Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 34.9s

Precision: binary64

Cost: 26304

Math

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

↓

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

FPCore

(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)) (- (log z) (fma (log t) (- 0.5 a) t))))

C

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)) + (log(z) - fma(log(t), (0.5 - a), t));
}

Julia

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(log(z) - fma(log(t), Float64(0.5 - a), t)))
end

Wolfram

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[Log[z], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * N[(0.5 - a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.3
Target	0.3
Herbie	0.3

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

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. Simplified 0.3

   \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   Proof
   (+.f64 (log.f64 (+.f64 x y)) (-.f64 (log.f64 z) (fma.f64 (log.f64 t) (-.f64 1/2 a) t))): 0 points increase in error, 0 points decrease in error
   (+.f64 (log.f64 (+.f64 x y)) (-.f64 (log.f64 z) (fma.f64 (log.f64 t) (Rewrite<= unsub-neg_binary64 (+.f64 1/2 (neg.f64 a))) t))): 0 points increase in error, 0 points decrease in error
   (+.f64 (log.f64 (+.f64 x y)) (-.f64 (log.f64 z) (fma.f64 (log.f64 t) (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 a) 1/2)) t))): 0 points increase in error, 0 points decrease in error
   (+.f64 (log.f64 (+.f64 x y)) (-.f64 (log.f64 z) (fma.f64 (log.f64 t) (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 a)) 1/2) t))): 0 points increase in error, 0 points decrease in error
   (+.f64 (log.f64 (+.f64 x y)) (-.f64 (log.f64 z) (fma.f64 (log.f64 t) (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 a 1/2))) t))): 0 points increase in error, 0 points decrease in error
   (+.f64 (log.f64 (+.f64 x y)) (-.f64 (log.f64 z) (fma.f64 (log.f64 t) (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 a 1/2))) t))): 0 points increase in error, 0 points decrease in error
   (+.f64 (log.f64 (+.f64 x y)) (-.f64 (log.f64 z) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (log.f64 t) (neg.f64 (-.f64 a 1/2))) t)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (log.f64 (+.f64 x y)) (-.f64 (log.f64 z) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (-.f64 a 1/2)) (log.f64 t))) t))): 0 points increase in error, 0 points decrease in error
   (+.f64 (log.f64 (+.f64 x y)) (-.f64 (log.f64 z) (Rewrite<= +-commutative_binary64 (+.f64 t (*.f64 (neg.f64 (-.f64 a 1/2)) (log.f64 t)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (log.f64 (+.f64 x y)) (-.f64 (log.f64 z) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 t (*.f64 (-.f64 a 1/2) (log.f64 t)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) (-.f64 t (*.f64 (-.f64 a 1/2) (log.f64 t))))): 10 points increase in error, 10 points decrease in error
   (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a 1/2) (log.f64 t)))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.3

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

Alternatives

Alternative 1
Error	1.2
Cost	20424

\[\begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a + -0.5 \leq -0.6:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a + -0.5 \leq -0.5:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + \left(\log t \cdot -0.5 - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	12.4
Cost	20040

\[\begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a \leq -0.24513292104280143:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.116410142397736 \cdot 10^{-18}:\\ \;\;\;\;\left(\log z + \left(\log y - \log t \cdot 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.3
Cost	20032

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

Alternative 4
Error	15.0
Cost	19976

\[\begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a \leq -0.24513292104280143:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.116410142397736 \cdot 10^{-18}:\\ \;\;\;\;\left(\log z + \log \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	12.2
Cost	19908

\[\begin{array}{l} \mathbf{if}\;t \leq 0.18645025240305188:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \]

Alternative 6
Error	20.1
Cost	19904

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

Alternative 7
Error	8.7
Cost	13896

\[\begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a \leq -162345336576131.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 203815.09080347212:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a + -0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	14.8
Cost	13772

\[\begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a \leq -0.24513292104280143:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.131262678680714 \cdot 10^{-260}:\\ \;\;\;\;\log \left(z \cdot \left(\left(x + y\right) \cdot \sqrt{\frac{1}{t}}\right)\right) - t\\ \mathbf{elif}\;a \leq 3.116410142397736 \cdot 10^{-18}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	17.4
Cost	13768

\[\begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a \leq -162345336576131.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 203815.09080347212:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot \left(a + -0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	18.0
Cost	13644

\[\begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a \leq -9.403220079671976 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.0285700114875274 \cdot 10^{-277}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{elif}\;a \leq 1.7338104982083062 \cdot 10^{-96}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	17.7
Cost	13644

\[\begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a \leq -0.24513292104280143:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.0285700114875274 \cdot 10^{-277}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 1.7338104982083062 \cdot 10^{-96}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	17.3
Cost	13640

\[\begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a \leq -0.24513292104280143:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.116410142397736 \cdot 10^{-18}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	18.7
Cost	13576

\[\begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a \leq -0.24513292104280143:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.116410142397736 \cdot 10^{-18}:\\ \;\;\;\;\log \left({t}^{-0.5} \cdot \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	18.8
Cost	13576

\[\begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a \leq -0.24513292104280143:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.116410142397736 \cdot 10^{-18}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	14.4
Cost	6984

\[\begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a \leq -9.403220079671976 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.5842936917718207 \cdot 10^{-24}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	24.0
Cost	6856

\[\begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a \leq -162345336576131.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.1871528846494894 \cdot 10^{+52}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	16.1
Cost	6720

\[\log t \cdot a - t \]

Alternative 18
Error	39.7
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2022317 
(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))))