{z}}}}\right)}, x\right)}\]

  7. Applied add-cube-cbrt1.8

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{\color{blue}{\left(\sqrt[3]{a + t} \cdot \sqrt[3]{a + t}\right) \cdot \sqrt[3]{a + t}}}}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{t}{\sqrt[3]{z}}}\right)}, x\right)}\]

  8. Applied sqrt-prod1.8

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\color{blue}{\sqrt{\sqrt[3]{a + t} \cdot \sqrt[3]{a + t}} \cdot \sqrt{\sqrt[3]{a + t}}}}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{t}{\sqrt[3]{z}}}\right)}, x\right)}\]

  9. Applied times-frac1.5

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \color{blue}{\frac{\sqrt{\sqrt[3]{a + t} \cdot \sqrt[3]{a + t}}}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{\sqrt{\sqrt[3]{a + t}}}{\frac{t}{\sqrt[3]{z}}}}\right)}, x\right)}\]

  10. Final simplification1.5

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{\sqrt[3]{a + t}}}{\frac{t}{\sqrt[3]{z}}} \cdot \frac{\sqrt{\sqrt[3]{a + t} \cdot \sqrt[3]{a + t}}}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019162 +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)))))))))))