Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A

Average Error: 0.15% → 0.15%

Time: 12.6s

Precision: binary64

Cost: 19648

Math

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

↓

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

FPCore

(FPCore (x y z t) :precision binary64 (+ (- (- (* x (log y)) y) z) (log t)))

↓

(FPCore (x y z t) :precision binary64 (- (- (fma x (log y) (log t)) y) z))

C

double code(double x, double y, double z, double t) {
	return (((x * log(y)) - y) - z) + log(t);
}

↓

double code(double x, double y, double z, double t) {
	return (fma(x, log(y), log(t)) - y) - z;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * log(y)) - y) - z) + log(t))
end

↓

function code(x, y, z, t)
	return Float64(Float64(fma(x, log(y), log(t)) - y) - z)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 0.15

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

2. Simplified 0.15

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

   [Start] 0.15	\[ \left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   +-lft-identity [<=] 0.15	\[ \color{blue}{0 + \left(\left(\left(x \cdot \log y - y\right) - z\right) + \log t\right)} \]
   +-commutative [=>] 0.15	\[ 0 + \color{blue}{\left(\log t + \left(\left(x \cdot \log y - y\right) - z\right)\right)} \]
   associate-+r- [=>] 0.15	\[ 0 + \color{blue}{\left(\left(\log t + \left(x \cdot \log y - y\right)\right) - z\right)} \]
   associate-+r- [=>] 0.15	\[ \color{blue}{\left(0 + \left(\log t + \left(x \cdot \log y - y\right)\right)\right) - z} \]
   +-lft-identity [=>] 0.15	\[ \color{blue}{\left(\log t + \left(x \cdot \log y - y\right)\right)} - z \]
   associate-+r- [=>] 0.15	\[ \color{blue}{\left(\left(\log t + x \cdot \log y\right) - y\right)} - z \]
   +-commutative [=>] 0.15	\[ \left(\color{blue}{\left(x \cdot \log y + \log t\right)} - y\right) - z \]
   fma-def [=>] 0.15	\[ \left(\color{blue}{\mathsf{fma}\left(x, \log y, \log t\right)} - y\right) - z \]

3. Final simplification 0.15

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

Alternatives

Alternative 1
Error	1.25%
Cost	13513

\[\begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;z \leq -4 \cdot 10^{+16} \lor \neg \left(z \leq 3.6 \cdot 10^{-26}\right):\\ \;\;\;\;t_1 - z\\ \mathbf{else}:\\ \;\;\;\;\log t + t_1\\ \end{array} \]

Alternative 2
Error	0.15%
Cost	13376

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

Alternative 3
Error	23.08%
Cost	7248

\[\begin{array}{l} t_1 := \left(-y\right) - z\\ t_2 := x \cdot \log y - z\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-96}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	28.85%
Cost	7120

\[\begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(-y\right) - z\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-98}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+109}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	0.82%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -4800000000000 \lor \neg \left(x \leq 1.35\right):\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]

Alternative 6
Error	0.83%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -4800000000000:\\ \;\;\;\;\left(\frac{x}{\frac{1}{\log y}} - y\right) - z\\ \mathbf{elif}\;x \leq 1.12:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \end{array} \]

Alternative 7
Error	44.56%
Cost	6992

\[\begin{array}{l} t_1 := \left(-y\right) - z\\ \mathbf{if}\;z \leq -1.08 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-122}:\\ \;\;\;\;\log t\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-226}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-82}:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	10.2%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+57} \lor \neg \left(x \leq 3 \cdot 10^{+25}\right):\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]

Alternative 9
Error	27.85%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+89} \lor \neg \left(x \leq 1.9 \cdot 10^{+111}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]

Alternative 10
Error	41.71%
Cost	256

\[\left(-y\right) - z \]

Alternative 11
Error	69.39%
Cost	128

\[-y \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))