Average Error: 0.1 → 0.0
Time: 2.1s
Precision: binary64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
↓
\[\mathsf{fma}\left(-4, \frac{z - x}{y}, 2\right) \]
(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))
↓
(FPCore (x y z) :precision binary64 (fma -4.0 (/ (- z x) y) 2.0))
double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
}
↓
double code(double x, double y, double z) {
	return fma(-4.0, ((z - x) / y), 2.0);
}

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y))
end
↓
function code(x, y, z)
	return fma(-4.0, Float64(Float64(z - x) / y), 2.0)
end

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := N[(-4.0 * N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] + 2.0), $MachinePrecision]

1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(-4, \frac{z - x}{y}, 2\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{y}, x - z, 2\right)} \]

  3. Taylor expanded in y around 0 0.0

    \[\leadsto \color{blue}{\left(2 + 4 \cdot \frac{x}{y}\right) - 4 \cdot \frac{z}{y}} \]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{z - x}{y}, 2\right)} \]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(-4, \frac{z - x}{y}, 2\right) \]

Reproduce

herbie shell --seed 2022146 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))