Average Error: 7.8 → 7.8
Time: 6.4s
Precision: binary64
\[[z, t] = \mathsf{sort}([z, t]) \\]
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
↓
\[\frac{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) + \mathsf{fma}\left(t \cdot -9, z, t \cdot \left(z \cdot 9\right)\right)}{a \cdot 2} \]
(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 (* t (* z -9.0))) (fma (* t -9.0) z (* t (* z 9.0))))
  (* a 2.0)))
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, (t * (z * -9.0))) + fma((t * -9.0), z, (t * (z * 9.0)))) / (a * 2.0);
}

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end
↓
function code(x, y, z, t, a)
	return Float64(Float64(fma(x, y, Float64(t * Float64(z * -9.0))) + fma(Float64(t * -9.0), z, Float64(t * Float64(z * 9.0)))) / Float64(a * 2.0))
end

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y + N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t * -9.0), $MachinePrecision] * z + N[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\frac{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) + \mathsf{fma}\left(t \cdot -9, z, t \cdot \left(z \cdot 9\right)\right)}{a \cdot 2}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.8
Target5.6
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 egg-rr7.8

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

  3. Final simplification7.8

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

Reproduce

herbie shell --seed 2022134 
(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)))