Average Error: 3.1 → 0.7
Time: 8.2s
Precision: binary64
\[[y, z, t] = \mathsf{sort}([y, z, t]) \\]
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq 1.5397415417505267 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(t \cdot y\right), z, x + x\right) + 27 \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\right) + b \cdot \left(27 \cdot a\right)\\ \end{array} \]
\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 1.5397415417505267 \cdot 10^{-88}:\\
\;\;\;\;\mathsf{fma}\left(-9 \cdot \left(t \cdot y\right), z, x + x\right) + 27 \cdot \left(a \cdot b\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot 2 - t \cdot \left(z \cdot \left(y \cdot 9\right)\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 1.5397415417505267e-88)
   (+ (fma (* -9.0 (* t y)) z (+ x x)) (* 27.0 (* a b)))
   (+ (- (* x 2.0) (* t (* z (* y 9.0)))) (* 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 <= 1.5397415417505267e-88) {
		tmp = fma((-9.0 * (t * y)), z, (x + x)) + (27.0 * (a * b));
	} else {
		tmp = ((x * 2.0) - (t * (z * (y * 9.0)))) + (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

Target

Original3.1
Target3.5
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y < 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if t < 1.53974154175052667e-88

    1. Initial program 7.0

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

    2. Taylor expanded in y around 0 1.0

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

    3. Taylor expanded in a around 0 1.0

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

    4. Applied egg-rr0.2

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

    if 1.53974154175052667e-88 < t

    1. Initial program 1.0

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5397415417505267 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(t \cdot y\right), z, x + x\right) + 27 \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\right) + b \cdot \left(27 \cdot a\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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)))