Average Error: 0.2 → 0.4
Time: 6.5s
Precision: binary64
\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\left(\log y + \mathsf{fma}\left(\log t, -0.5 + a, \log z\right)\right) + \left(\frac{x}{y} - t\right) \]
(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) (fma (log t) (+ -0.5 a) (log z))) (- (/ x y) 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) + fma(log(t), (-0.5 + a), log(z))) + ((x / y) - 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(Float64(log(y) + fma(log(t), Float64(-0.5 + a), log(z))) + Float64(Float64(x / y) - 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[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(-0.5 + a), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]

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

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
Herbie0.4
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right) \]

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. Taylor expanded in x around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2022170 
(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))))