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

↓

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

\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 r413924 = x;
        double r413925 = y;
        double r413926 = 2.0;
        double r413927 = z;
        double r413928 = t;
        double r413929 = a;
        double r413930 = r413928 + r413929;
        double r413931 = sqrt(r413930);
        double r413932 = r413927 * r413931;
        double r413933 = r413932 / r413928;
        double r413934 = b;
        double r413935 = c;
        double r413936 = r413934 - r413935;
        double r413937 = 5.0;
        double r413938 = 6.0;
        double r413939 = r413937 / r413938;
        double r413940 = r413929 + r413939;
        double r413941 = 3.0;
        double r413942 = r413928 * r413941;
        double r413943 = r413926 / r413942;
        double r413944 = r413940 - r413943;
        double r413945 = r413936 * r413944;
        double r413946 = r413933 - r413945;
        double r413947 = r413926 * r413946;
        double r413948 = exp(r413947);
        double r413949 = r413925 * r413948;
        double r413950 = r413924 + r413949;
        double r413951 = r413924 / r413950;
        return r413951;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r413952 = x;
        double r413953 = y;
        double r413954 = 2.0;
        double r413955 = exp(r413954);
        double r413956 = 3.0;
        double r413957 = r413954 / r413956;
        double r413958 = t;
        double r413959 = r413957 / r413958;
        double r413960 = a;
        double r413961 = 5.0;
        double r413962 = 6.0;
        double r413963 = r413961 / r413962;
        double r413964 = r413960 + r413963;
        double r413965 = r413959 - r413964;
        double r413966 = b;
        double r413967 = c;
        double r413968 = r413966 - r413967;
        double r413969 = z;
        double r413970 = cbrt(r413958);
        double r413971 = r413970 * r413970;
        double r413972 = r413969 / r413971;
        double r413973 = r413958 + r413960;
        double r413974 = sqrt(r413973);
        double r413975 = r413974 / r413970;
        double r413976 = r413972 * r413975;
        double r413977 = fma(r413965, r413968, r413976);
        double r413978 = pow(r413955, r413977);
        double r413979 = fma(r413953, r413978, r413952);
        double r413980 = r413952 / r413979;
        return r413980;
}

Target

Original	3.8
Target	2.8
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 3.8
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
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 2020045 +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