Average Error: 9.2 → 0.5
Time: 5.2s
Precision: binary64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(x \cdot \log y + \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(\sqrt[3]{z} \cdot \left(\log 1 + \left(\frac{y}{1} \cdot \left(\frac{y}{1} \cdot -0.5\right) - y \cdot 1\right)\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(x \cdot \log y + \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(\sqrt[3]{z} \cdot \left(\log 1 + \left(\frac{y}{1} \cdot \left(\frac{y}{1} \cdot -0.5\right) - y \cdot 1\right)\right)\right)\right) - t
(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))
↓
(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* x (log y))
   (*
    (* (cbrt z) (cbrt z))
    (* (cbrt z) (+ (log 1.0) (- (* (/ y 1.0) (* (/ y 1.0) -0.5)) (* y 1.0))))))
  t))
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (x * ((double) log(y)))) + ((double) (z * ((double) log(((double) (1.0 - y)))))))) - t));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (x * ((double) log(y)))) + ((double) (((double) (((double) cbrt(z)) * ((double) cbrt(z)))) * ((double) (((double) cbrt(z)) * ((double) (((double) log(1.0)) + ((double) (((double) ((y / 1.0) * ((double) ((y / 1.0) * -0.5)))) - ((double) (y * 1.0)))))))))))) - t));
}

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

Original9.2
Target0.3
Herbie0.5
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program Error: 9.2 bits

    \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]

  2. Taylor expanded around 0 Error: 0.3 bits

    \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1 - \left(1 \cdot y + 0.5 \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}\right) - t\]

  3. SimplifiedError: 0.3 bits

    \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1 + \left(\left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot -0.5 - 1 \cdot y\right)\right)}\right) - t\]
  4. Using strategy rm

  5. Applied add-cube-cbrtError: 0.5 bits

    \[\leadsto \left(x \cdot \log y + \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} \cdot \left(\log 1 + \left(\left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot -0.5 - 1 \cdot y\right)\right)\right) - t\]

  6. Applied associate-*l*Error: 0.5 bits

    \[\leadsto \left(x \cdot \log y + \color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(\sqrt[3]{z} \cdot \left(\log 1 + \left(\left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot -0.5 - 1 \cdot y\right)\right)\right)}\right) - t\]

  7. SimplifiedError: 0.5 bits

    \[\leadsto \left(x \cdot \log y + \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \color{blue}{\left(\left(\log 1 + \left(\frac{y}{1} \cdot \left(\frac{y}{1} \cdot -0.5\right) - y \cdot 1\right)\right) \cdot \sqrt[3]{z}\right)}\right) - t\]

  8. Final simplificationError: 0.5 bits

    \[\leadsto \left(x \cdot \log y + \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(\sqrt[3]{z} \cdot \left(\log 1 + \left(\frac{y}{1} \cdot \left(\frac{y}{1} \cdot -0.5\right) - y \cdot 1\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))