bug333 (missed optimization)

Average Accuracy: 8.6% → 100.0%

Time: 4.1s

Precision: binary64

Cost: 13504

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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) / (sqrt((x + 1.0d0)) + sqrt((1.0d0 - x)))
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 + x) / (Math.sqrt((x + 1.0)) + Math.sqrt((1.0 - x)));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x)
	tmp = (x + x) / (sqrt((x + 1.0)) + sqrt((1.0 - x)));
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 + x), $MachinePrecision] / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	8.6%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 8.6%

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

2. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(x + x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}}} \]
   Proof

   [Start] 8.6	\[ \sqrt{1 + x} - \sqrt{1 - x} \]
   flip-- [=>] 8.6	\[ \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{1 - x} \cdot \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}}} \]
   div-inv [=>] 8.6	\[ \color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{1 - x} \cdot \sqrt{1 - x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}}} \]
   add-sqr-sqrt [<=] 8.6	\[ \left(\color{blue}{\left(1 + x\right)} - \sqrt{1 - x} \cdot \sqrt{1 - x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} \]
   add-sqr-sqrt [<=] 8.7	\[ \left(\left(1 + x\right) - \color{blue}{\left(1 - x\right)}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} \]
   associate--r- [=>] 21.2	\[ \color{blue}{\left(\left(\left(1 + x\right) - 1\right) + x\right)} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} \]
   add-exp-log [=>] 21.2	\[ \left(\left(\color{blue}{e^{\log \left(1 + x\right)}} - 1\right) + x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} \]
   expm1-udef [<=] 21.2	\[ \left(\color{blue}{\mathsf{expm1}\left(\log \left(1 + x\right)\right)} + x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} \]
   log1p-def [=>] 100.0	\[ \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right) + x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} \]
   expm1-log1p-u [<=] 100.0	\[ \left(\color{blue}{x} + x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} \]

3. Simplified 100.0%

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

   [Start] 100.0	\[ \left(x + x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} \]
   associate-*r/ [=>] 100.0	\[ \color{blue}{\frac{\left(x + x\right) \cdot 1}{\sqrt{1 + x} + \sqrt{1 - x}}} \]
   associate-/l* [=>] 100.0	\[ \color{blue}{\frac{x + x}{\frac{\sqrt{1 + x} + \sqrt{1 - x}}{1}}} \]
   /-rgt-identity [=>] 100.0	\[ \frac{x + x}{\color{blue}{\sqrt{1 + x} + \sqrt{1 - x}}} \]
   +-commutative [=>] 100.0	\[ \frac{x + x}{\sqrt{\color{blue}{x + 1}} + \sqrt{1 - x}} \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	6784

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

Alternative 2
Accuracy	98.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023126 
(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))))