Asymptote C

Percentage Accurate: 53.7% → 99.8%

Time: 2.3min

Alternatives: 10

Speedup: 0.7×

Specification

Math

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

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (- (/ 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)
use fmin_fmax_functions
    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]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	(x / (x + (1))) - ((x + (1)) / (x - (1)))
END code

Initial Program: 53.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (- (/ 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)
use fmin_fmax_functions
    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]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	(x / (x + (1))) - ((x + (1)) / (x - (1)))
END code

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0:\\ \;\;\;\;\frac{-1 \cdot \frac{1 + 3 \cdot \frac{1}{x}}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(x, x, -1\right)}{\mathsf{fma}\left(-3, x, -1\right)}}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.0)
  (/ (- (* -1.0 (/ (+ 1.0 (* 3.0 (/ 1.0 x))) x)) 3.0) x)
  (/ 1.0 (/ (fma x x -1.0) (fma -3.0 x -1.0)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.0) {
		tmp = ((-1.0 * ((1.0 + (3.0 * (1.0 / x))) / x)) - 3.0) / x;
	} else {
		tmp = 1.0 / (fma(x, x, -1.0) / fma(-3.0, x, -1.0));
	}
	return tmp;
}

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(-1.0 * N[(N[(1.0 + N[(3.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(N[(x * x + -1.0), $MachinePrecision] / N[(-3.0 * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF (((x / (x + (1))) - ((x + (1)) / (x - (1)))) <= (0)) THEN ((((-1) * (((1) + ((3) * ((1) / x))) / x)) - (3)) / x) ELSE ((1) / (((x * x) + (-1)) / (((-3) * x) + (-1)))) ENDIF IN
	tmp
END code

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)))) < 0.0

   1. Initial program 53.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 51.4%

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

   if 0.0 < (-.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 53.7%

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

   3. Applied rewrites 57.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 76.7%

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

   8. Applied rewrites 76.7%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(x, x, -1\right)}{\mathsf{fma}\left(-3, x, -1\right)}}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.0)
  (/ (- (/ -1.0 x) 3.0) x)
  (/ 1.0 (/ (fma x x -1.0) (fma -3.0 x -1.0)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.0) {
		tmp = ((-1.0 / x) - 3.0) / x;
	} else {
		tmp = 1.0 / (fma(x, x, -1.0) / fma(-3.0, x, -1.0));
	}
	return tmp;
}

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(-1.0 / x), $MachinePrecision] - 3.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(N[(x * x + -1.0), $MachinePrecision] / N[(-3.0 * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF (((x / (x + (1))) - ((x + (1)) / (x - (1)))) <= (0)) THEN ((((-1) / x) - (3)) / x) ELSE ((1) / (((x * x) + (-1)) / (((-3) * x) + (-1)))) ENDIF IN
	tmp
END code

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)))) < 0.0

   1. Initial program 53.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 50.4%

      \[\leadsto \frac{\frac{-1}{x} - 3}{x} \]

   if 0.0 < (-.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 53.7%

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

   3. Applied rewrites 57.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 76.7%

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

   8. Applied rewrites 76.7%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.0)
  (/ (- (/ -1.0 x) 3.0) x)
  (/ (fma -3.0 x -1.0) (fma x x -1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.0) {
		tmp = ((-1.0 / x) - 3.0) / 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(x + 1.0) / Float64(x - 1.0))) <= 0.0)
		tmp = Float64(Float64(Float64(-1.0 / x) - 3.0) / 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[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(-1.0 / x), $MachinePrecision] - 3.0), $MachinePrecision] / x), $MachinePrecision], N[(N[(-3.0 * x + -1.0), $MachinePrecision] / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF (((x / (x + (1))) - ((x + (1)) / (x - (1)))) <= (0)) THEN ((((-1) / x) - (3)) / x) ELSE ((((-3) * x) + (-1)) / ((x * x) + (-1))) ENDIF IN
	tmp
END code

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)))) < 0.0

   1. Initial program 53.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 50.4%

      \[\leadsto \frac{\frac{-1}{x} - 3}{x} \]

   if 0.0 < (-.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 53.7%

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

   3. Applied rewrites 57.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 76.8%

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

   8. Applied rewrites 76.8%

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

Alternative 4: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.5:\\ \;\;\;\;\frac{-3}{x} + \frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(x, 3, 1\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.5)
  (+ (/ -3.0 x) (/ -1.0 (* x x)))
  (* (fma x x 1.0) (fma x 3.0 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.5) {
		tmp = (-3.0 / x) + (-1.0 / (x * x));
	} else {
		tmp = fma(x, x, 1.0) * fma(x, 3.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0))) <= 0.5)
		tmp = Float64(Float64(-3.0 / x) + Float64(-1.0 / Float64(x * x)));
	else
		tmp = Float64(fma(x, x, 1.0) * fma(x, 3.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], N[(N[(-3.0 / x), $MachinePrecision] + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] * N[(x * 3.0 + 1.0), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF (((x / (x + (1))) - ((x + (1)) / (x - (1)))) <= (5e-1)) THEN (((-3) / x) + ((-1) / (x * x))) ELSE (((x * x) + (1)) * ((x * (3)) + (1))) ENDIF IN
	tmp
END code

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)))) < 0.5

   1. Initial program 53.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 50.4%

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

   if 0.5 < (-.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 53.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 + x \cdot \left(3 + x \cdot \left(1 + 3 \cdot x\right)\right) \]

   4. Applied rewrites 49.8%

      \[\leadsto 1 + x \cdot \left(3 + x \cdot \left(1 + 3 \cdot x\right)\right) \]

   6. Applied rewrites 49.8%

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

Alternative 5: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.5:\\ \;\;\;\;\frac{\frac{-1}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(x, 3, 1\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.5)
  (/ (- (/ -1.0 x) 3.0) x)
  (* (fma x x 1.0) (fma x 3.0 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.5) {
		tmp = ((-1.0 / x) - 3.0) / x;
	} else {
		tmp = fma(x, x, 1.0) * fma(x, 3.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0))) <= 0.5)
		tmp = Float64(Float64(Float64(-1.0 / x) - 3.0) / x);
	else
		tmp = Float64(fma(x, x, 1.0) * fma(x, 3.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], N[(N[(N[(-1.0 / x), $MachinePrecision] - 3.0), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] * N[(x * 3.0 + 1.0), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF (((x / (x + (1))) - ((x + (1)) / (x - (1)))) <= (5e-1)) THEN ((((-1) / x) - (3)) / x) ELSE (((x * x) + (1)) * ((x * (3)) + (1))) ENDIF IN
	tmp
END code

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)))) < 0.5

   1. Initial program 53.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 50.4%

      \[\leadsto \frac{\frac{-1}{x} - 3}{x} \]

   if 0.5 < (-.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 53.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 + x \cdot \left(3 + x \cdot \left(1 + 3 \cdot x\right)\right) \]

   4. Applied rewrites 49.8%

      \[\leadsto 1 + x \cdot \left(3 + x \cdot \left(1 + 3 \cdot x\right)\right) \]

   6. Applied rewrites 49.8%

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

Alternative 6: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.5:\\ \;\;\;\;\frac{\frac{-1}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x - -3, 1\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.5)
  (/ (- (/ -1.0 x) 3.0) x)
  (fma x (- x -3.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.5) {
		tmp = ((-1.0 / x) - 3.0) / x;
	} else {
		tmp = fma(x, (x - -3.0), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0))) <= 0.5)
		tmp = Float64(Float64(Float64(-1.0 / x) - 3.0) / x);
	else
		tmp = fma(x, Float64(x - -3.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], N[(N[(N[(-1.0 / x), $MachinePrecision] - 3.0), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(x - -3.0), $MachinePrecision] + 1.0), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF (((x / (x + (1))) - ((x + (1)) / (x - (1)))) <= (5e-1)) THEN ((((-1) / x) - (3)) / x) ELSE ((x * (x - (-3))) + (1)) ENDIF IN
	tmp
END code

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)))) < 0.5

   1. Initial program 53.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 50.4%

      \[\leadsto \frac{\frac{-1}{x} - 3}{x} \]

   if 0.5 < (-.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 53.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 + x \cdot \left(3 + x\right) \]

   4. Applied rewrites 50.2%

      \[\leadsto 1 + x \cdot \left(3 + x\right) \]

   6. Applied rewrites 50.2%

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

Alternative 7: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.5:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x - -3, 1\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.5)
  (/ -3.0 x)
  (fma x (- x -3.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.5) {
		tmp = -3.0 / x;
	} else {
		tmp = fma(x, (x - -3.0), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0))) <= 0.5)
		tmp = Float64(-3.0 / x);
	else
		tmp = fma(x, Float64(x - -3.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], N[(-3.0 / x), $MachinePrecision], N[(x * N[(x - -3.0), $MachinePrecision] + 1.0), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF (((x / (x + (1))) - ((x + (1)) / (x - (1)))) <= (5e-1)) THEN ((-3) / x) ELSE ((x * (x - (-3))) + (1)) ENDIF IN
	tmp
END code

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)))) < 0.5

   1. Initial program 53.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 51.1%

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

   if 0.5 < (-.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 53.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 + x \cdot \left(3 + x\right) \]

   4. Applied rewrites 50.2%

      \[\leadsto 1 + x \cdot \left(3 + x\right) \]

   6. Applied rewrites 50.2%

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

Alternative 8: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.5:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3, x, 1\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.5)
  (/ -3.0 x)
  (fma 3.0 x 1.0)))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.5) {
		tmp = -3.0 / x;
	} else {
		tmp = fma(3.0, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0))) <= 0.5)
		tmp = Float64(-3.0 / x);
	else
		tmp = fma(3.0, x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], N[(-3.0 / x), $MachinePrecision], N[(3.0 * x + 1.0), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF (((x / (x + (1))) - ((x + (1)) / (x - (1)))) <= (5e-1)) THEN ((-3) / x) ELSE (((3) * x) + (1)) ENDIF IN
	tmp
END code

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)))) < 0.5

   1. Initial program 53.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 51.1%

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

   if 0.5 < (-.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 53.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 + 3 \cdot x \]

   4. Applied rewrites 49.7%

      \[\leadsto 1 + 3 \cdot x \]

   6. Applied rewrites 49.7%

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

Alternative 9: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.5:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.5)
  (/ -3.0 x)
  1.0))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.5) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x / (x + 1.0d0)) - ((x + 1.0d0) / (x - 1.0d0))) <= 0.5d0) then
        tmp = (-3.0d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.5) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.5:
		tmp = -3.0 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0))) <= 0.5)
		tmp = Float64(-3.0 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.5)
		tmp = -3.0 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], N[(-3.0 / x), $MachinePrecision], 1.0]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF (((x / (x + (1))) - ((x + (1)) / (x - (1)))) <= (5e-1)) THEN ((-3) / x) ELSE (1) ENDIF IN
	tmp
END code

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)))) < 0.5

   1. Initial program 53.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 51.1%

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

   if 0.5 < (-.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 53.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 50.4%

      \[\leadsto \frac{\frac{-1}{x} - 3}{x} \]

   8. Applied rewrites 27.2%

      \[\leadsto \frac{-0.3333333333333333 - x}{\left(0.3333333333333333 \cdot x\right) \cdot x} \]

   9. Taylor expanded in x around 0

      \[\leadsto 1 \]

   11. Applied rewrites 50.2%

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

Alternative 10: 50.2% accurate, 19.0× speedup

Math

\[1 \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    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

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	1
END code

Derivation

1. Initial program 53.7%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 50.4%

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

6. Applied rewrites 50.4%

   \[\leadsto \frac{\frac{-1}{x} - 3}{x} \]

8. Applied rewrites 27.2%

   \[\leadsto \frac{-0.3333333333333333 - x}{\left(0.3333333333333333 \cdot x\right) \cdot x} \]

9. Taylor expanded in x around 0

   \[\leadsto 1 \]

11. Applied rewrites 50.2%

    \[\leadsto 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2026074 +o generate:egglog
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))