Average Error: 7.5 → 7.4
Time: 8.5s
Precision: binary64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
↓
\[\frac{\frac{\mathsf{fma}\left(-9, z \cdot t, x \cdot y\right)}{2}}{a} \]
(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 -9.0 (* z t) (* x y)) 2.0) 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(-9.0, (z * t), (x * y)) / 2.0) / a;
}

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(-9.0, Float64(z * t), Float64(x * y)) / 2.0) / a)
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[(-9.0 * N[(z * t), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] / a), $MachinePrecision]

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.5
Target5.2
Herbie7.4
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \]

Derivation

  1. Initial program 7.5

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

  2. Simplified7.5

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

  3. Applied egg-rr7.7

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

  4. Applied egg-rr7.9

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

  5. Applied egg-rr7.4

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

  6. Final simplification7.4

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

Reproduce

herbie shell --seed 2022131 
(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)))