Average Error: 4.9 → 4.7
Time: 19.4s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.699374283935706957857036182841032710783 \cdot 10^{-111}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, \frac{1}{z}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\ \mathbf{elif}\;t \le -5.007773027212033065092460998980032370379 \cdot 10^{-202}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{y}{\sqrt[3]{z}}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\ \mathbf{elif}\;t \le 2.841522777850616113336553397998959361956 \cdot 10^{-176}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(x \cdot \left(\sqrt[3]{-\frac{t}{1 - z}} \cdot \sqrt[3]{-\frac{t}{1 - z}}\right)\right) \cdot \sqrt[3]{-\frac{t}{1 - z}}\\ \mathbf{elif}\;t \le 3.284223438702235885793954830185385259072 \cdot 10^{74}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, \frac{1}{z}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.699374283935706957857036182841032710783 \cdot 10^{-111}:\\
\;\;\;\;\mathsf{fma}\left(x \cdot y, \frac{1}{z}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\

\mathbf{elif}\;t \le -5.007773027212033065092460998980032370379 \cdot 10^{-202}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{y}{\sqrt[3]{z}}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\

\mathbf{elif}\;t \le 2.841522777850616113336553397998959361956 \cdot 10^{-176}:\\
\;\;\;\;\frac{x \cdot y}{z} + \left(x \cdot \left(\sqrt[3]{-\frac{t}{1 - z}} \cdot \sqrt[3]{-\frac{t}{1 - z}}\right)\right) \cdot \sqrt[3]{-\frac{t}{1 - z}}\\

\mathbf{elif}\;t \le 3.284223438702235885793954830185385259072 \cdot 10^{74}:\\
\;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x \cdot y, \frac{1}{z}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r420168 = x;
        double r420169 = y;
        double r420170 = z;
        double r420171 = r420169 / r420170;
        double r420172 = t;
        double r420173 = 1.0;
        double r420174 = r420173 - r420170;
        double r420175 = r420172 / r420174;
        double r420176 = r420171 - r420175;
        double r420177 = r420168 * r420176;
        return r420177;
}
↓
double f(double x, double y, double z, double t) {
        double r420178 = t;
        double r420179 = -1.699374283935707e-111;
        bool r420180 = r420178 <= r420179;
        double r420181 = x;
        double r420182 = y;
        double r420183 = r420181 * r420182;
        double r420184 = 1.0;
        double r420185 = z;
        double r420186 = r420184 / r420185;
        double r420187 = 1.0;
        double r420188 = r420187 - r420185;
        double r420189 = r420178 / r420188;
        double r420190 = -r420189;
        double r420191 = r420181 * r420190;
        double r420192 = fma(r420183, r420186, r420191);
        double r420193 = -5.007773027212033e-202;
        bool r420194 = r420178 <= r420193;
        double r420195 = cbrt(r420185);
        double r420196 = r420195 * r420195;
        double r420197 = r420181 / r420196;
        double r420198 = r420182 / r420195;
        double r420199 = fma(r420197, r420198, r420191);
        double r420200 = 2.841522777850616e-176;
        bool r420201 = r420178 <= r420200;
        double r420202 = r420183 / r420185;
        double r420203 = cbrt(r420190);
        double r420204 = r420203 * r420203;
        double r420205 = r420181 * r420204;
        double r420206 = r420205 * r420203;
        double r420207 = r420202 + r420206;
        double r420208 = 3.284223438702236e+74;
        bool r420209 = r420178 <= r420208;
        double r420210 = r420185 / r420182;
        double r420211 = r420181 / r420210;
        double r420212 = r420211 + r420191;
        double r420213 = r420209 ? r420212 : r420192;
        double r420214 = r420201 ? r420207 : r420213;
        double r420215 = r420194 ? r420199 : r420214;
        double r420216 = r420180 ? r420192 : r420215;
        return r420216;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.9
Target4.5
Herbie4.7
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if t < -1.699374283935707e-111 or 3.284223438702236e+74 < t

    1. Initial program 4.2

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
    2. Using strategy rm

    3. Applied sub-neg4.2

      \[\leadsto x \cdot \color{blue}{\left(\frac{y}{z} + \left(-\frac{t}{1 - z}\right)\right)}\]

    4. Applied distribute-lft-in4.2

      \[\leadsto \color{blue}{x \cdot \frac{y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)}\]

    5. Simplified4.0

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\]
    6. Using strategy rm

    7. Applied div-inv4.0

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\]

    8. Applied fma-def4.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot y, \frac{1}{z}, x \cdot \left(-\frac{t}{1 - z}\right)\right)}\]

    if -1.699374283935707e-111 < t < -5.007773027212033e-202

    1. Initial program 5.5

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
    2. Using strategy rm

    3. Applied sub-neg5.5

      \[\leadsto x \cdot \color{blue}{\left(\frac{y}{z} + \left(-\frac{t}{1 - z}\right)\right)}\]

    4. Applied distribute-lft-in5.5

      \[\leadsto \color{blue}{x \cdot \frac{y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)}\]

    5. Simplified7.3

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\]
    6. Using strategy rm

    7. Applied add-cube-cbrt8.0

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} + x \cdot \left(-\frac{t}{1 - z}\right)\]

    8. Applied times-frac6.6

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}} + x \cdot \left(-\frac{t}{1 - z}\right)\]

    9. Applied fma-def6.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{y}{\sqrt[3]{z}}, x \cdot \left(-\frac{t}{1 - z}\right)\right)}\]

    if -5.007773027212033e-202 < t < 2.841522777850616e-176

    1. Initial program 6.8

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
    2. Using strategy rm

    3. Applied sub-neg6.8

      \[\leadsto x \cdot \color{blue}{\left(\frac{y}{z} + \left(-\frac{t}{1 - z}\right)\right)}\]

    4. Applied distribute-lft-in6.8

      \[\leadsto \color{blue}{x \cdot \frac{y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)}\]

    5. Simplified5.8

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\]
    6. Using strategy rm

    7. Applied add-cube-cbrt5.8

      \[\leadsto \frac{x \cdot y}{z} + x \cdot \color{blue}{\left(\left(\sqrt[3]{-\frac{t}{1 - z}} \cdot \sqrt[3]{-\frac{t}{1 - z}}\right) \cdot \sqrt[3]{-\frac{t}{1 - z}}\right)}\]

    8. Applied associate-*r*5.8

      \[\leadsto \frac{x \cdot y}{z} + \color{blue}{\left(x \cdot \left(\sqrt[3]{-\frac{t}{1 - z}} \cdot \sqrt[3]{-\frac{t}{1 - z}}\right)\right) \cdot \sqrt[3]{-\frac{t}{1 - z}}}\]

    if 2.841522777850616e-176 < t < 3.284223438702236e+74

    1. Initial program 4.6

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
    2. Using strategy rm

    3. Applied sub-neg4.6

      \[\leadsto x \cdot \color{blue}{\left(\frac{y}{z} + \left(-\frac{t}{1 - z}\right)\right)}\]

    4. Applied distribute-lft-in4.6

      \[\leadsto \color{blue}{x \cdot \frac{y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)}\]

    5. Simplified5.7

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\]
    6. Using strategy rm

    7. Applied associate-/l*4.3

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} + x \cdot \left(-\frac{t}{1 - z}\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification4.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.699374283935706957857036182841032710783 \cdot 10^{-111}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, \frac{1}{z}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\ \mathbf{elif}\;t \le -5.007773027212033065092460998980032370379 \cdot 10^{-202}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{y}{\sqrt[3]{z}}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\ \mathbf{elif}\;t \le 2.841522777850616113336553397998959361956 \cdot 10^{-176}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(x \cdot \left(\sqrt[3]{-\frac{t}{1 - z}} \cdot \sqrt[3]{-\frac{t}{1 - z}}\right)\right) \cdot \sqrt[3]{-\frac{t}{1 - z}}\\ \mathbf{elif}\;t \le 3.284223438702235885793954830185385259072 \cdot 10^{74}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, \frac{1}{z}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z)))))))

  (* x (- (/ y z) (/ t (- 1 z)))))