Average Error: 0.1 → 0.1
Time: 8.3s
Precision: binary64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
↓
\[\left(y - \mathsf{fma}\left(y + 0.5, \log y, z\right)\right) + x \]
(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))
↓
(FPCore (x y z) :precision binary64 (+ (- y (fma (+ y 0.5) (log y) z)) x))
double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + y) - z;
}
↓
double code(double x, double y, double z) {
	return (y - fma((y + 0.5), log(y), z)) + x;
}

function code(x, y, z)
	return Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
end
↓
function code(x, y, z)
	return Float64(Float64(y - fma(Float64(y + 0.5), log(y), z)) + x)
end

code[x_, y_, z_] := N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]
↓
code[x_, y_, z_] := N[(N[(y - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(y - \mathsf{fma}\left(y + 0.5, \log y, z\right)\right) + x

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.1
\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y \]

Derivation

  1. Initial program 0.1

    \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

  2. Simplified0.1

    \[\leadsto \color{blue}{y - \left(\mathsf{fma}\left(y + 0.5, \log y, z\right) - x\right)} \]

  3. Applied associate--r-_binary640.1

    \[\leadsto \color{blue}{\left(y - \mathsf{fma}\left(y + 0.5, \log y, z\right)\right) + x} \]

  4. Final simplification0.1

    \[\leadsto \left(y - \mathsf{fma}\left(y + 0.5, \log y, z\right)\right) + x \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))