Average Error: 4.6 → 2.1
Time: 9.4s
Precision: binary64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -6.3697161759007095 \cdot 10^{188} \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 2.2141006646816808 \cdot 10^{168}\right):\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x \cdot \frac{y}{z}} \cdot \sqrt[3]{x \cdot \frac{y}{z}}\right) \cdot \sqrt[3]{x \cdot \frac{y}{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.6
Target4.4
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -6.3697161759007095e188 or 2.2141006646816808e168 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 16.2

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

    3. Applied sub-neg16.2

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

    4. Applied distribute-lft-in16.2

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

    6. Applied associate-*r/1.7

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

    8. Applied distribute-neg-frac1.7

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

    9. Applied associate-*r/2.3

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

    if -6.3697161759007095e188 < (- (/ y z) (/ t (- 1.0 z))) < 2.2141006646816808e168

    1. Initial program 1.5

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

    3. Applied sub-neg1.5

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

    4. Applied distribute-lft-in1.5

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

    6. Applied add-cube-cbrt2.0

      \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \frac{y}{z}} \cdot \sqrt[3]{x \cdot \frac{y}{z}}\right) \cdot \sqrt[3]{x \cdot \frac{y}{z}}} + x \cdot \left(-\frac{t}{1 - z}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -6.3697161759007095 \cdot 10^{188} \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 2.2141006646816808 \cdot 10^{168}\right):\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x \cdot \frac{y}{z}} \cdot \sqrt[3]{x \cdot \frac{y}{z}}\right) \cdot \sqrt[3]{x \cdot \frac{y}{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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 (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (neg (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))