Average Error: 0.3 → 0.0
Time: 3.1s
Precision: binary64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
↓
\[\mathsf{fma}\left(\frac{x}{y}, 4, 4\right) + \frac{z}{y} \cdot -4 \]
(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
↓
(FPCore (x y z) :precision binary64 (+ (fma (/ x y) 4.0 4.0) (* (/ z y) -4.0)))
double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}
↓
double code(double x, double y, double z) {
	return fma((x / y), 4.0, 4.0) + ((z / y) * -4.0);
}

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end
↓
function code(x, y, z)
	return Float64(fma(Float64(x / y), 4.0, 4.0) + Float64(Float64(z / y) * -4.0))
end

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := N[(N[(N[(x / y), $MachinePrecision] * 4.0 + 4.0), $MachinePrecision] + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]

1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{x}{y}, 4, 4\right) + \frac{z}{y} \cdot -4

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.3

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]

  2. Taylor expanded in x around 0 0.0

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

  3. Applied egg-rr0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022166 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))