Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A

Average Error: 0.1 → 0.1

Time: 4.6s

Precision: binary64

Cost: 448

Math

\[x \cdot \left(1 - x \cdot y\right) \]

↓

\[x - x \cdot \left(x \cdot y\right) \]

FPCore

(FPCore (x y) :precision binary64 (* x (- 1.0 (* x y))))

↓

(FPCore (x y) :precision binary64 (- x (* x (* x y))))

C

double code(double x, double y) {
	return x * (1.0 - (x * y));
}

↓

double code(double x, double y) {
	return x - (x * (x * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 - (x * y))
end function

↓

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

Java

public static double code(double x, double y) {
	return x * (1.0 - (x * y));
}

↓

public static double code(double x, double y) {
	return x - (x * (x * y));
}

Python

def code(x, y):
	return x * (1.0 - (x * y))

↓

def code(x, y):
	return x - (x * (x * y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 - Float64(x * y)))
end

↓

function code(x, y)
	return Float64(x - Float64(x * Float64(x * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 - (x * y));
end

↓

function tmp = code(x, y)
	tmp = x - (x * (x * y));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(x - N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

   \[x \cdot \left(1 - x \cdot y\right) \]

2. Taylor expanded in x around 0 7.9

   \[\leadsto \color{blue}{-1 \cdot \left(y \cdot {x}^{2}\right) + x} \]

3. Simplified 0.1

   \[\leadsto \color{blue}{x - x \cdot \left(x \cdot y\right)} \]
   Proof
   (-.f64 x (*.f64 x (*.f64 x y))): 0 points increase in error, 0 points decrease in error
   (-.f64 x (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x x) y))): 37 points increase in error, 12 points decrease in error
   (-.f64 x (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) y)): 0 points increase in error, 0 points decrease in error
   (-.f64 x (Rewrite<= *-commutative_binary64 (*.f64 y (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unsub-neg_binary64 (+.f64 x (neg.f64 (*.f64 y (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 y (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (*.f64 y (pow.f64 x 2))) x)): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.1

   \[\leadsto x - x \cdot \left(x \cdot y\right) \]

Alternatives

Alternative 1
Error	14.7
Cost	648

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.1
Cost	448

\[x \cdot \left(1 - x \cdot y\right) \]

Alternative 3
Error	21.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))