Average Error: 7.8 → 7.8
Time: 24.3s
Precision: binary64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
↓
\[\frac{\mathsf{fma}\left(t, z \cdot -9, y \cdot x\right)}{\frac{a}{0.5}} \]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\frac{\mathsf{fma}\left(t, z \cdot -9, y \cdot x\right)}{\frac{a}{0.5}}

(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
 (/ (fma t (* z -9.0) (* y x)) (/ a 0.5)))
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 fma(t, (z * -9.0), (y * x)) / (a / 0.5);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.8
Target5.7
Herbie7.8
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \]

Derivation

  1. Initial program 7.8

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

  2. Applied clear-num_binary648.0

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

  3. Simplified8.0

    \[\leadsto \frac{1}{\color{blue}{\frac{a}{\frac{\mathsf{fma}\left(t, z \cdot -9, y \cdot x\right)}{2}}}} \]

  4. Applied div-inv_binary648.0

    \[\leadsto \frac{1}{\frac{a}{\color{blue}{\mathsf{fma}\left(t, z \cdot -9, y \cdot x\right) \cdot \frac{1}{2}}}} \]

  5. Applied *-un-lft-identity_binary648.0

    \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot a}}{\mathsf{fma}\left(t, z \cdot -9, y \cdot x\right) \cdot \frac{1}{2}}} \]

  6. Applied times-frac_binary648.1

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(t, z \cdot -9, y \cdot x\right)} \cdot \frac{a}{\frac{1}{2}}}} \]

  7. Applied associate-/r*_binary647.9

    \[\leadsto \color{blue}{\frac{\frac{1}{\frac{1}{\mathsf{fma}\left(t, z \cdot -9, y \cdot x\right)}}}{\frac{a}{\frac{1}{2}}}} \]

  8. Applied inv-pow_binary647.9

    \[\leadsto \frac{\frac{1}{\color{blue}{{\left(\mathsf{fma}\left(t, z \cdot -9, y \cdot x\right)\right)}^{-1}}}}{\frac{a}{\frac{1}{2}}} \]

  9. Applied pow-flip_binary647.8

    \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(t, z \cdot -9, y \cdot x\right)\right)}^{\left(--1\right)}}}{\frac{a}{\frac{1}{2}}} \]

  10. Simplified7.8

    \[\leadsto \frac{{\left(\mathsf{fma}\left(t, z \cdot -9, y \cdot x\right)\right)}^{\color{blue}{1}}}{\frac{a}{\frac{1}{2}}} \]

  11. Final simplification7.8

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

Reproduce

herbie shell --seed 2021313 
(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)))