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

↓

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

\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{z}{\frac{t}{\sqrt{t + a}}}\right)\right)}, x\right)}

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r264573 = x;
        double r264574 = y;
        double r264575 = 2.0;
        double r264576 = z;
        double r264577 = t;
        double r264578 = a;
        double r264579 = r264577 + r264578;
        double r264580 = sqrt(r264579);
        double r264581 = r264576 * r264580;
        double r264582 = r264581 / r264577;
        double r264583 = b;
        double r264584 = c;
        double r264585 = r264583 - r264584;
        double r264586 = 5.0;
        double r264587 = 6.0;
        double r264588 = r264586 / r264587;
        double r264589 = r264578 + r264588;
        double r264590 = 3.0;
        double r264591 = r264577 * r264590;
        double r264592 = r264575 / r264591;
        double r264593 = r264589 - r264592;
        double r264594 = r264585 * r264593;
        double r264595 = r264582 - r264594;
        double r264596 = r264575 * r264595;
        double r264597 = exp(r264596);
        double r264598 = r264574 * r264597;
        double r264599 = r264573 + r264598;
        double r264600 = r264573 / r264599;
        return r264600;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r264601 = x;
        double r264602 = y;
        double r264603 = 2.0;
        double r264604 = exp(r264603);
        double r264605 = t;
        double r264606 = r264603 / r264605;
        double r264607 = 3.0;
        double r264608 = r264606 / r264607;
        double r264609 = a;
        double r264610 = 5.0;
        double r264611 = 6.0;
        double r264612 = r264610 / r264611;
        double r264613 = r264609 + r264612;
        double r264614 = r264608 - r264613;
        double r264615 = b;
        double r264616 = c;
        double r264617 = r264615 - r264616;
        double r264618 = z;
        double r264619 = r264605 + r264609;
        double r264620 = sqrt(r264619);
        double r264621 = r264605 / r264620;
        double r264622 = r264618 / r264621;
        double r264623 = fma(r264614, r264617, r264622);
        double r264624 = pow(r264604, r264623);
        double r264625 = fma(r264602, r264624, r264601);
        double r264626 = r264601 / r264625;
        return r264626;
}

Target

Original	3.8
Target	3.0
Herbie	2.2

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

Derivation

Initial program 3.8
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Simplified2.7
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}}\]
Using strategy rm
Applied associate-/l*2.2
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \color{blue}{\frac{z}{\frac{t}{\sqrt{t + a}}}}\right)\right)}, x\right)}\]
Final simplification2.2
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\frac{t}{\sqrt{t + a}}}\right)\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019235 +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 -2.1183266448915811e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.83333333333333337 c)) (* a b))))))) (if (< t 5.19658877065154709e-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