Average Error: 4.7 → 2.1
Time: 7.7s
Precision: binary64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}} - x \cdot \frac{t}{{\left(\sqrt[3]{1 - z}\right)}^{3}}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}} - x \cdot \frac{t}{{\left(\sqrt[3]{1 - z}\right)}^{3}}
(FPCore (x y z t) :precision binary64 (* x (- (/ y z) (/ t (- 1.0 z)))))
↓
(FPCore (x y z t)
 :precision binary64
 (-
  (*
   (* x (/ (* (cbrt y) (cbrt y)) (* (cbrt z) (cbrt z))))
   (/ (cbrt y) (cbrt z)))
  (* x (/ t (pow (cbrt (- 1.0 z)) 3.0)))))
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) {
	return ((x * ((cbrt(y) * cbrt(y)) / (cbrt(z) * cbrt(z)))) * (cbrt(y) / cbrt(z))) - (x * (t / pow(cbrt(1.0 - z), 3.0)));
}

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

Original4.7
Target4.4
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Initial program 4.7

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

  3. Applied add-cube-cbrt_binary644.9

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

  4. Applied add-cube-cbrt_binary645.1

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

  5. Applied times-frac_binary645.1

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

  6. Applied cancel-sign-sub-inv_binary645.1

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

  7. Applied distribute-rgt-in_binary645.1

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

  8. Simplified5.1

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

  9. Simplified4.9

    \[\leadsto x \cdot \frac{y}{z} + \color{blue}{x \cdot \left(-\frac{t}{{\left(\sqrt[3]{1 - z}\right)}^{3}}\right)}\]
  10. Using strategy rm

  11. Applied add-cube-cbrt_binary645.4

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

  12. Applied add-cube-cbrt_binary645.5

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

  13. Applied times-frac_binary645.5

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

  14. Applied associate-*r*_binary642.1

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

  15. Final simplification2.1

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

Alternatives

Reproduce

herbie shell --seed 2021118 
(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)))))