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

↓

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

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

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r337254 = x;
        double r337255 = y;
        double r337256 = 2.0;
        double r337257 = z;
        double r337258 = t;
        double r337259 = a;
        double r337260 = r337258 + r337259;
        double r337261 = sqrt(r337260);
        double r337262 = r337257 * r337261;
        double r337263 = r337262 / r337258;
        double r337264 = b;
        double r337265 = c;
        double r337266 = r337264 - r337265;
        double r337267 = 5.0;
        double r337268 = 6.0;
        double r337269 = r337267 / r337268;
        double r337270 = r337259 + r337269;
        double r337271 = 3.0;
        double r337272 = r337258 * r337271;
        double r337273 = r337256 / r337272;
        double r337274 = r337270 - r337273;
        double r337275 = r337266 * r337274;
        double r337276 = r337263 - r337275;
        double r337277 = r337256 * r337276;
        double r337278 = exp(r337277);
        double r337279 = r337255 * r337278;
        double r337280 = r337254 + r337279;
        double r337281 = r337254 / r337280;
        return r337281;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r337282 = x;
        double r337283 = y;
        double r337284 = 2.0;
        double r337285 = exp(r337284);
        double r337286 = 3.0;
        double r337287 = r337284 / r337286;
        double r337288 = t;
        double r337289 = r337287 / r337288;
        double r337290 = a;
        double r337291 = 5.0;
        double r337292 = 6.0;
        double r337293 = r337291 / r337292;
        double r337294 = r337290 + r337293;
        double r337295 = r337289 - r337294;
        double r337296 = b;
        double r337297 = c;
        double r337298 = r337296 - r337297;
        double r337299 = z;
        double r337300 = cbrt(r337288);
        double r337301 = r337300 * r337300;
        double r337302 = r337299 / r337301;
        double r337303 = r337288 + r337290;
        double r337304 = sqrt(r337303);
        double r337305 = r337304 / r337300;
        double r337306 = r337302 * r337305;
        double r337307 = fma(r337295, r337298, r337306);
        double r337308 = pow(r337285, r337307);
        double r337309 = fma(r337283, r337308, r337282);
        double r337310 = r337282 / r337309;
        return r337310;
}

Target

Original	4.0
Target	3.1
Herbie	1.4

\[\begin{array}{l} \mathbf{if}\;t \lt -2.1183266448915811 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.19658877065154709 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

Initial program 4.0
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Simplified2.6
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{3}}{t} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}}\]
Using strategy rm
Applied add-cube-cbrt2.6
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{3}}{t} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\right)\right)}, x\right)}\]
Applied times-frac1.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{3}}{t} - \left(a + \frac{5}{6}\right), b - c, \color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}}\right)\right)}, x\right)}\]
Final simplification1.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{3}}{t} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2 (/ (- (* (* z (sqrt (+ t a))) (* (* 3 t) (- a (/ 5 6)))) (* (- (* (+ (/ 5 6) a) (* 3 t)) 2) (* (- a (/ 5 6)) (* (- b c) t)))) (* (* (* t t) 3) (- a (/ 5 6))))))))) (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3))))))))))))

  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Target

Derivation

Reproduce