Average Error: 0.3 → 0.3
Time: 7.0s
Precision: binary64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\log \left(y + x\right) + \left(\mathsf{fma}\left(\log t, a, \log z\right) - \mathsf{fma}\left(\log t, 0.5, t\right)\right) \]
(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
↓
(FPCore (x y z t a)
 :precision binary64
 (+ (log (+ y x)) (- (fma (log t) a (log z)) (fma (log t) 0.5 t))))
double code(double x, double y, double z, double t, double a) {
	return ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
}
↓
double code(double x, double y, double z, double t, double a) {
	return log((y + x)) + (fma(log(t), a, log(z)) - fma(log(t), 0.5, t));
}

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

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

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.3
Target0.3
Herbie0.3
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right) \]

Derivation

  1. Initial program 0.3

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

  2. Applied egg-rr16.5

    \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{\frac{\mathsf{fma}\left(a, a, -0.25\right) \cdot \log t}{a + 0.5}} \]

  3. Taylor expanded in a around 0 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2022166 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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