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

Average Accuracy: 99.9% → 99.9%

Time: 10.6s

Precision: binary64

Cost: 19648

Math

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

↓

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

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(fma(x, log(y), 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[(x * N[Log[y], $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Simplified 99.9%

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

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

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.2%
Cost	13380

\[\begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;y \leq 480:\\ \;\;\;\;\left(\log t + t_1\right) - z\\ \mathbf{else}:\\ \;\;\;\;t_1 - \left(y + z\right)\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	13376

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

Alternative 3
Accuracy	74.8%
Cost	7512

\[\begin{array}{l} t_1 := \log t - z\\ t_2 := x \cdot \log y - y\\ t_3 := \left(-y\right) - z\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-220}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-249}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-77}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+132}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	70.7%
Cost	7384

\[\begin{array}{l} t_1 := \log t - z\\ t_2 := x \cdot \log y\\ t_3 := \left(-y\right) - z\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-219}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-249}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-70}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+134}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	99.1%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+15} \lor \neg \left(x \leq 0.006\right):\\ \;\;\;\;x \cdot \log y - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]

Alternative 6
Accuracy	88.2%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+100} \lor \neg \left(x \leq 5.3 \cdot 10^{+132}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]

Alternative 7
Accuracy	72.0%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+100} \lor \neg \left(x \leq 1.22 \cdot 10^{+135}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]

Alternative 8
Accuracy	49.8%
Cost	392

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+56}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+39}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Accuracy	59.2%
Cost	256

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

Alternative 10
Accuracy	29.8%
Cost	128

\[-y \]

Reproduce

herbie shell --seed 2023151 
(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)))