Average Error: 8.0 → 7.9
Time: 6.1s
Precision: binary64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
↓
\[\frac{0.5 \cdot \left(y \cdot x\right) - 4.5 \cdot \left(t \cdot z\right)}{a} \]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\frac{0.5 \cdot \left(y \cdot x\right) - 4.5 \cdot \left(t \cdot z\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 (* y x)) (* 4.5 (* 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 * (y * x)) - (4.5 * (t * z))) / a;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.0
Target5.8
Herbie7.9
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \]

Derivation

  1. Initial program 8.0

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

  2. Simplified8.0

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

  3. Taylor expanded in a around inf 7.9

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

  4. Final simplification7.9

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

Reproduce

herbie shell --seed 2022061 
(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)))