?

Average Error: 0.3 → 0.3
Time: 27.9s
Precision: binary64
Cost: 26304

?

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

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(z) - t) + fma(Float64(a + -0.5), log(t), log(Float64(x + y))))
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[z], $MachinePrecision] - t), $MachinePrecision] + N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error?

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. Simplified0.3

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

    [Start]0.3

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

    associate--l+ [=>]0.3

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

    +-commutative [=>]0.3

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

    associate-+l+ [=>]0.3

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

    +-commutative [<=]0.3

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

    fma-def [=>]0.3

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

    sub-neg [=>]0.3

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

    metadata-eval [=>]0.3

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

  3. Final simplification0.3

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

Alternatives

Alternative 1
Error1.4
Cost20168
\[\begin{array}{l} t_1 := \log z - t\\ \mathbf{if}\;a \leq -270000:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, t_1\right)\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{-26}:\\ \;\;\;\;t_1 + \left(\log \left(x + y\right) + -0.5 \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \]
Alternative 2
Error1.0
Cost20036
\[\begin{array}{l} \mathbf{if}\;t \leq 112:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + \left(a + -0.5\right) \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, \log z - t\right)\\ \end{array} \]
Alternative 3
Error0.3
Cost20032
\[\left(\left(\log z - t\right) + \log \left(x + y\right)\right) + \left(a + -0.5\right) \cdot \log t \]
Alternative 4
Error0.3
Cost20032
\[\left(\left(\log z + \log \left(x + y\right)\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]
Alternative 5
Error12.6
Cost19908
\[\begin{array}{l} \mathbf{if}\;t \leq 112:\\ \;\;\;\;\log z + \left(\log y + \left(a + -0.5\right) \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, \log z - t\right)\\ \end{array} \]
Alternative 6
Error8.8
Cost19652
\[\begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \left(a + -0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, \log z - t\right)\\ \end{array} \]
Alternative 7
Error17.2
Cost13840
\[\begin{array}{l} t_1 := \log z - t\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{-126}:\\ \;\;\;\;t_1 + a \cdot \log t\\ \mathbf{elif}\;a \leq -3.2 \cdot 10^{-199}:\\ \;\;\;\;-0.5 \cdot \log t + \log \left(z \cdot y\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-174}:\\ \;\;\;\;t_1 + \log \left(x + y\right)\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{-102}:\\ \;\;\;\;\log \left(\frac{z \cdot y}{{t}^{\left(0.5 - a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \]
Alternative 8
Error17.0
Cost13776
\[\begin{array}{l} t_1 := -0.5 \cdot \log t + \log \left(z \cdot y\right)\\ t_2 := \log z - t\\ \mathbf{if}\;a \leq -2.5 \cdot 10^{-126}:\\ \;\;\;\;t_2 + a \cdot \log t\\ \mathbf{elif}\;a \leq -5 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-174}:\\ \;\;\;\;t_2 + \log \left(x + y\right)\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \]
Alternative 9
Error8.8
Cost13636
\[\begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \left(a + -0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]
Alternative 10
Error14.1
Cost13513
\[\begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{-8} \lor \neg \left(a \leq 8.2 \cdot 10^{-27}\right):\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \end{array} \]
Alternative 11
Error17.3
Cost13508
\[\begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;\log \left(z \cdot y\right) + \left(a + -0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]
Alternative 12
Error15.0
Cost13248
\[\left(\log z - t\right) + a \cdot \log t \]
Alternative 13
Error23.0
Cost6857
\[\begin{array}{l} \mathbf{if}\;a \leq -7500000000 \lor \neg \left(a \leq 6.5 \cdot 10^{+80}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log z - t\\ \end{array} \]
Alternative 14
Error14.9
Cost6848
\[\left(a + -0.5\right) \cdot \log t - t \]
Alternative 15
Error37.5
Cost6724
\[\begin{array}{l} \mathbf{if}\;t \leq 105:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
Alternative 16
Error24.6
Cost6724
\[\begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{+48}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
Alternative 17
Error39.7
Cost128
\[-t \]

Error

Reproduce?

herbie shell --seed 2023055 
(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))))