Average Error: 2.7 → 1.1
Time: 10.5s
Precision: binary64
\[[y, z, t] = \mathsf{sort}([y, z, t]) \\]
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
↓
\[\begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ \mathbf{if}\;z \leq 1.7086528660097502 \cdot 10^{-208}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(x \cdot 2 - t \cdot \left(9 \cdot \left(z \cdot y\right)\right)\right)\\ \end{array} \]
\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}
t_1 := \left(a \cdot 27\right) \cdot b\\
\mathbf{if}\;z \leq 1.7086528660097502 \cdot 10^{-208}:\\
\;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right) + t_1\\

\mathbf{else}:\\
\;\;\;\;t_1 + \left(x \cdot 2 - t \cdot \left(9 \cdot \left(z \cdot y\right)\right)\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
 (let* ((t_1 (* (* a 27.0) b)))
   (if (<= z 1.7086528660097502e-208)
     (+ (- (* x 2.0) (* 9.0 (* y (* z t)))) t_1)
     (+ t_1 (- (* x 2.0) (* t (* 9.0 (* z y))))))))
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 t_1 = (a * 27.0) * b;
	double tmp;
	if (z <= 1.7086528660097502e-208) {
		tmp = ((x * 2.0) - (9.0 * (y * (z * t)))) + t_1;
	} else {
		tmp = t_1 + ((x * 2.0) - (t * (9.0 * (z * y))));
	}
	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 z < 1.7086528660097502e-208

    1. Initial program 3.5

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

    2. Taylor expanded in y around 0 0.9

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

    if 1.7086528660097502e-208 < z

    1. Initial program 1.4

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

    2. Applied associate-*l*_binary641.4

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

    3. Applied *-un-lft-identity_binary641.4

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

    4. Applied associate-*l*_binary641.4

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

    5. Simplified1.4

      \[\leadsto \left(x \cdot 2 - \left(1 \cdot \color{blue}{\left(9 \cdot \left(y \cdot z\right)\right)}\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7086528660097502 \cdot 10^{-208}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 27\right) \cdot b + \left(x \cdot 2 - t \cdot \left(9 \cdot \left(z \cdot y\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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)))