2frac (problem 3.3.1)

Percentage Accurate: 77.7% → 99.6%

Time: 4.1s

Alternatives: 6

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} - \frac{1}{x} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} - \frac{1}{x} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left({x}^{-1}, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{x}\\ \mathbf{if}\;x \leq -13500:\\ \;\;\;\;\frac{t\_0}{x}\\ \mathbf{elif}\;x \leq 1700:\\ \;\;\;\;\frac{1}{x - -1} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot -1, 1, -1\right)}{\left(\left(-1 \cdot x\right) \cdot x\right) \cdot -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (fma (* (/ (fma (pow x -1.0) 1.0 -1.0) x) -1.0) 1.0 -1.0) x)))
   (if (<= x -13500.0)
     (/ t_0 x)
     (if (<= x 1700.0)
       (- (/ 1.0 (- x -1.0)) (/ 1.0 x))
       (/ (fma (* t_0 -1.0) 1.0 -1.0) (* (* (* -1.0 x) x) -1.0))))))

C

double code(double x) {
	double t_0 = fma(((fma(pow(x, -1.0), 1.0, -1.0) / x) * -1.0), 1.0, -1.0) / x;
	double tmp;
	if (x <= -13500.0) {
		tmp = t_0 / x;
	} else if (x <= 1700.0) {
		tmp = (1.0 / (x - -1.0)) - (1.0 / x);
	} else {
		tmp = fma((t_0 * -1.0), 1.0, -1.0) / (((-1.0 * x) * x) * -1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(fma(Float64(Float64(fma((x ^ -1.0), 1.0, -1.0) / x) * -1.0), 1.0, -1.0) / x)
	tmp = 0.0
	if (x <= -13500.0)
		tmp = Float64(t_0 / x);
	elseif (x <= 1700.0)
		tmp = Float64(Float64(1.0 / Float64(x - -1.0)) - Float64(1.0 / x));
	else
		tmp = Float64(fma(Float64(t_0 * -1.0), 1.0, -1.0) / Float64(Float64(Float64(-1.0 * x) * x) * -1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] * 1.0 + -1.0), $MachinePrecision] / x), $MachinePrecision] * -1.0), $MachinePrecision] * 1.0 + -1.0), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -13500.0], N[(t$95$0 / x), $MachinePrecision], If[LessEqual[x, 1700.0], N[(N[(1.0 / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * -1.0), $MachinePrecision] * 1.0 + -1.0), $MachinePrecision] / N[(N[(N[(-1.0 * x), $MachinePrecision] * x), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -13500

   1. Initial program 52.2%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{\frac{1}{x} - 1}{x} - 1}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 99.7%

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

   if -13500 < x < 1700

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   if 1700 < x

   1. Initial program 62.1%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{x} - 1}{x} - 1}{x} - 1}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. frac-2neg N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left({x}^{-1}, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{x} \cdot -1, 1, -1\right) \cdot -1}{\left(-1 \cdot x\right) \cdot x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -13500:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left({x}^{-1}, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{x}}{x}\\ \mathbf{elif}\;x \leq 1700:\\ \;\;\;\;\frac{1}{x - -1} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left({x}^{-1}, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{\left(\left(-1 \cdot x\right) \cdot x\right) \cdot -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{-1 \cdot -1}{\mathsf{fma}\left(-1 \cdot x, 1, -1\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* -1.0 -1.0) (* (fma (* -1.0 x) 1.0 -1.0) x)))

C

double code(double x) {
	return (-1.0 * -1.0) / (fma((-1.0 * x), 1.0, -1.0) * x);
}

Julia

function code(x)
	return Float64(Float64(-1.0 * -1.0) / Float64(fma(Float64(-1.0 * x), 1.0, -1.0) * x))
end

Wolfram

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

Derivation

1. Initial program 79.4%

   \[\frac{1}{x + 1} - \frac{1}{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. lift-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. frac-sub N/A

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

   10. lower-/.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. distribute-neg-in N/A

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

   17. *-lft-identity N/A

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

   18. *-commutative N/A

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

   19. metadata-eval N/A

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

   20. lower-fma.f64 N/A

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

   21. mul-1-neg N/A

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

   22. lower-*.f64 N/A

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

4. Applied rewrites 79.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.9%

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

8. Final simplification 99.9%

   \[\leadsto \frac{-1 \cdot -1}{\mathsf{fma}\left(-1 \cdot x, 1, -1\right) \cdot x} \]
9. Add Preprocessing

Alternative 3: 51.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-1 \cdot x\right) \cdot x\\ \mathbf{if}\;\frac{1}{x - -1} - \frac{1}{x} \leq -1:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-1 \cdot \left(x - 1\right)\right) \cdot x, 1, -1\right)}{t\_0 \cdot x}}{-1 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left({x}^{-1}, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{t\_0 \cdot -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* -1.0 x) x)))
   (if (<= (- (/ 1.0 (- x -1.0)) (/ 1.0 x)) -1.0)
     (/ (/ (fma (* (* -1.0 (- x 1.0)) x) 1.0 -1.0) (* t_0 x)) (* -1.0 x))
     (/
      (fma
       (* (/ (fma (* (/ (fma (pow x -1.0) 1.0 -1.0) x) -1.0) 1.0 -1.0) x) -1.0)
       1.0
       -1.0)
      (* t_0 -1.0)))))

C

double code(double x) {
	double t_0 = (-1.0 * x) * x;
	double tmp;
	if (((1.0 / (x - -1.0)) - (1.0 / x)) <= -1.0) {
		tmp = (fma(((-1.0 * (x - 1.0)) * x), 1.0, -1.0) / (t_0 * x)) / (-1.0 * x);
	} else {
		tmp = fma(((fma(((fma(pow(x, -1.0), 1.0, -1.0) / x) * -1.0), 1.0, -1.0) / x) * -1.0), 1.0, -1.0) / (t_0 * -1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(-1.0 * x) * x)
	tmp = 0.0
	if (Float64(Float64(1.0 / Float64(x - -1.0)) - Float64(1.0 / x)) <= -1.0)
		tmp = Float64(Float64(fma(Float64(Float64(-1.0 * Float64(x - 1.0)) * x), 1.0, -1.0) / Float64(t_0 * x)) / Float64(-1.0 * x));
	else
		tmp = Float64(fma(Float64(Float64(fma(Float64(Float64(fma((x ^ -1.0), 1.0, -1.0) / x) * -1.0), 1.0, -1.0) / x) * -1.0), 1.0, -1.0) / Float64(t_0 * -1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(-1.0 * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(N[(N[(-1.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * 1.0 + -1.0), $MachinePrecision] / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] * 1.0 + -1.0), $MachinePrecision] / x), $MachinePrecision] * -1.0), $MachinePrecision] * 1.0 + -1.0), $MachinePrecision] / x), $MachinePrecision] * -1.0), $MachinePrecision] * 1.0 + -1.0), $MachinePrecision] / N[(t$95$0 * -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -1

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{\frac{1}{x} - 1}{x} - 1}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 6.5%

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

   7. Applied rewrites 6.5%

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

   if -1 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 70.6%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{x} - 1}{x} - 1}{x} - 1}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. frac-2neg N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 68.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left({x}^{-1}, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{x} \cdot -1, 1, -1\right) \cdot -1}{\left(-1 \cdot x\right) \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x - -1} - \frac{1}{x} \leq -1:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-1 \cdot \left(x - 1\right)\right) \cdot x, 1, -1\right)}{\left(\left(-1 \cdot x\right) \cdot x\right) \cdot x}}{-1 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left({x}^{-1}, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{\left(\left(-1 \cdot x\right) \cdot x\right) \cdot -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left({x}^{-1}, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (fma (* (/ (fma (pow x -1.0) 1.0 -1.0) x) -1.0) 1.0 -1.0) x) x))

C

double code(double x) {
	return (fma(((fma(pow(x, -1.0), 1.0, -1.0) / x) * -1.0), 1.0, -1.0) / x) / x;
}

Julia

function code(x)
	return Float64(Float64(fma(Float64(Float64(fma((x ^ -1.0), 1.0, -1.0) / x) * -1.0), 1.0, -1.0) / x) / x)
end

Wolfram

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

Derivation

1. Initial program 79.4%

   \[\frac{1}{x + 1} - \frac{1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{\frac{1}{x} - 1}{x} - 1}{{x}^{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

5. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left({x}^{-1}, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{x}}{x}} \]
6. Add Preprocessing

Alternative 5: 49.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left({x}^{-1}, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{\left(\left(-1 \cdot x\right) \cdot x\right) \cdot -1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma
   (* (/ (fma (* (/ (fma (pow x -1.0) 1.0 -1.0) x) -1.0) 1.0 -1.0) x) -1.0)
   1.0
   -1.0)
  (* (* (* -1.0 x) x) -1.0)))

C

double code(double x) {
	return fma(((fma(((fma(pow(x, -1.0), 1.0, -1.0) / x) * -1.0), 1.0, -1.0) / x) * -1.0), 1.0, -1.0) / (((-1.0 * x) * x) * -1.0);
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(fma(Float64(Float64(fma((x ^ -1.0), 1.0, -1.0) / x) * -1.0), 1.0, -1.0) / x) * -1.0), 1.0, -1.0) / Float64(Float64(Float64(-1.0 * x) * x) * -1.0))
end

Wolfram

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

Derivation

1. Initial program 79.4%

   \[\frac{1}{x + 1} - \frac{1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{x} - 1}{x} - 1}{x} - 1}{{x}^{2}}} \]
4. Step-by-step derivation

   1. frac-2neg N/A

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

   2. lower-/.f64 N/A

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

5. Applied rewrites 47.8%

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

6. Final simplification 47.8%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left({x}^{-1}, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{x} \cdot -1, 1, -1\right)}{\left(\left(-1 \cdot x\right) \cdot x\right) \cdot -1} \]
7. Add Preprocessing

Alternative 6: 15.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(-1 \cdot \left(x - 1\right)\right) \cdot x, 1, -1\right) \cdot x, 1, 1\right)}{{x}^{3}}}{\left(\left(-1 \cdot x\right) \cdot x\right) \cdot -1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/ (fma (* (fma (* (* -1.0 (- x 1.0)) x) 1.0 -1.0) x) 1.0 1.0) (pow x 3.0))
  (* (* (* -1.0 x) x) -1.0)))

C

double code(double x) {
	return (fma((fma(((-1.0 * (x - 1.0)) * x), 1.0, -1.0) * x), 1.0, 1.0) / pow(x, 3.0)) / (((-1.0 * x) * x) * -1.0);
}

Julia

function code(x)
	return Float64(Float64(fma(Float64(fma(Float64(Float64(-1.0 * Float64(x - 1.0)) * x), 1.0, -1.0) * x), 1.0, 1.0) / (x ^ 3.0)) / Float64(Float64(Float64(-1.0 * x) * x) * -1.0))
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(N[(-1.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * 1.0 + -1.0), $MachinePrecision] * x), $MachinePrecision] * 1.0 + 1.0), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(-1.0 * x), $MachinePrecision] * x), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.4%

   \[\frac{1}{x + 1} - \frac{1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{x} - 1}{x} - 1}{x} - 1}{{x}^{2}}} \]
4. Step-by-step derivation

   1. frac-2neg N/A

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

   2. lower-/.f64 N/A

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

5. Applied rewrites 47.8%

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

7. Applied rewrites 13.0%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(-1 \cdot \left(x - 1\right)\right) \cdot x, 1, -1\right) \cdot x, 1, 1\right)}{{x}^{3}} \cdot -1}{\left(\color{blue}{-1} \cdot x\right) \cdot x} \]

8. Final simplification 13.0%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(-1 \cdot \left(x - 1\right)\right) \cdot x, 1, -1\right) \cdot x, 1, 1\right)}{{x}^{3}}}{\left(\left(-1 \cdot x\right) \cdot x\right) \cdot -1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64

  :alt
  (! :herbie-platform c (/ 1 (* x (- -1 x))))

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