Average Error: 7.6 → 7.5
Time: 6.4s
Precision: binary64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
↓
\[0.5 \cdot \frac{\mathsf{fma}\left(y, x, -9 \cdot \left(t \cdot z\right)\right)}{a} \]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
0.5 \cdot \frac{\mathsf{fma}\left(y, x, -9 \cdot \left(t \cdot z\right)\right)}{a}

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))
↓
(FPCore (x y z t a)
 :precision binary64
 (* 0.5 (/ (fma y x (* -9.0 (* t z))) a)))
double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}
↓
double code(double x, double y, double z, double t, double a) {
	return 0.5 * (fma(y, x, (-9.0 * (t * z))) / a);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.6
Target5.6
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \]

Derivation

  1. Initial program 7.6

    \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

  2. Simplified7.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(-9, z \cdot t, x \cdot y\right) \cdot \frac{0.5}{a}} \]

  3. Taylor expanded in a around 0 7.5

    \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x - 9 \cdot \left(t \cdot z\right)}{a}} \]

  4. Applied fma-neg_binary647.5

    \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(y, x, -9 \cdot \left(t \cdot z\right)\right)}}{a} \]

  5. Simplified7.5

    \[\leadsto 0.5 \cdot \frac{\mathsf{fma}\left(y, x, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}{a} \]

  6. Applied pow1_binary647.5

    \[\leadsto 0.5 \cdot \frac{\color{blue}{{\left(\mathsf{fma}\left(y, x, -9 \cdot \left(t \cdot z\right)\right)\right)}^{1}}}{a} \]

  7. Final simplification7.5

    \[\leadsto 0.5 \cdot \frac{\mathsf{fma}\left(y, x, -9 \cdot \left(t \cdot z\right)\right)}{a} \]

Reproduce

herbie shell --seed 2021329 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))