Asymptote C

Percentage Accurate: 54.7% → 100.0%

Time: 7.5s

Alternatives: 5

Speedup: 1.5×

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: 54.7% 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: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+34}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 50000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e+34)
   (/ -3.0 x)
   (if (<= x 50000000000000.0)
     (/ (fma -3.0 x -1.0) (fma x x -1.0))
     (/ -3.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e+34) {
		tmp = -3.0 / x;
	} else if (x <= 50000000000000.0) {
		tmp = fma(-3.0, x, -1.0) / fma(x, x, -1.0);
	} else {
		tmp = -3.0 / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e+34)
		tmp = Float64(-3.0 / x);
	elseif (x <= 50000000000000.0)
		tmp = Float64(fma(-3.0, x, -1.0) / fma(x, x, -1.0));
	else
		tmp = Float64(-3.0 / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e+34], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 50000000000000.0], N[(N[(-3.0 * x + -1.0), $MachinePrecision] / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision], N[(-3.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999946e33 or 5e13 < x

   1. Initial program 5.5%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if -9.99999999999999946e33 < x < 5e13

   1. Initial program 95.2%

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

      1. frac-sub N/A

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

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

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. metadata-eval 95.2

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

   4. Applied egg-rr 95.2%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 100.0

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

   7. Simplified 100.0%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 5e-13)
   (/ (+ -3.0 (/ -1.0 x)) x)
   (/ (fma -3.0 x -1.0) (fma x x -1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 5e-13) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = fma(-3.0, x, -1.0) / fma(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))) <= 5e-13)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = Float64(fma(-3.0, x, -1.0) / fma(x, x, -1.0));
	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], 5e-13], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(-3.0 * x + -1.0), $MachinePrecision] / N[(x * 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)))) < 4.9999999999999999e-13

   1. Initial program 7.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. associate-*r/ N/A

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

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

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

      3. neg-mul-1 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 99.2

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

   5. Simplified 99.2%

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

   if 4.9999999999999999e-13 < (-.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 99.1%

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

      1. frac-sub N/A

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

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

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. metadata-eval 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 100.0

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

   7. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ -3.0 x)
   (if (<= x 1.0) (fma x (fma x (fma 3.0 x 1.0) 3.0) 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 = fma(x, fma(x, fma(3.0, x, 1.0), 3.0), 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 = fma(x, fma(x, fma(3.0, x, 1.0), 3.0), 1.0);
	else
		tmp = Float64(-3.0 / x);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 98.6

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

   5. Simplified 98.6%

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

   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

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 99.4

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

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(3, x, 1\right), 3\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.4% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) (/ -3.0 x) (if (<= x 1.0) (fma 3.0 x 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 = fma(3.0, x, 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 = fma(3.0, x, 1.0);
	else
		tmp = Float64(-3.0 / x);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 98.6

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

   5. Simplified 98.6%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 99.3

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

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3, x, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 51.0% accurate, 35.0× 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 52.1%

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

3. Taylor expanded in x around 0

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

5. Simplified 49.2%

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

Reproduce

herbie shell --seed 2024196 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))