bug333 (missed optimization)

Average Error: 58.6 → 0.2

Time: 3.4s

Precision: binary64

Cost: 13504

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

Math

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

↓

\[0.125 \cdot {x}^{3} + \left(x + 0.0546875 \cdot {x}^{5}\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (+ (* 0.125 (pow x 3.0)) (+ x (* 0.0546875 (pow x 5.0)))))

C

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

↓

double code(double x) {
	return (0.125 * pow(x, 3.0)) + (x + (0.0546875 * pow(x, 5.0)));
}

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 = (0.125d0 * (x ** 3.0d0)) + (x + (0.0546875d0 * (x ** 5.0d0)))
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 (0.125 * Math.pow(x, 3.0)) + (x + (0.0546875 * Math.pow(x, 5.0)));
}

Python

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

↓

def code(x):
	return (0.125 * math.pow(x, 3.0)) + (x + (0.0546875 * math.pow(x, 5.0)))

Julia

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

↓

function code(x)
	return Float64(Float64(0.125 * (x ^ 3.0)) + Float64(x + Float64(0.0546875 * (x ^ 5.0))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (0.125 * (x ^ 3.0)) + (x + (0.0546875 * (x ^ 5.0)));
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[(N[(0.125 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.0546875 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	58.6
Target	0.0
Herbie	0.2

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

Derivation

1. Initial program 58.6

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

2. Simplified 58.6

   \[\leadsto \color{blue}{\sqrt{x - -1} - \sqrt{1 - x}} \]
   Proof

   [Start] 58.6	\[ \sqrt{1 + x} - \sqrt{1 - x} \]
   rational.json-simplify-17 [=>] 58.6	\[ \sqrt{\color{blue}{x - -1}} - \sqrt{1 - x} \]

3. Taylor expanded in x around 0 0.2

   \[\leadsto \color{blue}{0.0546875 \cdot {x}^{5} + \left(0.125 \cdot {x}^{3} + x\right)} \]

4. Simplified 0.2

   \[\leadsto \color{blue}{0.125 \cdot {x}^{3} + \left(x + 0.0546875 \cdot {x}^{5}\right)} \]
   Proof

   [Start] 0.2	\[ 0.0546875 \cdot {x}^{5} + \left(0.125 \cdot {x}^{3} + x\right) \]
   rational.json-simplify-41 [=>] 0.2	\[ \color{blue}{0.125 \cdot {x}^{3} + \left(x + 0.0546875 \cdot {x}^{5}\right)} \]

5. Final simplification 0.2

   \[\leadsto 0.125 \cdot {x}^{3} + \left(x + 0.0546875 \cdot {x}^{5}\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	6784

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

Alternative 2
Error	0.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023075 
(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))))