2frac (problem 3.3.1)

Average Accuracy: 77.6% → 99.9%

Time: 4.6s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 77.6%

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

2. Applied egg-rr 52.4%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{1 + x} + \frac{-1}{x}} \cdot \sqrt[3]{{\left(\frac{1}{1 + x} + \frac{-1}{x}\right)}^{2}}} \]
   Proof

   [Start] 77.6	\[ \frac{1}{x + 1} - \frac{1}{x} \]
   add-cube-cbrt [=>] 76.6	\[ \color{blue}{\left(\sqrt[3]{\frac{1}{x + 1} - \frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x + 1} - \frac{1}{x}}\right) \cdot \sqrt[3]{\frac{1}{x + 1} - \frac{1}{x}}} \]
   associate-*l* [=>] 76.6	\[ \color{blue}{\sqrt[3]{\frac{1}{x + 1} - \frac{1}{x}} \cdot \left(\sqrt[3]{\frac{1}{x + 1} - \frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x + 1} - \frac{1}{x}}\right)} \]
   sub-neg [=>] 76.6	\[ \sqrt[3]{\color{blue}{\frac{1}{x + 1} + \left(-\frac{1}{x}\right)}} \cdot \left(\sqrt[3]{\frac{1}{x + 1} - \frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x + 1} - \frac{1}{x}}\right) \]
   +-commutative [=>] 76.6	\[ \sqrt[3]{\frac{1}{\color{blue}{1 + x}} + \left(-\frac{1}{x}\right)} \cdot \left(\sqrt[3]{\frac{1}{x + 1} - \frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x + 1} - \frac{1}{x}}\right) \]
   distribute-neg-frac [=>] 76.6	\[ \sqrt[3]{\frac{1}{1 + x} + \color{blue}{\frac{-1}{x}}} \cdot \left(\sqrt[3]{\frac{1}{x + 1} - \frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x + 1} - \frac{1}{x}}\right) \]
   metadata-eval [=>] 76.6	\[ \sqrt[3]{\frac{1}{1 + x} + \frac{\color{blue}{-1}}{x}} \cdot \left(\sqrt[3]{\frac{1}{x + 1} - \frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x + 1} - \frac{1}{x}}\right) \]
   cbrt-unprod [=>] 52.4	\[ \sqrt[3]{\frac{1}{1 + x} + \frac{-1}{x}} \cdot \color{blue}{\sqrt[3]{\left(\frac{1}{x + 1} - \frac{1}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{1}{x}\right)}} \]
   pow2 [=>] 52.4	\[ \sqrt[3]{\frac{1}{1 + x} + \frac{-1}{x}} \cdot \sqrt[3]{\color{blue}{{\left(\frac{1}{x + 1} - \frac{1}{x}\right)}^{2}}} \]
   sub-neg [=>] 52.4	\[ \sqrt[3]{\frac{1}{1 + x} + \frac{-1}{x}} \cdot \sqrt[3]{{\color{blue}{\left(\frac{1}{x + 1} + \left(-\frac{1}{x}\right)\right)}}^{2}} \]
   +-commutative [=>] 52.4	\[ \sqrt[3]{\frac{1}{1 + x} + \frac{-1}{x}} \cdot \sqrt[3]{{\left(\frac{1}{\color{blue}{1 + x}} + \left(-\frac{1}{x}\right)\right)}^{2}} \]
   distribute-neg-frac [=>] 52.4	\[ \sqrt[3]{\frac{1}{1 + x} + \frac{-1}{x}} \cdot \sqrt[3]{{\left(\frac{1}{1 + x} + \color{blue}{\frac{-1}{x}}\right)}^{2}} \]
   metadata-eval [=>] 52.4	\[ \sqrt[3]{\frac{1}{1 + x} + \frac{-1}{x}} \cdot \sqrt[3]{{\left(\frac{1}{1 + x} + \frac{\color{blue}{-1}}{x}\right)}^{2}} \]

3. Simplified 63.3%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{x \cdot \left(x + 1\right)}} \cdot \sqrt[3]{{\left(\frac{-1}{x \cdot \left(x + 1\right)}\right)}^{2}}} \]
   Proof

   [Start] 52.4	\[ \sqrt[3]{\frac{1}{1 + x} + \frac{-1}{x}} \cdot \sqrt[3]{{\left(\frac{1}{1 + x} + \frac{-1}{x}\right)}^{2}} \]

4. Applied egg-rr 99.9%

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

   [Start] 63.3	\[ \sqrt[3]{\frac{-1}{x \cdot \left(x + 1\right)}} \cdot \sqrt[3]{{\left(\frac{-1}{x \cdot \left(x + 1\right)}\right)}^{2}} \]
   *-commutative [=>] 63.3	\[ \color{blue}{\sqrt[3]{{\left(\frac{-1}{x \cdot \left(x + 1\right)}\right)}^{2}} \cdot \sqrt[3]{\frac{-1}{x \cdot \left(x + 1\right)}}} \]
   cbrt-prod [<=] 50.9	\[ \color{blue}{\sqrt[3]{{\left(\frac{-1}{x \cdot \left(x + 1\right)}\right)}^{2} \cdot \frac{-1}{x \cdot \left(x + 1\right)}}} \]
   unpow2 [=>] 50.9	\[ \sqrt[3]{\color{blue}{\left(\frac{-1}{x \cdot \left(x + 1\right)} \cdot \frac{-1}{x \cdot \left(x + 1\right)}\right)} \cdot \frac{-1}{x \cdot \left(x + 1\right)}} \]
   add-cbrt-cube [<=] 99.3	\[ \color{blue}{\frac{-1}{x \cdot \left(x + 1\right)}} \]
   associate-/r* [=>] 99.9	\[ \color{blue}{\frac{\frac{-1}{x}}{x + 1}} \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	97.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 2
Accuracy	98.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	448

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

Alternative 4
Accuracy	99.3%
Cost	448

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

Alternative 5
Accuracy	51.6%
Cost	192

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

Reproduce

herbie shell --seed 2023153 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))