Average Error: 0.2 → 0.2
Time: 30.2s
Precision: binary64
Cost: 26496
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\left(\left(-\log \left(\frac{1}{z}\right)\right) + \mathsf{fma}\left(\log t, a + -0.5, \log \left(y + x\right)\right)\right) - t \]
(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
 (- (+ (- (log (/ 1.0 z))) (fma (log t) (+ a -0.5) (log (+ y x)))) 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((1.0 / z)) + fma(log(t), (a + -0.5), log((y + x)))) - 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(Float64(-log(Float64(1.0 / z))) + fma(log(t), Float64(a + -0.5), log(Float64(y + x)))) - 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[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]) + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[N[(y + x), $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
↓
\left(\left(-\log \left(\frac{1}{z}\right)\right) + \mathsf{fma}\left(\log t, a + -0.5, \log \left(y + x\right)\right)\right) - t

Error

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. Taylor expanded in a around 0 0.2

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

  3. Taylor expanded in z around inf 0.2

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

  4. Simplified0.2

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

Alternatives

Alternative 1
Error0.3
Cost26304
\[\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right) - \left(t - \log z\right) \]
Alternative 2
Error12.2
Cost20296
\[\begin{array}{l} t_1 := \left(\log z + a \cdot \log t\right) - t\\ \mathbf{if}\;a - 0.5 \leq -0.50000000000002:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a - 0.5 \leq -0.5:\\ \;\;\;\;\left(\log x - t\right) + \left(\log z + -0.5 \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error12.2
Cost20296
\[\begin{array}{l} t_1 := \left(\log z + a \cdot \log t\right) - t\\ \mathbf{if}\;a - 0.5 \leq -0.50000000000002:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a - 0.5 \leq -0.5:\\ \;\;\;\;\left(\log z + \left(\log x + -0.5 \cdot \log t\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error12.2
Cost20296
\[\begin{array}{l} t_1 := \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right) - t\\ \mathbf{if}\;a - 0.5 \leq -0.50000000000002:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a - 0.5 \leq -0.5:\\ \;\;\;\;\left(\log z + \left(\log x + -0.5 \cdot \log t\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error0.9
Cost20036
\[\begin{array}{l} \mathbf{if}\;t \leq 360:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t + \left(\log \left(y + x\right) + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \end{array} \]
Alternative 6
Error0.3
Cost20032
\[\left(\left(a + -0.5\right) \cdot \log t - \left(t - \log \left(x + y\right)\right)\right) + \log z \]
Alternative 7
Error13.4
Cost19908
\[\begin{array}{l} \mathbf{if}\;t \leq 6.1 \cdot 10^{-142}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t + \left(\log z + \log x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \end{array} \]
Alternative 8
Error14.8
Cost13248
\[\left(\log z + a \cdot \log t\right) - t \]
Alternative 9
Error24.3
Cost6724
\[\begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{+47}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
Alternative 10
Error16.4
Cost6720
\[a \cdot \log t - t \]
Alternative 11
Error39.2
Cost128
\[-t \]

Error

Reproduce

herbie shell --seed 2023010 
(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))))