Average Error: 6.1 → 2.3
Time: 6.0s
Precision: binary64
\[x - \frac{y \cdot \left(z - t\right)}{a} \]
↓
\[\mathsf{fma}\left(1, x, t \cdot \frac{y}{a} - \frac{z}{\frac{a}{y}}\right) \]
(FPCore (x y z t a) :precision binary64 (- x (/ (* y (- z t)) a)))
↓
(FPCore (x y z t a)
 :precision binary64
 (fma 1.0 x (- (* t (/ y a)) (/ z (/ a y)))))
double code(double x, double y, double z, double t, double a) {
	return x - ((y * (z - t)) / a);
}
↓
double code(double x, double y, double z, double t, double a) {
	return fma(1.0, x, ((t * (y / a)) - (z / (a / y))));
}

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y * Float64(z - t)) / a))
end
↓
function code(x, y, z, t, a)
	return fma(1.0, x, Float64(Float64(t * Float64(y / a)) - Float64(z / Float64(a / y))))
end

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_] := N[(1.0 * x + N[(N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x - \frac{y \cdot \left(z - t\right)}{a}
↓
\mathsf{fma}\left(1, x, t \cdot \frac{y}{a} - \frac{z}{\frac{a}{y}}\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.1
Target0.7
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

  1. Initial program 6.1

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

  2. Taylor expanded in y around 0 6.1

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

  3. Simplified2.2

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

  4. Applied egg-rr2.2

    \[\leadsto x - \color{blue}{\frac{z - t}{\frac{a}{y}}} \]

  5. Applied egg-rr5.1

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

  6. Applied egg-rr2.3

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

  7. Final simplification2.3

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

Reproduce

herbie shell --seed 2022150 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

  (- x (/ (* y (- z t)) a)))