2sqrt (example 3.1)

Average Error: 29.6 → 0.2

Time: 8.5s

Precision: binary64

Cost: 26176

Math

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

↓

\[\frac{1}{\mathsf{fma}\left({x}^{0.25}, {x}^{0.25}, \sqrt{1 + x}\right)} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (/ 1.0 (fma (pow x 0.25) (pow x 0.25) (sqrt (+ 1.0 x)))))

C

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

↓

double code(double x) {
	return 1.0 / fma(pow(x, 0.25), pow(x, 0.25), sqrt((1.0 + x)));
}

Julia

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

↓

function code(x)
	return Float64(1.0 / fma((x ^ 0.25), (x ^ 0.25), sqrt(Float64(1.0 + x))))
end

Wolfram

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

↓

code[x_] := N[(1.0 / N[(N[Power[x, 0.25], $MachinePrecision] * N[Power[x, 0.25], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	29.6
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 29.6

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

2. Applied egg-rr 29.0

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

3. Simplified 0.2

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

   [Start] 29.0	\[ \left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
   *-commutative [=>] 29.0	\[ \color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}} \cdot \left(x + \left(1 - x\right)\right)} \]
   associate-*l/ [=>] 29.0	\[ \color{blue}{\frac{1 \cdot \left(x + \left(1 - x\right)\right)}{\sqrt{x + 1} + \sqrt{x}}} \]
   *-lft-identity [=>] 29.0	\[ \frac{\color{blue}{x + \left(1 - x\right)}}{\sqrt{x + 1} + \sqrt{x}} \]
   +-commutative [=>] 29.0	\[ \frac{\color{blue}{\left(1 - x\right) + x}}{\sqrt{x + 1} + \sqrt{x}} \]
   associate-+l- [=>] 0.2	\[ \frac{\color{blue}{1 - \left(x - x\right)}}{\sqrt{x + 1} + \sqrt{x}} \]
   +-inverses [=>] 0.2	\[ \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]
   metadata-eval [=>] 0.2	\[ \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}} \]
   +-commutative [=>] 0.2	\[ \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} \]

4. Applied egg-rr 0.2

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({x}^{0.25}, {x}^{0.25}, \sqrt{1 + x}\right)}} \]

5. Final simplification 0.2

   \[\leadsto \frac{1}{\mathsf{fma}\left({x}^{0.25}, {x}^{0.25}, \sqrt{1 + x}\right)} \]

Alternatives

Alternative 1
Error	0.3
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq 29500000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	13248

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

Alternative 3
Error	1.0
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\frac{1}{x \cdot 0.5 + \left(1 + \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 4
Error	1.3
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1}{1 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 5
Error	2.1
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.3:\\ \;\;\;\;1 + x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 6
Error	31.0
Cost	448

\[\frac{1}{x + \left(1 + x\right)} \]

Alternative 7
Error	31.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023034 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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