Average Error: 0.0 → 0.0
Time: 10.0s
Precision: binary64
\[x + \left(y - z\right) \cdot \left(t - x\right) \]
↓
\[\mathsf{fma}\left(y, t, \mathsf{fma}\left(x, z - y, x\right) - t \cdot z\right) \]
(FPCore (x y z t) :precision binary64 (+ x (* (- y z) (- t x))))
↓
(FPCore (x y z t) :precision binary64 (fma y t (- (fma x (- z y) x) (* t z))))
double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - x));
}
↓
double code(double x, double y, double z, double t) {
	return fma(y, t, (fma(x, (z - y), x) - (t * z)));
}

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

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

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original0.0
Target0.0
Herbie0.0
\[x + \left(t \cdot \left(y - z\right) + \left(-x\right) \cdot \left(y - z\right)\right) \]

Derivation

  1. Initial program 0.0

    \[x + \left(y - z\right) \cdot \left(t - x\right) \]

  2. Simplified0.0

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

  3. Taylor expanded in y around 0 0.0

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

  4. Applied egg-rr0.0

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

  5. Taylor expanded in z around 0 0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2022166 
(FPCore (x y z t)
  :name "Data.Metrics.Snapshot:quantile from metrics-0.3.0.2"
  :precision binary64

  :herbie-target
  (+ x (+ (* t (- y z)) (* (- x) (- y z))))

  (+ x (* (- y z) (- t x))))