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

Average Accuracy: 99.9% → 99.9%

Time: 13.8s

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 99.9%

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

2. Simplified 99.9%

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

   [Start] 99.9	\[ \left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   +-lft-identity [<=] 99.9	\[ \color{blue}{0 + \left(\left(\left(x \cdot \log y - y\right) - z\right) + \log t\right)} \]
   +-commutative [=>] 99.9	\[ 0 + \color{blue}{\left(\log t + \left(\left(x \cdot \log y - y\right) - z\right)\right)} \]
   associate-+r- [=>] 99.9	\[ 0 + \color{blue}{\left(\left(\log t + \left(x \cdot \log y - y\right)\right) - z\right)} \]
   associate-+r- [=>] 99.9	\[ \color{blue}{\left(0 + \left(\log t + \left(x \cdot \log y - y\right)\right)\right) - z} \]
   +-lft-identity [=>] 99.9	\[ \color{blue}{\left(\log t + \left(x \cdot \log y - y\right)\right)} - z \]
   associate-+r- [=>] 99.9	\[ \color{blue}{\left(\left(\log t + x \cdot \log y\right) - y\right)} - z \]
   +-commutative [=>] 99.9	\[ \left(\color{blue}{\left(x \cdot \log y + \log t\right)} - y\right) - z \]
   fma-def [=>] 99.9	\[ \left(\color{blue}{\mathsf{fma}\left(x, \log y, \log t\right)} - y\right) - z \]

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13376

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

Alternative 2
Accuracy	72.8%
Cost	7512

\[\begin{array}{l} t_1 := \left(-y\right) - z\\ t_2 := \log t - z\\ t_3 := x \cdot \log y - z\\ \mathbf{if}\;x \leq -8 \cdot 10^{+68}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-173}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-149}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Accuracy	67.8%
Cost	7384

\[\begin{array}{l} t_1 := \log t - z\\ t_2 := x \cdot \log y\\ t_3 := \left(-y\right) - z\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-10}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-243}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+132}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 3.9 \cdot 10^{-9}\right):\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]

Alternative 5
Accuracy	88.9%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+68} \lor \neg \left(x \leq 4.8 \cdot 10^{+128}\right):\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]

Alternative 6
Accuracy	71.0%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+69} \lor \neg \left(x \leq 1.35 \cdot 10^{+129}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]

Alternative 7
Accuracy	55.4%
Cost	6729

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-232} \lor \neg \left(z \leq 1.8 \cdot 10^{-214}\right):\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t\\ \end{array} \]

Alternative 8
Accuracy	57.1%
Cost	256

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

Alternative 9
Accuracy	29.3%
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2023141 
(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)))