Average Error: 2.7 → 1.1
Time: 9.4s
Precision: binary64
\[[y, z, t]=\mathsf{sort}([y, z, t])\]
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq 1.681822738427985 \cdot 10^{+109}:\\ \;\;\;\;\left(27 \cdot \left(a \cdot b\right) - \left(t \cdot \left(9 \cdot y\right)\right) \cdot z\right) + \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - t \cdot \left(\left(9 \cdot y\right) \cdot z\right)\right) + b \cdot \left(27 \cdot a\right)\\ \end{array} \]
\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;t \leq 1.681822738427985 \cdot 10^{+109}:\\
\;\;\;\;\left(27 \cdot \left(a \cdot b\right) - \left(t \cdot \left(9 \cdot y\right)\right) \cdot z\right) + \left(x + x\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot 2 - t \cdot \left(\left(9 \cdot y\right) \cdot z\right)\right) + b \cdot \left(27 \cdot a\right)\\


\end{array}

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))
↓
(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 1.681822738427985e+109)
   (+ (- (* 27.0 (* a b)) (* (* t (* 9.0 y)) z)) (+ x x))
   (+ (- (* x 2.0) (* t (* (* 9.0 y) z))) (* b (* 27.0 a)))))
double code(double x, double y, double z, double t, double a, double b) {
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 1.681822738427985e+109) {
		tmp = ((27.0 * (a * b)) - ((t * (9.0 * y)) * z)) + (x + x);
	} else {
		tmp = ((x * 2.0) - (t * ((9.0 * y) * z))) + (b * (27.0 * a));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.7
Target3.0
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;y < 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if t < 1.68182273842798502e109

    1. Initial program 3.7

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

    2. Taylor expanded around 0 0.7

      \[\leadsto \color{blue}{\left(27 \cdot \left(a \cdot b\right) + 2 \cdot x\right) - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)} \]

    3. Simplified0.7

      \[\leadsto \color{blue}{\left(27 \cdot \left(a \cdot b\right) - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(x + x\right)} \]
    4. Using strategy rm

    5. Applied associate-*r*_binary640.8

      \[\leadsto \left(27 \cdot \left(a \cdot b\right) - \color{blue}{\left(9 \cdot y\right) \cdot \left(t \cdot z\right)}\right) + \left(x + x\right) \]

    6. Simplified0.8

      \[\leadsto \left(27 \cdot \left(a \cdot b\right) - \color{blue}{\left(y \cdot 9\right)} \cdot \left(t \cdot z\right)\right) + \left(x + x\right) \]
    7. Using strategy rm

    8. Applied associate-*r*_binary641.2

      \[\leadsto \left(27 \cdot \left(a \cdot b\right) - \color{blue}{\left(\left(y \cdot 9\right) \cdot t\right) \cdot z}\right) + \left(x + x\right) \]

    9. Simplified1.2

      \[\leadsto \left(27 \cdot \left(a \cdot b\right) - \color{blue}{\left(t \cdot \left(9 \cdot y\right)\right)} \cdot z\right) + \left(x + x\right) \]

    if 1.68182273842798502e109 < t

    1. Initial program 0.8

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
    2. Using strategy rm

    3. Applied *-un-lft-identity_binary640.8

      \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot \color{blue}{\left(1 \cdot z\right)}\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

    4. Applied associate-*r*_binary640.8

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(\left(y \cdot 9\right) \cdot 1\right) \cdot z\right)} \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

    5. Simplified0.8

      \[\leadsto \left(x \cdot 2 - \left(\color{blue}{\left(9 \cdot y\right)} \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.681822738427985 \cdot 10^{+109}:\\ \;\;\;\;\left(27 \cdot \left(a \cdot b\right) - \left(t \cdot \left(9 \cdot y\right)\right) \cdot z\right) + \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - t \cdot \left(\left(9 \cdot y\right) \cdot z\right)\right) + b \cdot \left(27 \cdot a\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021204 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b))) (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b)))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))