bug333 (missed optimization)

Average Error: 58.9 → 0.1

Time: 6.0s

Precision: binary64

Cost: 20224

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

Math

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

↓

\[\left(x - {x}^{3} \cdot -0.125\right) + \left(0.0546875 \cdot {x}^{5} - {x}^{7} \cdot -0.0322265625\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (+
  (- x (* (pow x 3.0) -0.125))
  (- (* 0.0546875 (pow x 5.0)) (* (pow x 7.0) -0.0322265625))))

C

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

↓

double code(double x) {
	return (x - (pow(x, 3.0) * -0.125)) + ((0.0546875 * pow(x, 5.0)) - (pow(x, 7.0) * -0.0322265625));
}

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 - ((x ** 3.0d0) * (-0.125d0))) + ((0.0546875d0 * (x ** 5.0d0)) - ((x ** 7.0d0) * (-0.0322265625d0)))
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 - (Math.pow(x, 3.0) * -0.125)) + ((0.0546875 * Math.pow(x, 5.0)) - (Math.pow(x, 7.0) * -0.0322265625));
}

Python

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

↓

def code(x):
	return (x - (math.pow(x, 3.0) * -0.125)) + ((0.0546875 * math.pow(x, 5.0)) - (math.pow(x, 7.0) * -0.0322265625))

Julia

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

↓

function code(x)
	return Float64(Float64(x - Float64((x ^ 3.0) * -0.125)) + Float64(Float64(0.0546875 * (x ^ 5.0)) - Float64((x ^ 7.0) * -0.0322265625)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (x - ((x ^ 3.0) * -0.125)) + ((0.0546875 * (x ^ 5.0)) - ((x ^ 7.0) * -0.0322265625));
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[(x - N[(N[Power[x, 3.0], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0546875 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[x, 7.0], $MachinePrecision] * -0.0322265625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	58.9
Target	0.0
Herbie	0.1

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

Derivation

1. Initial program 58.9

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

2. Simplified 58.9

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

   [Start] 58.9	\[ \sqrt{1 + x} - \sqrt{1 - x} \]
   rational_best_45_simplify-73 [=>] 58.9	\[ \sqrt{\color{blue}{x + 1}} - \sqrt{1 - x} \]
   rational_best_45_simplify-19 [=>] 58.9	\[ \sqrt{\color{blue}{x - -1}} - \sqrt{1 - x} \]

3. Taylor expanded in x around 0 0.1

   \[\leadsto \color{blue}{0.0546875 \cdot {x}^{5} + \left(0.0322265625 \cdot {x}^{7} + \left(0.125 \cdot {x}^{3} + x\right)\right)} \]

4. Simplified 0.1

   \[\leadsto \color{blue}{0.125 \cdot {x}^{3} + \left(0.0546875 \cdot {x}^{5} + \left(x + 0.0322265625 \cdot {x}^{7}\right)\right)} \]
   Proof

   [Start] 0.1	\[ 0.0546875 \cdot {x}^{5} + \left(0.0322265625 \cdot {x}^{7} + \left(0.125 \cdot {x}^{3} + x\right)\right) \]
   rational_best_45_simplify-80 [=>] 0.1	\[ 0.0546875 \cdot {x}^{5} + \color{blue}{\left(0.125 \cdot {x}^{3} + \left(0.0322265625 \cdot {x}^{7} + x\right)\right)} \]
   rational_best_45_simplify-80 [=>] 0.1	\[ \color{blue}{0.125 \cdot {x}^{3} + \left(0.0546875 \cdot {x}^{5} + \left(0.0322265625 \cdot {x}^{7} + x\right)\right)} \]
   rational_best_45_simplify-73 [=>] 0.1	\[ 0.125 \cdot {x}^{3} + \left(0.0546875 \cdot {x}^{5} + \color{blue}{\left(x + 0.0322265625 \cdot {x}^{7}\right)}\right) \]

5. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\left(\left(x + 0.0546875 \cdot {x}^{5}\right) - {x}^{3} \cdot -0.125\right) - {x}^{7} \cdot -0.0322265625} \]

6. Simplified 0.1

   \[\leadsto \color{blue}{\left(x - {x}^{3} \cdot -0.125\right) + \left(0.0546875 \cdot {x}^{5} - {x}^{7} \cdot -0.0322265625\right)} \]
   Proof

   [Start] 0.1	\[ \left(\left(x + 0.0546875 \cdot {x}^{5}\right) - {x}^{3} \cdot -0.125\right) - {x}^{7} \cdot -0.0322265625 \]
   rational_best_45_simplify-109 [=>] 0.1	\[ \color{blue}{\left(0.0546875 \cdot {x}^{5} + \left(x - {x}^{3} \cdot -0.125\right)\right)} - {x}^{7} \cdot -0.0322265625 \]
   rational_best_45_simplify-109 [=>] 0.1	\[ \color{blue}{\left(x - {x}^{3} \cdot -0.125\right) + \left(0.0546875 \cdot {x}^{5} - {x}^{7} \cdot -0.0322265625\right)} \]

7. Final simplification 0.1

   \[\leadsto \left(x - {x}^{3} \cdot -0.125\right) + \left(0.0546875 \cdot {x}^{5} - {x}^{7} \cdot -0.0322265625\right) \]

Alternatives

Alternative 1
Error	0.1
Cost	20224

\[0.125 \cdot {x}^{3} + \left(0.0546875 \cdot {x}^{5} + \left(x + 0.0322265625 \cdot {x}^{7}\right)\right) \]

Alternative 2
Error	0.1
Cost	13504

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

Alternative 3
Error	0.2
Cost	6784

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

Alternative 4
Error	0.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023098 
(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))))