3frac (problem 3.3.3)

Average Error: 9.7 → 0.3

Time: 7.5s

Precision: binary64

Cost: 704

Math

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

↓

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

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 -1.0) (- -1.0 x)))))

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 + -1.0) * (-1.0 - x)));
}

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 + (-1.0d0)) * ((-1.0d0) - x)))
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 + -1.0) * (-1.0 - x)));
}

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 + -1.0) * (-1.0 - x)))

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 + -1.0) * Float64(-1.0 - x))))
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 + -1.0) * (-1.0 - x)));
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 + -1.0), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	9.7
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 9.7

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

2. Simplified 9.7

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

   [Start] 9.7	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   associate-+l- [=>] 9.7	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 9.7	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
   neg-mul-1 [=>] 9.7	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   metadata-eval [<=] 9.7	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   cancel-sign-sub-inv [<=] 9.7	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   +-commutative [=>] 9.7	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   *-lft-identity [=>] 9.7	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 9.7	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
   metadata-eval [=>] 9.7	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

3. Applied egg-rr 25.5

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

4. Simplified 25.5

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

   [Start] 25.5	\[ \frac{-1 \cdot \mathsf{fma}\left(x, x, -x\right) - \left(-1 - x\right) \cdot \left(-2 + \left(2 \cdot x - x\right)\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   sub-neg [=>] 25.5	\[ \frac{\color{blue}{-1 \cdot \mathsf{fma}\left(x, x, -x\right) + \left(-\left(-1 - x\right) \cdot \left(-2 + \left(2 \cdot x - x\right)\right)\right)}}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   +-commutative [=>] 25.5	\[ \frac{\color{blue}{\left(-\left(-1 - x\right) \cdot \left(-2 + \left(2 \cdot x - x\right)\right)\right) + -1 \cdot \mathsf{fma}\left(x, x, -x\right)}}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   distribute-rgt-neg-in [=>] 25.5	\[ \frac{\color{blue}{\left(-1 - x\right) \cdot \left(-\left(-2 + \left(2 \cdot x - x\right)\right)\right)} + -1 \cdot \mathsf{fma}\left(x, x, -x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   neg-sub0 [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \color{blue}{\left(0 - \left(-2 + \left(2 \cdot x - x\right)\right)\right)} + -1 \cdot \mathsf{fma}\left(x, x, -x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   associate--r+ [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \color{blue}{\left(\left(0 - -2\right) - \left(2 \cdot x - x\right)\right)} + -1 \cdot \mathsf{fma}\left(x, x, -x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   metadata-eval [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(\color{blue}{2} - \left(2 \cdot x - x\right)\right) + -1 \cdot \mathsf{fma}\left(x, x, -x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   sub-neg [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - \color{blue}{\left(2 \cdot x + \left(-x\right)\right)}\right) + -1 \cdot \mathsf{fma}\left(x, x, -x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   neg-mul-1 [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - \left(2 \cdot x + \color{blue}{-1 \cdot x}\right)\right) + -1 \cdot \mathsf{fma}\left(x, x, -x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   distribute-rgt-out [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - \color{blue}{x \cdot \left(2 + -1\right)}\right) + -1 \cdot \mathsf{fma}\left(x, x, -x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   metadata-eval [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x \cdot \color{blue}{1}\right) + -1 \cdot \mathsf{fma}\left(x, x, -x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   *-rgt-identity [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - \color{blue}{x}\right) + -1 \cdot \mathsf{fma}\left(x, x, -x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   fma-udef [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + -1 \cdot \color{blue}{\left(x \cdot x + \left(-x\right)\right)}}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   distribute-lft-in [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + \color{blue}{\left(-1 \cdot \left(x \cdot x\right) + -1 \cdot \left(-x\right)\right)}}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   associate-*l* [<=] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + \left(\color{blue}{\left(-1 \cdot x\right) \cdot x} + -1 \cdot \left(-x\right)\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   neg-mul-1 [<=] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + \left(\color{blue}{\left(-x\right)} \cdot x + -1 \cdot \left(-x\right)\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   *-commutative [<=] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + \left(\left(-x\right) \cdot x + \color{blue}{\left(-x\right) \cdot -1}\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   +-commutative [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + \color{blue}{\left(\left(-x\right) \cdot -1 + \left(-x\right) \cdot x\right)}}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   *-commutative [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + \left(\color{blue}{-1 \cdot \left(-x\right)} + \left(-x\right) \cdot x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   neg-mul-1 [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + \left(-1 \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(-x\right) \cdot x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   associate-*r* [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + \left(\color{blue}{\left(-1 \cdot -1\right) \cdot x} + \left(-x\right) \cdot x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   metadata-eval [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + \left(\color{blue}{1} \cdot x + \left(-x\right) \cdot x\right)}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   distribute-rgt-in [<=] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + \color{blue}{x \cdot \left(1 + \left(-x\right)\right)}}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   sub-neg [<=] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + x \cdot \color{blue}{\left(1 - x\right)}}{\left(-1 - x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   *-commutative [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + x \cdot \left(1 - x\right)}{\color{blue}{\mathsf{fma}\left(x, x, -x\right) \cdot \left(-1 - x\right)}} \]
   fma-udef [=>] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + x \cdot \left(1 - x\right)}{\color{blue}{\left(x \cdot x + \left(-x\right)\right)} \cdot \left(-1 - x\right)} \]
   sub-neg [<=] 25.5	\[ \frac{\left(-1 - x\right) \cdot \left(2 - x\right) + x \cdot \left(1 - x\right)}{\color{blue}{\left(x \cdot x - x\right)} \cdot \left(-1 - x\right)} \]

5. Taylor expanded in x around 0 0.3

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

6. Applied egg-rr 27.1

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2}{\mathsf{fma}\left(x, x, -x\right) \cdot \left(-1 - x\right)}\right)} - 1} \]

7. Simplified 0.1

   \[\leadsto \color{blue}{\frac{\frac{-2}{x \cdot x - x}}{-1 - x}} \]
   Proof

   [Start] 27.1	\[ e^{\mathsf{log1p}\left(\frac{-2}{\mathsf{fma}\left(x, x, -x\right) \cdot \left(-1 - x\right)}\right)} - 1 \]
   expm1-def [=>] 17.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-2}{\mathsf{fma}\left(x, x, -x\right) \cdot \left(-1 - x\right)}\right)\right)} \]
   expm1-log1p [=>] 0.3	\[ \color{blue}{\frac{-2}{\mathsf{fma}\left(x, x, -x\right) \cdot \left(-1 - x\right)}} \]
   associate-/r* [=>] 0.1	\[ \color{blue}{\frac{\frac{-2}{\mathsf{fma}\left(x, x, -x\right)}}{-1 - x}} \]
   fma-neg [<=] 0.1	\[ \frac{\frac{-2}{\color{blue}{x \cdot x - x}}}{-1 - x} \]

8. Applied egg-rr 27.1

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2}{\left(x \cdot \left(x + -1\right)\right) \cdot \left(-1 - x\right)}\right)} - 1} \]

9. Simplified 0.3

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

   [Start] 27.1	\[ e^{\mathsf{log1p}\left(\frac{-2}{\left(x \cdot \left(x + -1\right)\right) \cdot \left(-1 - x\right)}\right)} - 1 \]
   expm1-def [=>] 17.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-2}{\left(x \cdot \left(x + -1\right)\right) \cdot \left(-1 - x\right)}\right)\right)} \]
   expm1-log1p [=>] 0.3	\[ \color{blue}{\frac{-2}{\left(x \cdot \left(x + -1\right)\right) \cdot \left(-1 - x\right)}} \]
   associate-*l* [=>] 0.3	\[ \frac{-2}{\color{blue}{x \cdot \left(\left(x + -1\right) \cdot \left(-1 - x\right)\right)}} \]

10. Final simplification 0.3

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

Alternatives

Alternative 1
Error	1.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -0.76 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{-2}{x \cdot x}}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 2
Error	1.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -0.76:\\ \;\;\;\;\frac{\frac{\frac{-2}{x}}{x}}{-1 - x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x \cdot x}}{-1 - x}\\ \end{array} \]

Alternative 3
Error	10.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\frac{1}{x + 1} + \frac{-1}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \]

Alternative 4
Error	15.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 5
Error	10.4
Cost	457

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{0}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 6
Error	30.7
Cost	192

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

Alternative 7
Error	61.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023011 
(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))))