Average Error: 7.7 → 7.6
Time: 9.6s
Precision: binary64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
↓
\[-\frac{\mathsf{fma}\left(x, y \cdot -0.5, \left(t \cdot z\right) \cdot 4.5\right)}{a} \]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
-\frac{\mathsf{fma}\left(x, y \cdot -0.5, \left(t \cdot z\right) \cdot 4.5\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
 (- (/ (fma x (* y -0.5) (* (* t z) 4.5)) 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 -(fma(x, (y * -0.5), ((t * z) * 4.5)) / a);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.7
Target5.6
Herbie7.6
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \]

Derivation

  1. Initial program 7.7

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

  2. Simplified7.7

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

  3. Applied *-un-lft-identity_binary647.7

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

  4. Applied add-sqr-sqrt_binary648.4

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

  5. Applied times-frac_binary648.0

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

  6. Applied associate-*r*_binary648.0

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

  7. Simplified8.0

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

  8. Taylor expanded in a around -inf 8.3

    \[\leadsto \color{blue}{-1 \cdot \frac{9 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot \left(t \cdot z\right)\right) - {\left(\sqrt{0.5}\right)}^{2} \cdot \left(y \cdot x\right)}{a}} \]

  9. Simplified7.6

    \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(x, y \cdot -0.5, \left(t \cdot z\right) \cdot 4.5\right)}{a}} \]

  10. Final simplification7.6

    \[\leadsto -\frac{\mathsf{fma}\left(x, y \cdot -0.5, \left(t \cdot z\right) \cdot 4.5\right)}{a} \]

Reproduce

herbie shell --seed 2021280 
(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)))