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

Average Error: 0.1 → 0.1

Time: 10.3s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * log(y)) - y) - z) + log(t)
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((log(y) * x) - y) + (log(t) - z)
end function

Java

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

↓

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

Python

def code(x, y, z, t):
	return (((x * math.log(y)) - y) - z) + math.log(t)

↓

def code(x, y, z, t):
	return ((math.log(y) * x) - y) + (math.log(t) - 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(Float64(log(y) * x) - y) + Float64(log(t) - z))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * log(y)) - y) - z) + log(t);
end

↓

function tmp = code(x, y, z, t)
	tmp = ((log(y) * x) - y) + (log(t) - 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[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Taylor expanded in y around 0 0.1

   \[\leadsto \color{blue}{\left(\log y \cdot x + \left(-1 \cdot y + \log t\right)\right) - z} \]

3. Simplified 0.1

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

   [Start] 0.1	\[ \left(\log y \cdot x + \left(-1 \cdot y + \log t\right)\right) - z \]
   rational.json-simplify-2 [<=] 0.1	\[ \left(\color{blue}{x \cdot \log y} + \left(-1 \cdot y + \log t\right)\right) - z \]
   rational.json-simplify-1 [=>] 0.1	\[ \left(x \cdot \log y + \color{blue}{\left(\log t + -1 \cdot y\right)}\right) - z \]
   rational.json-simplify-41 [=>] 0.1	\[ \color{blue}{\left(\log t + \left(-1 \cdot y + x \cdot \log y\right)\right)} - z \]
   rational.json-simplify-1 [=>] 0.1	\[ \left(\log t + \color{blue}{\left(x \cdot \log y + -1 \cdot y\right)}\right) - z \]
   rational.json-simplify-2 [=>] 0.1	\[ \left(\log t + \left(x \cdot \log y + \color{blue}{y \cdot -1}\right)\right) - z \]
   rational.json-simplify-8 [<=] 0.1	\[ \left(\log t + \left(x \cdot \log y + \color{blue}{\left(-y\right)}\right)\right) - z \]
   rational.json-simplify-12 [=>] 0.1	\[ \left(\log t + \left(x \cdot \log y + \color{blue}{\left(0 - y\right)}\right)\right) - z \]
   rational.json-simplify-48 [<=] 0.1	\[ \left(\log t + \color{blue}{\left(\left(0 + x \cdot \log y\right) - y\right)}\right) - z \]
   rational.json-simplify-1 [<=] 0.1	\[ \left(\log t + \left(\color{blue}{\left(x \cdot \log y + 0\right)} - y\right)\right) - z \]
   rational.json-simplify-4 [=>] 0.1	\[ \left(\log t + \left(\color{blue}{x \cdot \log y} - y\right)\right) - z \]
   rational.json-simplify-48 [=>] 0.1	\[ \color{blue}{\left(x \cdot \log y - y\right) + \left(\log t - z\right)} \]
   rational.json-simplify-2 [=>] 0.1	\[ \left(\color{blue}{\log y \cdot x} - y\right) + \left(\log t - z\right) \]

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	6.7
Cost	13644

\[\begin{array}{l} t_1 := \log y \cdot x - y\\ t_2 := x \cdot \log y + \left(\log t - z\right)\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+72}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+86}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	33.2
Cost	7252

\[\begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+60}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-154}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-145}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+14}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3
Error	26.1
Cost	7252

\[\begin{array}{l} t_1 := \log t - y\\ t_2 := \log y \cdot x\\ t_3 := \log t - z\\ \mathbf{if}\;x \leq -60000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-231}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	34.6
Cost	7124

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+55}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-202}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-184}:\\ \;\;\;\;\log t\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-111}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq 10^{-73}:\\ \;\;\;\;\log t\\ \mathbf{elif}\;z \leq 10^{+24}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5
Error	26.5
Cost	7120

\[\begin{array}{l} t_1 := \log t - y\\ \mathbf{if}\;z \leq -1.42 \cdot 10^{+62}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-265}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-167}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Error	10.2
Cost	6984

\[\begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+136}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	6.7
Cost	6984

\[\begin{array}{l} t_1 := \log y \cdot x - y\\ \mathbf{if}\;x \leq -60000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+86}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	32.5
Cost	392

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+63}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+20}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Error	44.2
Cost	128

\[-y \]

Reproduce

herbie shell --seed 2023074 
(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)))