Average Error: 17.3 → 0.0
Time: 4.4s
Precision: binary64
\[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]
↓
\[\mathsf{fma}\left(y, x, -y \cdot z\right) \]
(FPCore (x y z)
 :precision binary64
 (+ (- (- (* x y) (* y z)) (* y y)) (* y y)))
↓
(FPCore (x y z) :precision binary64 (fma y x (- (* y z))))
double code(double x, double y, double z) {
	return (((x * y) - (y * z)) - (y * y)) + (y * y);
}
↓
double code(double x, double y, double z) {
	return fma(y, x, -(y * z));
}

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

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

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.3
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y \]

Derivation

  1. Initial program 17.3

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

  2. Simplified0.0

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

  3. Taylor expanded in x around 0 0.0

    \[\leadsto \color{blue}{y \cdot x - y \cdot z} \]

  4. Applied fma-neg_binary640.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022129 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, B"
  :precision binary64

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

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