bug333 (missed optimization)

Average Accuracy: 8.5% → 99.5%

Time: 3.0s

Precision: binary64

Cost: 6784

\[-1 \leq x \land x \leq 1\]

Math

\[\sqrt{1 + x} - \sqrt{1 - x} \]

↓

\[x + 0.125 \cdot {x}^{3} \]

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))

↓

(FPCore (x) :precision binary64 (+ x (* 0.125 (pow x 3.0))))

C

double code(double x) {
	return sqrt((1.0 + x)) - sqrt((1.0 - x));
}

↓

double code(double x) {
	return x + (0.125 * pow(x, 3.0));
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + (0.125d0 * (x ** 3.0d0))
end function

Java

public static double code(double x) {
	return Math.sqrt((1.0 + x)) - Math.sqrt((1.0 - x));
}

↓

public static double code(double x) {
	return x + (0.125 * Math.pow(x, 3.0));
}

Python

def code(x):
	return math.sqrt((1.0 + x)) - math.sqrt((1.0 - x))

↓

def code(x):
	return x + (0.125 * math.pow(x, 3.0))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 + x)) - sqrt(Float64(1.0 - x)))
end

↓

function code(x)
	return Float64(x + Float64(0.125 * (x ^ 3.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 + x)) - sqrt((1.0 - x));
end

↓

function tmp = code(x)
	tmp = x + (0.125 * (x ^ 3.0));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(x + N[(0.125 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	8.5%
Target	100.0%
Herbie	99.5%

\[\frac{2 \cdot x}{\sqrt{1 + x} + \sqrt{1 - x}} \]

Derivation

1. Initial program 8.5%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]

2. Taylor expanded in x around 0 99.5%

   \[\leadsto \color{blue}{0.125 \cdot {x}^{3} + x} \]

3. Final simplification 99.5%

   \[\leadsto x + 0.125 \cdot {x}^{3} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x)
  :name "bug333 (missed optimization)"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :herbie-target
  (/ (* 2.0 x) (+ (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))

  (- (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))