?

Average Error: 0.3 → 0.3
Time: 27.8s
Precision: binary64
Cost: 26304

?

\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\log \left(x + y\right) + \left(\mathsf{fma}\left(a + -0.5, \log t, \log z\right) - t\right) \]
(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 (+ x y)) (- (fma (+ a -0.5) (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 log((x + y)) + (fma((a + -0.5), 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(log(Float64(x + y)) + Float64(fma(Float64(a + -0.5), 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[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]

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

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

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

    associate--l+ [=>]0.3

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

    associate-+l- [<=]0.3

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

    +-commutative [=>]0.3

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

    associate-+r- [=>]0.3

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

    fma-def [=>]0.3

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

    sub-neg [=>]0.3

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

    metadata-eval [=>]0.3

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

  3. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost20160
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \frac{a + -0.5}{\frac{1}{\log t}} \]
Alternative 2
Error9.1
Cost20036
\[\begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + \log t \cdot \left(a + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y - t\right)\right) + a \cdot \log t\\ \end{array} \]
Alternative 3
Error0.3
Cost20032
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \log t \cdot \left(a + -0.5\right) \]
Alternative 4
Error12.5
Cost19908
\[\begin{array}{l} \mathbf{if}\;t \leq 25:\\ \;\;\;\;\log z + \left(\log y + \log t \cdot \left(a + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t + \left(\log z - t\right)\\ \end{array} \]
Alternative 5
Error12.5
Cost19908
\[\begin{array}{l} \mathbf{if}\;t \leq 25:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t + \left(\log z - t\right)\\ \end{array} \]
Alternative 6
Error20.5
Cost19908
\[\begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y - t\right)\right) + a \cdot \log t\\ \end{array} \]
Alternative 7
Error20.5
Cost19908
\[\begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y - t\right)\right) + a \cdot \log t\\ \end{array} \]
Alternative 8
Error20.2
Cost19904
\[\left(\log z + \left(\log y - t\right)\right) + \log t \cdot \left(a + -0.5\right) \]
Alternative 9
Error16.1
Cost13904
\[\begin{array}{l} t_1 := \log \left(x + y\right) + \left(a \cdot \log t - t\right)\\ \mathbf{if}\;a \leq -1.7 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-244}:\\ \;\;\;\;\log \left(\frac{y \cdot z}{{t}^{\left(0.5 - a\right)}}\right)\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-300}:\\ \;\;\;\;\log z + \left(\log y - t\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-263}:\\ \;\;\;\;-0.5 \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error15.8
Cost13904
\[\begin{array}{l} t_1 := \log \left(x + y\right) + \left(a \cdot \log t - t\right)\\ t_2 := -0.5 \cdot \log t\\ \mathbf{if}\;a \leq -6.9 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -7.5 \cdot 10^{-244}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + t_2\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{-299}:\\ \;\;\;\;\log z + \left(\log y - t\right)\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-263}:\\ \;\;\;\;t_2 + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 11
Error8.5
Cost13636
\[\begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t + \left(\log z - t\right)\\ \end{array} \]
Alternative 12
Error17.0
Cost13508
\[\begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t + \left(\log z - t\right)\\ \end{array} \]
Alternative 13
Error16.4
Cost13380
\[\begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-263}:\\ \;\;\;\;-0.5 \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t + \left(\log z - t\right)\\ \end{array} \]
Alternative 14
Error14.7
Cost13248
\[a \cdot \log t + \left(\log z - t\right) \]
Alternative 15
Error24.1
Cost6857
\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+30} \lor \neg \left(a \leq 1.85 \cdot 10^{+34}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
Alternative 16
Error14.6
Cost6848
\[\log t \cdot \left(a + -0.5\right) - t \]
Alternative 17
Error39.9
Cost128
\[-t \]

Error

Reproduce?

herbie shell --seed 2023039 
(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))))