Average Error: 0.2 → 0.2
Time: 12.2s
Precision: binary64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\mathsf{fma}\left(\log t, -0.5, \log \left(y + x\right) + \mathsf{fma}\left(a, \log t, \log z\right)\right) - t \]
(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
 (- (fma (log t) -0.5 (+ (log (+ y x)) (fma a (log t) (log z)))) 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 fma(log(t), -0.5, (log((y + x)) + fma(a, log(t), log(z)))) - 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(fma(log(t), -0.5, Float64(log(Float64(y + x)) + fma(a, log(t), log(z)))) - 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[(N[Log[t], $MachinePrecision] * -0.5 + N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.2
Target0.2
Herbie0.2
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right) \]

Derivation

  1. Initial program 0.2

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

  2. Applied egg-rr0.2

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

  3. Taylor expanded in z 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. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022150 
(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))))