Asymptote C

Percentage Accurate: 53.6% → 99.8%

Time: 11.7s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 10^{-5}:\\ \;\;\;\;\frac{-3 + \frac{-1 - \frac{3}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1 + x \cdot \left(x \cdot x\right)}, \left(1 + x \cdot x\right) - x, \frac{x + 1}{1 - x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 1e-5)
   (/ (+ -3.0 (/ (- -1.0 (/ 3.0 x)) x)) x)
   (fma
    (/ x (+ 1.0 (* x (* x x))))
    (- (+ 1.0 (* x x)) x)
    (/ (+ x 1.0) (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 1e-5) {
		tmp = (-3.0 + ((-1.0 - (3.0 / x)) / x)) / x;
	} else {
		tmp = fma((x / (1.0 + (x * (x * x)))), ((1.0 + (x * x)) - x), ((x + 1.0) / (1.0 - x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 1e-5)
		tmp = Float64(Float64(-3.0 + Float64(Float64(-1.0 - Float64(3.0 / x)) / x)) / x);
	else
		tmp = fma(Float64(x / Float64(1.0 + Float64(x * Float64(x * x)))), Float64(Float64(1.0 + Float64(x * x)) - x), Float64(Float64(x + 1.0) / Float64(1.0 - x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-5], N[(N[(-3.0 + N[(N[(-1.0 - N[(3.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / N[(1.0 + N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 1.00000000000000008e-5

   1. Initial program 7.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{1 + 3 \cdot \frac{1}{x}}{x} - 3}{x}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-3 - \frac{1 + \frac{3}{x}}{x}}{x}} \]

   if 1.00000000000000008e-5 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\frac{x \cdot x - 1 \cdot 1}{x - 1}}{\color{blue}{x} - 1}\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{x \cdot x - 1 \cdot 1}{\color{blue}{\left(x - 1\right) \cdot \left(x - 1\right)}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\left(x \cdot x - 1 \cdot 1\right), \color{blue}{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\left(x \cdot x - 1\right), \left(\left(x - \color{blue}{1}\right) \cdot \left(x - 1\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\left(x \cdot x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x - 1\right)} \cdot \left(x - 1\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x - 1\right)} \cdot \left(x - 1\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(\color{blue}{x} - 1\right) \cdot \left(x - 1\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \left(\left(x - \color{blue}{1}\right) \cdot \left(x - 1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \mathsf{*.f64}\left(\left(x - 1\right), \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{x} - 1\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{x} - 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x - 1\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      15. metadata-eval 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

      1. flip3-+ N/A

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

      2. associate-/r/ N/A

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

      3. fmm-def N/A

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

      4. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{x}{{x}^{3} + {1}^{3}}\right), \color{blue}{\left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)}, \left(\mathsf{neg}\left(\frac{x \cdot x + -1}{\left(x + -1\right) \cdot \left(x + -1\right)}\right)\right)\right) \]

   6. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 10^{-5}:\\ \;\;\;\;\frac{-3 + \frac{-1 - \frac{3}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1 + x \cdot \left(x \cdot x\right)}, \left(1 + x \cdot x\right) - x, \frac{x + 1}{1 - x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 10^{-5}:\\ \;\;\;\;\frac{-3 + \frac{-1 - \frac{3}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(x + 1\right) \cdot \frac{x + 1}{1 - x}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 1e-5)
   (/ (+ -3.0 (/ (- -1.0 (/ 3.0 x)) x)) x)
   (/ (+ x (* (+ x 1.0) (/ (+ x 1.0) (- 1.0 x)))) (+ x 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 1e-5) {
		tmp = (-3.0 + ((-1.0 - (3.0 / x)) / x)) / x;
	} else {
		tmp = (x + ((x + 1.0) * ((x + 1.0) / (1.0 - x)))) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x / (x + 1.0d0)) + (((-1.0d0) - x) / (x + (-1.0d0)))) <= 1d-5) then
        tmp = ((-3.0d0) + (((-1.0d0) - (3.0d0 / x)) / x)) / x
    else
        tmp = (x + ((x + 1.0d0) * ((x + 1.0d0) / (1.0d0 - x)))) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 1e-5) {
		tmp = (-3.0 + ((-1.0 - (3.0 / x)) / x)) / x;
	} else {
		tmp = (x + ((x + 1.0) * ((x + 1.0) / (1.0 - x)))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 1e-5:
		tmp = (-3.0 + ((-1.0 - (3.0 / x)) / x)) / x
	else:
		tmp = (x + ((x + 1.0) * ((x + 1.0) / (1.0 - x)))) / (x + 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 1e-5)
		tmp = Float64(Float64(-3.0 + Float64(Float64(-1.0 - Float64(3.0 / x)) / x)) / x);
	else
		tmp = Float64(Float64(x + Float64(Float64(x + 1.0) * Float64(Float64(x + 1.0) / Float64(1.0 - x)))) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 1e-5)
		tmp = (-3.0 + ((-1.0 - (3.0 / x)) / x)) / x;
	else
		tmp = (x + ((x + 1.0) * ((x + 1.0) / (1.0 - x)))) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-5], N[(N[(-3.0 + N[(N[(-1.0 - N[(3.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x + N[(N[(x + 1.0), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 1.00000000000000008e-5

   1. Initial program 7.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{1 + 3 \cdot \frac{1}{x}}{x} - 3}{x}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-3 - \frac{1 + \frac{3}{x}}{x}}{x}} \]

   if 1.00000000000000008e-5 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\frac{x \cdot x - 1 \cdot 1}{x - 1}}{\color{blue}{x} - 1}\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{x \cdot x - 1 \cdot 1}{\color{blue}{\left(x - 1\right) \cdot \left(x - 1\right)}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\left(x \cdot x - 1 \cdot 1\right), \color{blue}{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\left(x \cdot x - 1\right), \left(\left(x - \color{blue}{1}\right) \cdot \left(x - 1\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\left(x \cdot x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x - 1\right)} \cdot \left(x - 1\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x - 1\right)} \cdot \left(x - 1\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(\color{blue}{x} - 1\right) \cdot \left(x - 1\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \left(\left(x - \color{blue}{1}\right) \cdot \left(x - 1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \mathsf{*.f64}\left(\left(x - 1\right), \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{x} - 1\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{x} - 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x - 1\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      15. metadata-eval 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + x}{x + -1}} \cdot \frac{x}{x + -1} - \frac{1 + x}{x + -1}} \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1 \cdot \frac{x}{x + -1}}{\frac{1 + x}{x + -1}}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, x\right)}, \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(1 \cdot \frac{x}{x + -1}\right), \left(\frac{1 + x}{x + -1}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, x\right)}, \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot x}{x + -1}\right), \left(\frac{1 + x}{x + -1}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{x + -1}\right), \left(\frac{1 + x}{x + -1}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(x + -1\right)\right), \left(\frac{1 + x}{x + -1}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{1 + x}{x + -1}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\left(1 + x\right), \left(x + -1\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{x}\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\left(x + 1\right), \left(x + -1\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(x + -1\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      10. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{+.f64}\left(x, -1\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + -1}}{\frac{x + 1}{x + -1}}} - \frac{1 + x}{x + -1} \]
   9. Step-by-step derivation

      1. frac-sub N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. difference-of-sqr-1 N/A

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

      6. metadata-eval N/A

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

      7. flip-- N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{x + -1} \cdot \left(x + -1\right) - \frac{x + 1}{x + -1} \cdot \left(1 + x\right)\right), \color{blue}{\left(x + 1\right)}\right) \]

   10. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 10^{-5}:\\ \;\;\;\;\frac{-3 + \frac{-1 - \frac{3}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(x + 1\right) \cdot \frac{x + 1}{1 - x}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\frac{-3 + \frac{-1 - \frac{3}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 1e-5) (/ (+ -3.0 (/ (- -1.0 (/ 3.0 x)) x)) x) t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = (-3.0 + ((-1.0 - (3.0 / x)) / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (x + 1.0d0)) + (((-1.0d0) - x) / (x + (-1.0d0)))
    if (t_0 <= 1d-5) then
        tmp = ((-3.0d0) + (((-1.0d0) - (3.0d0 / x)) / x)) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = (-3.0 + ((-1.0 - (3.0 / x)) / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))
	tmp = 0
	if t_0 <= 1e-5:
		tmp = (-3.0 + ((-1.0 - (3.0 / x)) / x)) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 1e-5)
		tmp = Float64(Float64(-3.0 + Float64(Float64(-1.0 - Float64(3.0 / x)) / x)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 1e-5)
		tmp = (-3.0 + ((-1.0 - (3.0 / x)) / x)) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-5], N[(N[(-3.0 + N[(N[(-1.0 - N[(3.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 1.00000000000000008e-5

   1. Initial program 7.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{1 + 3 \cdot \frac{1}{x}}{x} - 3}{x}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-3 - \frac{1 + \frac{3}{x}}{x}}{x}} \]

   if 1.00000000000000008e-5 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 10^{-5}:\\ \;\;\;\;\frac{-3 + \frac{-1 - \frac{3}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3 - \frac{x + 3}{x \cdot x}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(1 + x \cdot x\right) \cdot \left(1 + x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ (- -3.0 (/ (+ x 3.0) (* x x))) x)
   (if (<= x 1.0)
     (* (+ 1.0 (* x x)) (+ 1.0 (* x 3.0)))
     (/ (+ -3.0 (/ -1.0 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-3.0 - ((x + 3.0) / (x * x))) / x;
	} else if (x <= 1.0) {
		tmp = (1.0 + (x * x)) * (1.0 + (x * 3.0));
	} else {
		tmp = (-3.0 + (-1.0 / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = ((-3.0d0) - ((x + 3.0d0) / (x * x))) / x
    else if (x <= 1.0d0) then
        tmp = (1.0d0 + (x * x)) * (1.0d0 + (x * 3.0d0))
    else
        tmp = ((-3.0d0) + ((-1.0d0) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-3.0 - ((x + 3.0) / (x * x))) / x;
	} else if (x <= 1.0) {
		tmp = (1.0 + (x * x)) * (1.0 + (x * 3.0));
	} else {
		tmp = (-3.0 + (-1.0 / x)) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = (-3.0 - ((x + 3.0) / (x * x))) / x
	elif x <= 1.0:
		tmp = (1.0 + (x * x)) * (1.0 + (x * 3.0))
	else:
		tmp = (-3.0 + (-1.0 / x)) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(-3.0 - Float64(Float64(x + 3.0) / Float64(x * x))) / x);
	elseif (x <= 1.0)
		tmp = Float64(Float64(1.0 + Float64(x * x)) * Float64(1.0 + Float64(x * 3.0)));
	else
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (-3.0 - ((x + 3.0) / (x * x))) / x;
	elseif (x <= 1.0)
		tmp = (1.0 + (x * x)) * (1.0 + (x * 3.0));
	else
		tmp = (-3.0 + (-1.0 / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 10.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{1 + 3 \cdot \frac{1}{x}}{x} - 3}{x}} \]

   5. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{-3 - \frac{1 + \frac{3}{x}}{x}}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-3, \color{blue}{\left(\frac{3 + x}{{x}^{2}}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-3, \mathsf{/.f64}\left(\left(3 + x\right), \left({x}^{2}\right)\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-3, \mathsf{/.f64}\left(\left(x + 3\right), \left({x}^{2}\right)\right)\right), x\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-3, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 3\right), \left({x}^{2}\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-3, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 3\right), \left(x \cdot x\right)\right)\right), x\right) \]

      5. *-lowering-*.f64 98.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-3, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 3\right), \mathsf{*.f64}\left(x, x\right)\right)\right), x\right) \]

   8. Simplified 98.2%

      \[\leadsto \frac{-3 - \color{blue}{\frac{x + 3}{x \cdot x}}}{x} \]

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(3 + x \cdot \left(1 + 3 \cdot x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto 1 + \left(x \cdot 3 + \color{blue}{x \cdot \left(x \cdot \left(1 + 3 \cdot x\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto 1 + \left(3 \cdot x + \color{blue}{x} \cdot \left(x \cdot \left(1 + 3 \cdot x\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \left(1 + 3 \cdot x\right) + \color{blue}{x \cdot \left(x \cdot \left(1 + 3 \cdot x\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \left(1 + 3 \cdot x\right) + \left(x \cdot x\right) \cdot \color{blue}{\left(1 + 3 \cdot x\right)} \]

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + 3 \cdot x\right), \color{blue}{\left({x}^{2} + 1\right)}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(3 \cdot x\right)\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \left(1 + \color{blue}{{x}^{2}}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      14. *-lowering-*.f64 99.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\left(1 + 3 \cdot x\right) \cdot \left(1 + x \cdot x\right)} \]

   if 1 < x

   1. Initial program 6.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{3 + \frac{1}{x}}{x}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{3 + \frac{1}{x}}{x}\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)}{\color{blue}{x}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)\right), \color{blue}{x}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-3 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{-1}{x}\right)\right), x\right) \]

      9. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(-1, x\right)\right), x\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-3 + \frac{-1}{x}}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3 - \frac{x + 3}{x \cdot x}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(1 + x \cdot x\right) \cdot \left(1 + x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3 + \frac{-1 - \frac{3}{x}}{x}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(1 + x \cdot x\right) \cdot \left(1 + x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ (+ -3.0 (/ (- -1.0 (/ 3.0 x)) x)) x)
   (if (<= x 1.0)
     (* (+ 1.0 (* x x)) (+ 1.0 (* x 3.0)))
     (/ (+ -3.0 (/ -1.0 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-3.0 + ((-1.0 - (3.0 / x)) / x)) / x;
	} else if (x <= 1.0) {
		tmp = (1.0 + (x * x)) * (1.0 + (x * 3.0));
	} else {
		tmp = (-3.0 + (-1.0 / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = ((-3.0d0) + (((-1.0d0) - (3.0d0 / x)) / x)) / x
    else if (x <= 1.0d0) then
        tmp = (1.0d0 + (x * x)) * (1.0d0 + (x * 3.0d0))
    else
        tmp = ((-3.0d0) + ((-1.0d0) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-3.0 + ((-1.0 - (3.0 / x)) / x)) / x;
	} else if (x <= 1.0) {
		tmp = (1.0 + (x * x)) * (1.0 + (x * 3.0));
	} else {
		tmp = (-3.0 + (-1.0 / x)) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = (-3.0 + ((-1.0 - (3.0 / x)) / x)) / x
	elif x <= 1.0:
		tmp = (1.0 + (x * x)) * (1.0 + (x * 3.0))
	else:
		tmp = (-3.0 + (-1.0 / x)) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(-3.0 + Float64(Float64(-1.0 - Float64(3.0 / x)) / x)) / x);
	elseif (x <= 1.0)
		tmp = Float64(Float64(1.0 + Float64(x * x)) * Float64(1.0 + Float64(x * 3.0)));
	else
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (-3.0 + ((-1.0 - (3.0 / x)) / x)) / x;
	elseif (x <= 1.0)
		tmp = (1.0 + (x * x)) * (1.0 + (x * 3.0));
	else
		tmp = (-3.0 + (-1.0 / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 10.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{1 + 3 \cdot \frac{1}{x}}{x} - 3}{x}} \]

   5. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{-3 - \frac{1 + \frac{3}{x}}{x}}{x}} \]

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(3 + x \cdot \left(1 + 3 \cdot x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto 1 + \left(x \cdot 3 + \color{blue}{x \cdot \left(x \cdot \left(1 + 3 \cdot x\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto 1 + \left(3 \cdot x + \color{blue}{x} \cdot \left(x \cdot \left(1 + 3 \cdot x\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \left(1 + 3 \cdot x\right) + \color{blue}{x \cdot \left(x \cdot \left(1 + 3 \cdot x\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \left(1 + 3 \cdot x\right) + \left(x \cdot x\right) \cdot \color{blue}{\left(1 + 3 \cdot x\right)} \]

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + 3 \cdot x\right), \color{blue}{\left({x}^{2} + 1\right)}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(3 \cdot x\right)\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \left(1 + \color{blue}{{x}^{2}}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      14. *-lowering-*.f64 99.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\left(1 + 3 \cdot x\right) \cdot \left(1 + x \cdot x\right)} \]

   if 1 < x

   1. Initial program 6.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{3 + \frac{1}{x}}{x}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{3 + \frac{1}{x}}{x}\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)}{\color{blue}{x}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)\right), \color{blue}{x}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-3 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{-1}{x}\right)\right), x\right) \]

      9. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(-1, x\right)\right), x\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-3 + \frac{-1}{x}}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3 + \frac{-1 - \frac{3}{x}}{x}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(1 + x \cdot x\right) \cdot \left(1 + x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(1 + x \cdot x\right) \cdot \left(1 + x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (+ -3.0 (/ -1.0 x)) x)))
   (if (<= x -1.0)
     t_0
     (if (<= x 1.0) (* (+ 1.0 (* x x)) (+ 1.0 (* x 3.0))) t_0))))

C

double code(double x) {
	double t_0 = (-3.0 + (-1.0 / x)) / x;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (1.0 + (x * x)) * (1.0 + (x * 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-3.0d0) + ((-1.0d0) / x)) / x
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = (1.0d0 + (x * x)) * (1.0d0 + (x * 3.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (-3.0 + (-1.0 / x)) / x;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (1.0 + (x * x)) * (1.0 + (x * 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (-3.0 + (-1.0 / x)) / x
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 1.0:
		tmp = (1.0 + (x * x)) * (1.0 + (x * 3.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(Float64(1.0 + Float64(x * x)) * Float64(1.0 + Float64(x * 3.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (-3.0 + (-1.0 / x)) / x;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = (1.0 + (x * x)) * (1.0 + (x * 3.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.0], N[(N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{3 + \frac{1}{x}}{x}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{3 + \frac{1}{x}}{x}\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)}{\color{blue}{x}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)\right), \color{blue}{x}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-3 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{-1}{x}\right)\right), x\right) \]

      9. /-lowering-/.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(-1, x\right)\right), x\right) \]

   5. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{-3 + \frac{-1}{x}}{x}} \]

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(3 + x \cdot \left(1 + 3 \cdot x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto 1 + \left(x \cdot 3 + \color{blue}{x \cdot \left(x \cdot \left(1 + 3 \cdot x\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto 1 + \left(3 \cdot x + \color{blue}{x} \cdot \left(x \cdot \left(1 + 3 \cdot x\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \left(1 + 3 \cdot x\right) + \color{blue}{x \cdot \left(x \cdot \left(1 + 3 \cdot x\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \left(1 + 3 \cdot x\right) + \left(x \cdot x\right) \cdot \color{blue}{\left(1 + 3 \cdot x\right)} \]

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + 3 \cdot x\right), \color{blue}{\left({x}^{2} + 1\right)}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(3 \cdot x\right)\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \left(1 + \color{blue}{{x}^{2}}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      14. *-lowering-*.f64 99.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\left(1 + 3 \cdot x\right) \cdot \left(1 + x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(1 + x \cdot x\right) \cdot \left(1 + x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (+ -3.0 (/ -1.0 x)) x)))
   (if (<= x -1.0) t_0 (if (<= x 1.0) (+ 1.0 (* x (+ x 3.0))) t_0))))

C

double code(double x) {
	double t_0 = (-3.0 + (-1.0 / x)) / x;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * (x + 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-3.0d0) + ((-1.0d0) / x)) / x
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 + (x * (x + 3.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (-3.0 + (-1.0 / x)) / x;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * (x + 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (-3.0 + (-1.0 / x)) / x
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 + (x * (x + 3.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 + Float64(x * Float64(x + 3.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (-3.0 + (-1.0 / x)) / x;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0 + (x * (x + 3.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.0], N[(1.0 + N[(x * N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{3 + \frac{1}{x}}{x}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{3 + \frac{1}{x}}{x}\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)}{\color{blue}{x}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)\right), \color{blue}{x}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-3 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{-1}{x}\right)\right), x\right) \]

      9. /-lowering-/.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(-1, x\right)\right), x\right) \]

   5. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{-3 + \frac{-1}{x}}{x}} \]

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(3 + x\right)} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(3 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(3 + x\right)}\right)\right) \]

      3. +-lowering-+.f64 99.1%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{1 + x \cdot \left(3 + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{0.1111111111111111 + x \cdot -0.3333333333333333}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 0.1111111111111111 (* x -0.3333333333333333)))))
   (if (<= x -1.0) t_0 (if (<= x 1.0) (+ 1.0 (* x (+ x 3.0))) t_0))))

C

double code(double x) {
	double t_0 = 1.0 / (0.1111111111111111 + (x * -0.3333333333333333));
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * (x + 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (0.1111111111111111d0 + (x * (-0.3333333333333333d0)))
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 + (x * (x + 3.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (0.1111111111111111 + (x * -0.3333333333333333));
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * (x + 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / (0.1111111111111111 + (x * -0.3333333333333333))
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 + (x * (x + 3.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(0.1111111111111111 + Float64(x * -0.3333333333333333)))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 + Float64(x * Float64(x + 3.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / (0.1111111111111111 + (x * -0.3333333333333333));
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0 + (x * (x + 3.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(0.1111111111111111 + N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.0], N[(1.0 + N[(x * N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{3 + \frac{1}{x}}{x}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{3 + \frac{1}{x}}{x}\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)}{\color{blue}{x}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)\right), \color{blue}{x}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-3 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{-1}{x}\right)\right), x\right) \]

      9. /-lowering-/.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(-1, x\right)\right), x\right) \]

   5. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{-3 + \frac{-1}{x}}{x}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{x}{-3 + \frac{-1}{x}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{-3 + \frac{-1}{x}}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left(-3 + \frac{-1}{x}\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(-3, \color{blue}{\left(\frac{-1}{x}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 98.4%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right)\right)\right) \]

   7. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{-3 + \frac{-1}{x}}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{9} \cdot \frac{1}{x} - \frac{1}{3}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\frac{1}{9} \cdot \frac{1}{x} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\frac{1}{9} \cdot \frac{1}{x} + \frac{-1}{3}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{1}{9} \cdot \frac{1}{x}\right) \cdot x + \color{blue}{\frac{-1}{3} \cdot x}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{9} \cdot \left(\frac{1}{x} \cdot x\right) + \color{blue}{\frac{-1}{3}} \cdot x\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{9} \cdot 1 + \frac{-1}{3} \cdot x\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{9} + \color{blue}{\frac{-1}{3}} \cdot x\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{9} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot x\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{9} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot x\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot x\right)\right)}\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \color{blue}{x}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \left(\frac{-1}{3} \cdot x\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \left(x \cdot \color{blue}{\frac{-1}{3}}\right)\right)\right) \]

      13. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{3}}\right)\right)\right) \]

   10. Simplified 98.2%

       \[\leadsto \frac{1}{\color{blue}{0.1111111111111111 + x \cdot -0.3333333333333333}} \]

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(3 + x\right)} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(3 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(3 + x\right)}\right)\right) \]

      3. +-lowering-+.f64 99.1%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{1 + x \cdot \left(3 + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{0.1111111111111111 + x \cdot -0.3333333333333333}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.1111111111111111 + x \cdot -0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ -3.0 x)
   (if (<= x 1.0) (+ 1.0 (* x (+ x 3.0))) (/ -3.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * (x + 3.0));
	} else {
		tmp = -3.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-3.0d0) / x
    else if (x <= 1.0d0) then
        tmp = 1.0d0 + (x * (x + 3.0d0))
    else
        tmp = (-3.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * (x + 3.0));
	} else {
		tmp = -3.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -3.0 / x
	elif x <= 1.0:
		tmp = 1.0 + (x * (x + 3.0))
	else:
		tmp = -3.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-3.0 / x);
	elseif (x <= 1.0)
		tmp = Float64(1.0 + Float64(x * Float64(x + 3.0)));
	else
		tmp = Float64(-3.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -3.0 / x;
	elseif (x <= 1.0)
		tmp = 1.0 + (x * (x + 3.0));
	else
		tmp = -3.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-3}{x}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 98.0%

         \[\leadsto \mathsf{/.f64}\left(-3, \color{blue}{x}\right) \]

   5. Simplified 98.0%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(3 + x\right)} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(3 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(3 + x\right)}\right)\right) \]

      3. +-lowering-+.f64 99.1%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{1 + x \cdot \left(3 + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) (/ -3.0 x) (if (<= x 1.0) (+ 1.0 (* x 3.0)) (/ -3.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * 3.0);
	} else {
		tmp = -3.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-3.0d0) / x
    else if (x <= 1.0d0) then
        tmp = 1.0d0 + (x * 3.0d0)
    else
        tmp = (-3.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * 3.0);
	} else {
		tmp = -3.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -3.0 / x
	elif x <= 1.0:
		tmp = 1.0 + (x * 3.0)
	else:
		tmp = -3.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-3.0 / x);
	elseif (x <= 1.0)
		tmp = Float64(1.0 + Float64(x * 3.0));
	else
		tmp = Float64(-3.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -3.0 / x;
	elseif (x <= 1.0)
		tmp = 1.0 + (x * 3.0);
	else
		tmp = -3.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-3}{x}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 98.0%

         \[\leadsto \mathsf{/.f64}\left(-3, \color{blue}{x}\right) \]

   5. Simplified 98.0%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 3 \cdot x} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(3 \cdot x\right)}\right) \]

      2. *-lowering-*.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, \color{blue}{x}\right)\right) \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{1 + 3 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) (/ -3.0 x) (if (<= x 1.0) 1.0 (/ -3.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = -3.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-3.0d0) / x
    else if (x <= 1.0d0) then
        tmp = 1.0d0
    else
        tmp = (-3.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = -3.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -3.0 / x
	elif x <= 1.0:
		tmp = 1.0
	else:
		tmp = -3.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-3.0 / x);
	elseif (x <= 1.0)
		tmp = 1.0;
	else
		tmp = Float64(-3.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -3.0 / x;
	elseif (x <= 1.0)
		tmp = 1.0;
	else
		tmp = -3.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 1.0], 1.0, N[(-3.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-3}{x}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 98.0%

         \[\leadsto \mathsf{/.f64}\left(-3, \color{blue}{x}\right) \]

   5. Simplified 98.0%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 97.6%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 50.2% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 46.7%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))