Average Error: 0.1 → 0.0
Time: 4.0s
Precision: binary64
\[\frac{\left(x + y\right) - z}{t \cdot 2} \]
↓
\[\mathsf{fma}\left(0.5, \frac{x}{t}, \frac{0.5 \cdot \left(y - z\right)}{t}\right) \]
(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))
↓
(FPCore (x y z t) :precision binary64 (fma 0.5 (/ x t) (/ (* 0.5 (- y z)) t)))
double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}
↓
double code(double x, double y, double z, double t) {
	return fma(0.5, (x / t), ((0.5 * (y - z)) / t));
}

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end
↓
function code(x, y, z, t)
	return fma(0.5, Float64(x / t), Float64(Float64(0.5 * Float64(y - z)) / t))
end

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := N[(0.5 * N[(x / t), $MachinePrecision] + N[(N[(0.5 * N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

\frac{\left(x + y\right) - z}{t \cdot 2}
↓
\mathsf{fma}\left(0.5, \frac{x}{t}, \frac{0.5 \cdot \left(y - z\right)}{t}\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

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

  2. Applied egg-rr1.3

    \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x + \left(y - z\right)}\right)}^{2}}{t} \cdot \frac{\sqrt[3]{x + \left(y - z\right)}}{2}} \]

  3. Taylor expanded in x around 0 0.1

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022153 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B"
  :precision binary64
  (/ (- (+ x y) z) (* t 2.0)))