Average Error: 0.1 → 0.1
Time: 4.1s
Precision: binary64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
↓
\[x \cdot 0.5 + \left(\mathsf{fma}\left(y, \log z, y\right) - y \cdot z\right) \]
(FPCore (x y z) :precision binary64 (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))
↓
(FPCore (x y z)
 :precision binary64
 (+ (* x 0.5) (- (fma y (log z) y) (* y z))))
double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + log(z)));
}
↓
double code(double x, double y, double z) {
	return (x * 0.5) + (fma(y, log(z), y) - (y * z));
}

function code(x, y, z)
	return Float64(Float64(x * 0.5) + Float64(y * Float64(Float64(1.0 - z) + log(z))))
end
↓
function code(x, y, z)
	return Float64(Float64(x * 0.5) + Float64(fma(y, log(z), y) - Float64(y * z)))
end

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] + N[(N[(y * N[Log[z], $MachinePrecision] + y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
x \cdot 0.5 + \left(\mathsf{fma}\left(y, \log z, y\right) - y \cdot z\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Initial program 0.1

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

  2. Taylor expanded in z around 0 0.1

    \[\leadsto x \cdot 0.5 + \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + \left(1 + \log z\right) \cdot y\right)} \]

  3. Taylor expanded in z around inf 0.1

    \[\leadsto x \cdot 0.5 + \left(-1 \cdot \left(y \cdot z\right) + \color{blue}{y \cdot \left(1 + -1 \cdot \log \left(\frac{1}{z}\right)\right)}\right) \]

  4. Simplified0.1

    \[\leadsto x \cdot 0.5 + \left(-1 \cdot \left(y \cdot z\right) + \color{blue}{\mathsf{fma}\left(y, \log z, y\right)}\right) \]

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022170 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

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

  (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))