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

Average Error: 0.1 → 0.1

Time: 15.3s

Precision: binary64

Cost: 19648

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t) :precision binary64 (fma (log y) x (- (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(log(y), x, (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 fma(log(y), x, Float64(log(t) - Float64(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[Log[y], $MachinePrecision] * x + N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   associate--l- [=>] 0.1	\[ \color{blue}{\left(x \cdot \log y - \left(y + z\right)\right)} + \log t \]
   associate-+l- [=>] 0.1	\[ \color{blue}{x \cdot \log y - \left(\left(y + z\right) - \log t\right)} \]

3. Taylor expanded in x around 0 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ \left(\log y \cdot x + \log t\right) - \left(y + z\right) \]
   remove-double-neg [<=] 0.1	\[ \left(\log y \cdot x + \color{blue}{\left(-\left(-\log t\right)\right)}\right) - \left(y + z\right) \]
   log-rec [<=] 0.1	\[ \left(\log y \cdot x + \left(-\color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) - \left(y + z\right) \]
   mul-1-neg [<=] 0.1	\[ \left(\log y \cdot x + \color{blue}{-1 \cdot \log \left(\frac{1}{t}\right)}\right) - \left(y + z\right) \]
   +-commutative [=>] 0.1	\[ \color{blue}{\left(-1 \cdot \log \left(\frac{1}{t}\right) + \log y \cdot x\right)} - \left(y + z\right) \]
   mul-1-neg [=>] 0.1	\[ \left(\color{blue}{\left(-\log \left(\frac{1}{t}\right)\right)} + \log y \cdot x\right) - \left(y + z\right) \]
   log-rec [=>] 0.1	\[ \left(\left(-\color{blue}{\left(-\log t\right)}\right) + \log y \cdot x\right) - \left(y + z\right) \]
   remove-double-neg [=>] 0.1	\[ \left(\color{blue}{\log t} + \log y \cdot x\right) - \left(y + z\right) \]
   *-commutative [<=] 0.1	\[ \left(\log t + \color{blue}{x \cdot \log y}\right) - \left(y + z\right) \]
   associate-+r- [<=] 0.1	\[ \color{blue}{\log t + \left(x \cdot \log y - \left(y + z\right)\right)} \]
   +-commutative [=>] 0.1	\[ \color{blue}{\left(x \cdot \log y - \left(y + z\right)\right) + \log t} \]
   associate-+l- [=>] 0.1	\[ \color{blue}{x \cdot \log y - \left(\left(y + z\right) - \log t\right)} \]
   sub-neg [=>] 0.1	\[ \color{blue}{x \cdot \log y + \left(-\left(\left(y + z\right) - \log t\right)\right)} \]
   *-commutative [=>] 0.1	\[ \color{blue}{\log y \cdot x} + \left(-\left(\left(y + z\right) - \log t\right)\right) \]
   fma-udef [<=] 0.1	\[ \color{blue}{\mathsf{fma}\left(\log y, x, -\left(\left(y + z\right) - \log t\right)\right)} \]
   neg-sub0 [=>] 0.1	\[ \mathsf{fma}\left(\log y, x, \color{blue}{0 - \left(\left(y + z\right) - \log t\right)}\right) \]
   associate-+l- [<=] 0.1	\[ \mathsf{fma}\left(\log y, x, \color{blue}{\left(0 - \left(y + z\right)\right) + \log t}\right) \]
   neg-sub0 [<=] 0.1	\[ \mathsf{fma}\left(\log y, x, \color{blue}{\left(-\left(y + z\right)\right)} + \log t\right) \]
   +-commutative [=>] 0.1	\[ \mathsf{fma}\left(\log y, x, \color{blue}{\log t + \left(-\left(y + z\right)\right)}\right) \]
   sub-neg [<=] 0.1	\[ \mathsf{fma}\left(\log y, x, \color{blue}{\log t - \left(y + z\right)}\right) \]

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	6.7
Cost	13512

\[\begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;y \leq 125:\\ \;\;\;\;\left(\log t + t_1\right) - z\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+90}:\\ \;\;\;\;t_1 + \left(\log t - y\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+109} \lor \neg \left(y \leq 5.7 \cdot 10^{+193}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t_1 - y\\ \end{array} \]

Alternative 2
Error	0.1
Cost	13376

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

Alternative 3
Error	0.1
Cost	13376

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

Alternative 4
Error	6.3
Cost	13188

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -z\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+33}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - y\\ \end{array} \]

Alternative 5
Error	19.6
Cost	7120

\[\begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \left(-z\right) - y\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-180}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{-153}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+119}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	20.0
Cost	7120

\[\begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \left(-z\right) - y\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -0.000106:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-260}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+118}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	27.8
Cost	6994

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-102} \lor \neg \left(z \leq -4.9 \cdot 10^{-163}\right) \land \left(z \leq 2.3 \cdot 10^{-231} \lor \neg \left(z \leq 3.2 \cdot 10^{-113}\right)\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log t\\ \end{array} \]

Alternative 8
Error	9.9
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+90} \lor \neg \left(x \leq 2.55 \cdot 10^{+118}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 9
Error	6.5
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+88} \lor \neg \left(x \leq 3.9 \cdot 10^{+33}\right):\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 10
Error	6.3
Cost	6984

\[\begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{+36}:\\ \;\;\;\;t_1 - z\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+33}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - y\\ \end{array} \]

Alternative 11
Error	18.6
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+89} \lor \neg \left(x \leq 1.25 \cdot 10^{+119}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 12
Error	34.0
Cost	524

\[\begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+26}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+96}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+145}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 13
Error	26.9
Cost	256

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

Alternative 14
Error	44.7
Cost	128

\[-y \]

Reproduce

herbie shell --seed 2023031 
(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)))