Average Error: 3.7 → 2.2
Time: 14.0s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
\frac{x}{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}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r89173 = x;
        double r89174 = y;
        double r89175 = 2.0;
        double r89176 = z;
        double r89177 = t;
        double r89178 = a;
        double r89179 = r89177 + r89178;
        double r89180 = sqrt(r89179);
        double r89181 = r89176 * r89180;
        double r89182 = r89181 / r89177;
        double r89183 = b;
        double r89184 = c;
        double r89185 = r89183 - r89184;
        double r89186 = 5.0;
        double r89187 = 6.0;
        double r89188 = r89186 / r89187;
        double r89189 = r89178 + r89188;
        double r89190 = 3.0;
        double r89191 = r89177 * r89190;
        double r89192 = r89175 / r89191;
        double r89193 = r89189 - r89192;
        double r89194 = r89185 * r89193;
        double r89195 = r89182 - r89194;
        double r89196 = r89175 * r89195;
        double r89197 = exp(r89196);
        double r89198 = r89174 * r89197;
        double r89199 = r89173 + r89198;
        double r89200 = r89173 / r89199;
        return r89200;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r89201 = x;
        double r89202 = y;
        double r89203 = 2.0;
        double r89204 = z;
        double r89205 = 1.0;
        double r89206 = r89204 / r89205;
        double r89207 = t;
        double r89208 = a;
        double r89209 = r89207 + r89208;
        double r89210 = sqrt(r89209);
        double r89211 = r89210 / r89207;
        double r89212 = b;
        double r89213 = c;
        double r89214 = r89212 - r89213;
        double r89215 = 5.0;
        double r89216 = 6.0;
        double r89217 = r89215 / r89216;
        double r89218 = r89208 + r89217;
        double r89219 = 3.0;
        double r89220 = r89207 * r89219;
        double r89221 = r89203 / r89220;
        double r89222 = r89218 - r89221;
        double r89223 = r89214 * r89222;
        double r89224 = -r89223;
        double r89225 = fma(r89206, r89211, r89224);
        double r89226 = r89203 * r89225;
        double r89227 = exp(r89226);
        double r89228 = r89202 * r89227;
        double r89229 = r89201 + r89228;
        double r89230 = r89201 / r89229;
        return r89230;
}

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

Derivation

  1. Initial program 3.7

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

  3. Applied *-un-lft-identity3.7

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{1 \cdot t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

  4. Applied times-frac3.3

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{1} \cdot \frac{\sqrt{t + a}}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

  5. Applied fma-neg2.2

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}}\]

  6. Final simplification2.2

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))