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

Average Accuracy: 99.9% → 99.9%

Time: 13.7s

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 99.9%

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

2. Simplified 99.9%

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

   [Start] 99.9	\[ \left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   associate-+l- [=>] 99.9	\[ \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]

3. Taylor expanded in x around 0 99.9%

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

4. Simplified 99.9%

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

   [Start] 99.9	\[ \left(\log y \cdot x + \log t\right) - \left(y + z\right) \]
   +-commutative [=>] 99.9	\[ \color{blue}{\left(\log t + \log y \cdot x\right)} - \left(y + z\right) \]
   remove-double-neg [<=] 99.9	\[ \left(\color{blue}{\left(-\left(-\log t\right)\right)} + \log y \cdot x\right) - \left(y + z\right) \]
   neg-mul-1 [=>] 99.9	\[ \left(\color{blue}{-1 \cdot \left(-\log t\right)} + \log y \cdot x\right) - \left(y + z\right) \]
   mul-1-neg [<=] 99.9	\[ \left(-1 \cdot \color{blue}{\left(-1 \cdot \log t\right)} + \log y \cdot x\right) - \left(y + z\right) \]
   associate-*r* [=>] 99.9	\[ \left(\color{blue}{\left(-1 \cdot -1\right) \cdot \log t} + \log y \cdot x\right) - \left(y + z\right) \]
   metadata-eval [=>] 99.9	\[ \left(\color{blue}{1} \cdot \log t + \log y \cdot x\right) - \left(y + z\right) \]
   *-commutative [<=] 99.9	\[ \left(\color{blue}{\log t \cdot 1} + \log y \cdot x\right) - \left(y + z\right) \]
   *-commutative [=>] 99.9	\[ \left(\log t \cdot 1 + \color{blue}{x \cdot \log y}\right) - \left(y + z\right) \]
   associate-+r- [<=] 99.9	\[ \color{blue}{\log t \cdot 1 + \left(x \cdot \log y - \left(y + z\right)\right)} \]
   *-rgt-identity [<=] 99.9	\[ \log t \cdot 1 + \color{blue}{\left(x \cdot \log y - \left(y + z\right)\right) \cdot 1} \]
   distribute-rgt-in [<=] 99.9	\[ \color{blue}{1 \cdot \left(\log t + \left(x \cdot \log y - \left(y + z\right)\right)\right)} \]
   *-lft-identity [=>] 99.9	\[ \color{blue}{\log t + \left(x \cdot \log y - \left(y + z\right)\right)} \]
   +-commutative [=>] 99.9	\[ \color{blue}{\left(x \cdot \log y - \left(y + z\right)\right) + \log t} \]
   associate-+l- [=>] 99.9	\[ \color{blue}{x \cdot \log y - \left(\left(y + z\right) - \log t\right)} \]
   associate-+r- [<=] 99.9	\[ x \cdot \log y - \color{blue}{\left(y + \left(z - \log t\right)\right)} \]
   unsub-neg [<=] 99.9	\[ \color{blue}{x \cdot \log y + \left(-\left(y + \left(z - \log t\right)\right)\right)} \]
   *-commutative [<=] 99.9	\[ \color{blue}{\log y \cdot x} + \left(-\left(y + \left(z - \log t\right)\right)\right) \]
   fma-udef [<=] 99.9	\[ \color{blue}{\mathsf{fma}\left(\log y, x, -\left(y + \left(z - \log t\right)\right)\right)} \]
   distribute-neg-in [=>] 99.9	\[ \mathsf{fma}\left(\log y, x, \color{blue}{\left(-y\right) + \left(-\left(z - \log t\right)\right)}\right) \]
   sub-neg [<=] 99.9	\[ \mathsf{fma}\left(\log y, x, \color{blue}{\left(-y\right) - \left(z - \log t\right)}\right) \]
   associate--r- [=>] 99.9	\[ \mathsf{fma}\left(\log y, x, \color{blue}{\left(\left(-y\right) - z\right) + \log t}\right) \]
   +-commutative [=>] 99.9	\[ \mathsf{fma}\left(\log y, x, \color{blue}{\log t + \left(\left(-y\right) - z\right)}\right) \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13376

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

Alternative 2
Accuracy	65.8%
Cost	7253

\[\begin{array}{l} t_1 := \log t - z\\ t_2 := \log y \cdot x\\ \mathbf{if}\;y \leq 2.6 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-166}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+67} \lor \neg \left(y \leq 2.5 \cdot 10^{+107}\right):\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	98.8%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+18} \lor \neg \left(x \leq 1.65 \cdot 10^{-16}\right):\\ \;\;\;\;\left(\log y \cdot x - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 4
Accuracy	84.7%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+141} \lor \neg \left(x \leq 9.5 \cdot 10^{+133}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 5
Accuracy	88.7%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+141} \lor \neg \left(x \leq 8.5 \cdot 10^{+79}\right):\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 6
Accuracy	89.1%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+141} \lor \neg \left(x \leq 1.9 \cdot 10^{+43}\right):\\ \;\;\;\;\log y \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 7
Accuracy	72.3%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+143} \lor \neg \left(x \leq 8.2 \cdot 10^{+132}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]

Alternative 8
Accuracy	48.3%
Cost	260

\[\begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+58}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 9
Accuracy	58.3%
Cost	256

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

Alternative 10
Accuracy	29.6%
Cost	128

\[-y \]

Reproduce

herbie shell --seed 2023133 
(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)))