3frac (problem 3.3.3)

Percentage Accurate: 96.8% → 100.0%

Time: 17.2s

Precision: binary64

Cost: 576

• could not determine a ground truth

  1. x = 2.4538325424698454e+120

Math

\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

↓

\[\frac{2}{x \cdot \left(x \cdot x + -1\right)} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

↓

(FPCore (x) :precision binary64 (/ 2.0 (* x (+ (* x x) -1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

↓

double code(double x) {
	return 2.0 / (x * ((x * x) + -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (x * ((x * x) + (-1.0d0)))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

↓

public static double code(double x) {
	return 2.0 / (x * ((x * x) + -1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

↓

def code(x):
	return 2.0 / (x * ((x * x) + -1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

↓

function code(x)
	return Float64(2.0 / Float64(x * Float64(Float64(x * x) + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

↓

function tmp = code(x)
	tmp = 2.0 / (x * ((x * x) + -1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	96.8%
Target	100.0%
Herbie	100.0%

\[\frac{2}{x \cdot \left(x \cdot x - 1\right)} \]

Derivation

1. Initial program 96.8%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

2. Applied egg-rr 96.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, x + -1, \left(x \cdot \left(1 + x\right)\right) \cdot 1\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \left(x + -1\right)}} \]
   Step-by-step derivation

   [Start] 96.8%	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   frac-sub [=>] 96.7%	\[ \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1} \]
   frac-add [=>] 97.7%	\[ \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}} \]
   fma-def [=>] 96.8%	\[ \frac{\color{blue}{\mathsf{fma}\left(1 \cdot x - \left(x + 1\right) \cdot 2, x - 1, \left(\left(x + 1\right) \cdot x\right) \cdot 1\right)}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \]
   /-rgt-identity [<=] 96.8%	\[ \frac{\mathsf{fma}\left(1 \cdot x - \color{blue}{\frac{x + 1}{1}} \cdot 2, x - 1, \left(\left(x + 1\right) \cdot x\right) \cdot 1\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \]
   *-un-lft-identity [<=] 96.8%	\[ \frac{\mathsf{fma}\left(\color{blue}{x} - \frac{x + 1}{1} \cdot 2, x - 1, \left(\left(x + 1\right) \cdot x\right) \cdot 1\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \]
   /-rgt-identity [=>] 96.8%	\[ \frac{\mathsf{fma}\left(x - \color{blue}{\left(x + 1\right)} \cdot 2, x - 1, \left(\left(x + 1\right) \cdot x\right) \cdot 1\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \]
   +-commutative [=>] 96.8%	\[ \frac{\mathsf{fma}\left(x - \color{blue}{\left(1 + x\right)} \cdot 2, x - 1, \left(\left(x + 1\right) \cdot x\right) \cdot 1\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \]
   sub-neg [=>] 96.8%	\[ \frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, \color{blue}{x + \left(-1\right)}, \left(\left(x + 1\right) \cdot x\right) \cdot 1\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \]
   metadata-eval [=>] 96.8%	\[ \frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, x + \color{blue}{-1}, \left(\left(x + 1\right) \cdot x\right) \cdot 1\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \]
   *-commutative [=>] 96.8%	\[ \frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, x + -1, \color{blue}{\left(x \cdot \left(x + 1\right)\right)} \cdot 1\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \]
   +-commutative [=>] 96.8%	\[ \frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, x + -1, \left(x \cdot \color{blue}{\left(1 + x\right)}\right) \cdot 1\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \]
   *-commutative [=>] 96.8%	\[ \frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, x + -1, \left(x \cdot \left(1 + x\right)\right) \cdot 1\right)}{\color{blue}{\left(x \cdot \left(x + 1\right)\right)} \cdot \left(x - 1\right)} \]
   +-commutative [=>] 96.8%	\[ \frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, x + -1, \left(x \cdot \left(1 + x\right)\right) \cdot 1\right)}{\left(x \cdot \color{blue}{\left(1 + x\right)}\right) \cdot \left(x - 1\right)} \]
   sub-neg [=>] 96.8%	\[ \frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, x + -1, \left(x \cdot \left(1 + x\right)\right) \cdot 1\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \color{blue}{\left(x + \left(-1\right)\right)}} \]
   metadata-eval [=>] 96.8%	\[ \frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, x + -1, \left(x \cdot \left(1 + x\right)\right) \cdot 1\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \left(x + \color{blue}{-1}\right)} \]

3. Simplified 96.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x - 2 \cdot \left(x + 1\right), x + -1, x \cdot \left(x + 1\right)\right)}{x \cdot \left(\left(x + 1\right) \cdot \left(x + -1\right)\right)}} \]
   Step-by-step derivation

   [Start] 96.8%	\[ \frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, x + -1, \left(x \cdot \left(1 + x\right)\right) \cdot 1\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \left(x + -1\right)} \]
   associate-*l* [=>] 96.9%	\[ \frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, x + -1, \left(x \cdot \left(1 + x\right)\right) \cdot 1\right)}{\color{blue}{x \cdot \left(\left(1 + x\right) \cdot \left(x + -1\right)\right)}} \]
   *-commutative [=>] 96.9%	\[ \frac{\mathsf{fma}\left(x - \color{blue}{2 \cdot \left(1 + x\right)}, x + -1, \left(x \cdot \left(1 + x\right)\right) \cdot 1\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(x + -1\right)\right)} \]
   +-commutative [=>] 96.9%	\[ \frac{\mathsf{fma}\left(x - 2 \cdot \color{blue}{\left(x + 1\right)}, x + -1, \left(x \cdot \left(1 + x\right)\right) \cdot 1\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(x + -1\right)\right)} \]
   *-rgt-identity [=>] 96.9%	\[ \frac{\mathsf{fma}\left(x - 2 \cdot \left(x + 1\right), x + -1, \color{blue}{x \cdot \left(1 + x\right)}\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(x + -1\right)\right)} \]
   +-commutative [=>] 96.9%	\[ \frac{\mathsf{fma}\left(x - 2 \cdot \left(x + 1\right), x + -1, x \cdot \color{blue}{\left(x + 1\right)}\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(x + -1\right)\right)} \]
   +-commutative [=>] 96.9%	\[ \frac{\mathsf{fma}\left(x - 2 \cdot \left(x + 1\right), x + -1, x \cdot \left(x + 1\right)\right)}{x \cdot \left(\color{blue}{\left(x + 1\right)} \cdot \left(x + -1\right)\right)} \]

4. Taylor expanded in x around 0 99.9%

   \[\leadsto \frac{\color{blue}{2}}{x \cdot \left(\left(x + 1\right) \cdot \left(x + -1\right)\right)} \]

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{2}{x \cdot \color{blue}{\left({x}^{2} - 1\right)}} \]

6. Simplified 100.0%

   \[\leadsto \frac{2}{x \cdot \color{blue}{\left(x \cdot x + -1\right)}} \]
   Step-by-step derivation

   [Start] 100.0%	\[ \frac{2}{x \cdot \left({x}^{2} - 1\right)} \]
   sub-neg [=>] 100.0%	\[ \frac{2}{x \cdot \color{blue}{\left({x}^{2} + \left(-1\right)\right)}} \]
   unpow2 [=>] 100.0%	\[ \frac{2}{x \cdot \left(\color{blue}{x \cdot x} + \left(-1\right)\right)} \]
   metadata-eval [=>] 100.0%	\[ \frac{2}{x \cdot \left(x \cdot x + \color{blue}{-1}\right)} \]

7. Final simplification 100.0%

   \[\leadsto \frac{2}{x \cdot \left(x \cdot x + -1\right)} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	576

\[\frac{2}{x \cdot \left(x \cdot x + -1\right)} \]

Alternative 2
Accuracy	94.6%
Cost	448

\[x \cdot -2 - \frac{2}{x} \]

Alternative 3
Accuracy	94.1%
Cost	192

\[\frac{-2}{x} \]

Alternative 4
Accuracy	3.3%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023277 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

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

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))