Average Error: 4.5 → 3.8
Time: 4.3s
Precision: binary64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -2.175766960824216 \cdot 10^{-196} \lor \neg \left(y \leq 1.2146613525368215 \cdot 10^{-138}\right):\\ \;\;\;\;\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}} + x \cdot \frac{t}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} + \frac{x \cdot t}{z - 1}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \leq -2.175766960824216 \cdot 10^{-196} \lor \neg \left(y \leq 1.2146613525368215 \cdot 10^{-138}\right):\\
\;\;\;\;\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}} + x \cdot \frac{t}{z - 1}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{y}{z} + \frac{x \cdot t}{z - 1}\\

\end{array}
(FPCore (x y z t) :precision binary64 (* x (- (/ y z) (/ t (- 1.0 z)))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.175766960824216e-196) (not (<= y 1.2146613525368215e-138)))
   (+ (* (/ x (* (cbrt z) (cbrt z))) (/ y (cbrt z))) (* x (/ t (- z 1.0))))
   (+ (* x (/ y z)) (/ (* x t) (- z 1.0)))))
double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - z)));
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.175766960824216e-196) || !(y <= 1.2146613525368215e-138)) {
		tmp = ((x / (cbrt(z) * cbrt(z))) * (y / cbrt(z))) + (x * (t / (z - 1.0)));
	} else {
		tmp = (x * (y / z)) + ((x * t) / (z - 1.0));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original4.5
Target4.3
Herbie3.8
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -2.17576696082421608e-196 or 1.2146613525368215e-138 < y

    1. Initial program 5.3

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

    3. Applied sub-neg_binary645.3

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

    4. Applied distribute-rgt-in_binary645.3

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

    5. Simplified5.3

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

    6. Simplified5.3

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

    8. Applied add-cube-cbrt_binary645.9

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

    9. Applied *-un-lft-identity_binary645.9

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

    10. Applied times-frac_binary645.9

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

    11. Applied associate-*r*_binary643.8

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

    12. Simplified3.8

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

    if -2.17576696082421608e-196 < y < 1.2146613525368215e-138

    1. Initial program 2.4

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

    3. Applied sub-neg_binary642.4

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

    4. Applied distribute-rgt-in_binary642.4

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

    5. Simplified2.4

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

    6. Simplified2.4

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

    8. Applied associate-*r/_binary643.7

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

  4. Final simplification3.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.175766960824216 \cdot 10^{-196} \lor \neg \left(y \leq 1.2146613525368215 \cdot 10^{-138}\right):\\ \;\;\;\;\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}} + x \cdot \frac{t}{z - 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} + \frac{x \cdot t}{z - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(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) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

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