Average Error: 0.0 → 0.0
Time: 3.2s
Precision: binary64
\[x + \left(y - z\right) \cdot \left(t - x\right) \]
↓
\[t \cdot \left(y - z\right) + \mathsf{fma}\left(x, z - y, x\right) \]
(FPCore (x y z t) :precision binary64 (+ x (* (- y z) (- t x))))
↓
(FPCore (x y z t) :precision binary64 (+ (* t (- y z)) (fma x (- z y) x)))
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 (t * (y - z)) + fma(x, (z - y), x);
}

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y - z) * Float64(t - x)))
end
↓
function code(x, y, z, t)
	return Float64(Float64(t * Float64(y - z)) + fma(x, Float64(z - y), x))
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[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] + N[(x * N[(z - y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

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

Error

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 t around 0 0.0

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

  4. Taylor expanded in y around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022186 
(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))))