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

↓

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

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

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r239979 = x;
        double r239980 = y;
        double r239981 = 2.0;
        double r239982 = z;
        double r239983 = t;
        double r239984 = a;
        double r239985 = r239983 + r239984;
        double r239986 = sqrt(r239985);
        double r239987 = r239982 * r239986;
        double r239988 = r239987 / r239983;
        double r239989 = b;
        double r239990 = c;
        double r239991 = r239989 - r239990;
        double r239992 = 5.0;
        double r239993 = 6.0;
        double r239994 = r239992 / r239993;
        double r239995 = r239984 + r239994;
        double r239996 = 3.0;
        double r239997 = r239983 * r239996;
        double r239998 = r239981 / r239997;
        double r239999 = r239995 - r239998;
        double r240000 = r239991 * r239999;
        double r240001 = r239988 - r240000;
        double r240002 = r239981 * r240001;
        double r240003 = exp(r240002);
        double r240004 = r239980 * r240003;
        double r240005 = r239979 + r240004;
        double r240006 = r239979 / r240005;
        return r240006;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r240007 = x;
        double r240008 = y;
        double r240009 = 2.0;
        double r240010 = exp(r240009);
        double r240011 = t;
        double r240012 = r240009 / r240011;
        double r240013 = 3.0;
        double r240014 = r240012 / r240013;
        double r240015 = a;
        double r240016 = 5.0;
        double r240017 = 6.0;
        double r240018 = r240016 / r240017;
        double r240019 = r240015 + r240018;
        double r240020 = r240014 - r240019;
        double r240021 = b;
        double r240022 = c;
        double r240023 = r240021 - r240022;
        double r240024 = z;
        double r240025 = cbrt(r240024);
        double r240026 = r240025 * r240025;
        double r240027 = cbrt(r240011);
        double r240028 = r240026 / r240027;
        double r240029 = r240025 / r240027;
        double r240030 = r240011 + r240015;
        double r240031 = sqrt(r240030);
        double r240032 = r240031 / r240027;
        double r240033 = r240029 * r240032;
        double r240034 = r240028 * r240033;
        double r240035 = fma(r240020, r240023, r240034);
        double r240036 = pow(r240010, r240035);
        double r240037 = fma(r240008, r240036, r240007);
        double r240038 = r240007 / r240037;
        return r240038;
}

Target

Original	4.1
Target	3.0
Herbie	1.4

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

Derivation

Initial program 4.1
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Simplified2.6
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}}\]
Using strategy rm
Applied add-cube-cbrt2.6
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\right)\right)}, x\right)}\]
Applied times-frac1.5
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}}\right)\right)}, x\right)}\]
Using strategy rm
Applied add-cube-cbrt1.5
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]
Applied times-frac1.5
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \color{blue}{\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]
Applied associate-*l*1.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \color{blue}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)}\right)\right)}, x\right)}\]
Final simplification1.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019325 +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)))))))))))

Error

Target

Derivation

Reproduce