Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.6% → 99.9%

Time: 8.6s

Alternatives: 11

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))

C

double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (6.0d0 * (x - 1.0d0)) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
end function

Java

public static double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * Math.sqrt(x)));
}

Python

def code(x):
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * math.sqrt(x)))

Julia

function code(x)
	return Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
end

Wolfram

code[x_] := N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))

C

double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (6.0d0 * (x - 1.0d0)) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
end function

Java

public static double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * Math.sqrt(x)));
}

Python

def code(x):
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * math.sqrt(x)))

Julia

function code(x)
	return Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
end

Wolfram

code[x_] := N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{6}{{\left(\frac{x + -1}{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)}\right)}^{-1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 6.0 (pow (/ (+ x -1.0) (+ x (fma 4.0 (sqrt x) 1.0))) -1.0)))

C

double code(double x) {
	return 6.0 / pow(((x + -1.0) / (x + fma(4.0, sqrt(x), 1.0))), -1.0);
}

Julia

function code(x)
	return Float64(6.0 / (Float64(Float64(x + -1.0) / Float64(x + fma(4.0, sqrt(x), 1.0))) ^ -1.0))
end

Wolfram

code[x_] := N[(6.0 / N[Power[N[(N[(x + -1.0), $MachinePrecision] / N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. div-inv 99.7%

      \[\leadsto \color{blue}{\left(6 \cdot \left(x - 1\right)\right) \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

   2. associate-*l* 99.8%

      \[\leadsto \color{blue}{6 \cdot \left(\left(x - 1\right) \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right)} \]

   3. sub-neg 99.8%

      \[\leadsto 6 \cdot \left(\color{blue}{\left(x + \left(-1\right)\right)} \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right) \]

   4. metadata-eval 99.8%

      \[\leadsto 6 \cdot \left(\left(x + \color{blue}{-1}\right) \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right) \]

   5. +-commutative 99.8%

      \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}\right) \]

   6. fma-udef 99.8%

      \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right) \]

3. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}\right)} \]
4. Step-by-step derivation

   1. un-div-inv 99.9%

      \[\leadsto 6 \cdot \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \]

   2. clear-num 99.9%

      \[\leadsto 6 \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}{x + -1}}} \]

   3. fma-udef 99.9%

      \[\leadsto 6 \cdot \frac{1}{\frac{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}{x + -1}} \]

   4. +-commutative 99.9%

      \[\leadsto 6 \cdot \frac{1}{\frac{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}}{x + -1}} \]

   5. flip-+ 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\color{blue}{\frac{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}}{x + -1}} \]

   6. unpow2 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\frac{\color{blue}{{\left(x + 1\right)}^{2}} - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   7. swap-sqr 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - \color{blue}{\left(4 \cdot 4\right) \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   8. metadata-eval 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - \color{blue}{16} \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   9. add-sqr-sqrt 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - 16 \cdot \color{blue}{x}}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   10. *-commutative 76.4%

       \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - \color{blue}{x \cdot 16}}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   11. +-commutative 76.4%

       \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - x \cdot 16}{\color{blue}{\left(1 + x\right)} - 4 \cdot \sqrt{x}}}{x + -1}} \]

   12. associate-+r- 76.4%

       \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - x \cdot 16}{\color{blue}{1 + \left(x - 4 \cdot \sqrt{x}\right)}}}{x + -1}} \]

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{6}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}{x + -1}}} \]
6. Step-by-step derivation

   1. clear-num 99.9%

      \[\leadsto \frac{6}{\color{blue}{\frac{1}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}}} \]

   2. fma-udef 99.9%

      \[\leadsto \frac{6}{\frac{1}{\frac{x + -1}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}}} \]

   3. associate-+l+ 99.9%

      \[\leadsto \frac{6}{\frac{1}{\frac{x + -1}{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) + 1}}}} \]

   4. un-div-inv 99.8%

      \[\leadsto \frac{6}{\frac{1}{\color{blue}{\left(x + -1\right) \cdot \frac{1}{\left(4 \cdot \sqrt{x} + x\right) + 1}}}} \]

   5. inv-pow 99.9%

      \[\leadsto \frac{6}{\color{blue}{{\left(\left(x + -1\right) \cdot \frac{1}{\left(4 \cdot \sqrt{x} + x\right) + 1}\right)}^{-1}}} \]

   6. un-div-inv 99.9%

      \[\leadsto \frac{6}{{\color{blue}{\left(\frac{x + -1}{\left(4 \cdot \sqrt{x} + x\right) + 1}\right)}}^{-1}} \]

   7. +-commutative 99.9%

      \[\leadsto \frac{6}{{\left(\frac{x + -1}{\color{blue}{\left(x + 4 \cdot \sqrt{x}\right)} + 1}\right)}^{-1}} \]

   8. associate-+r+ 99.9%

      \[\leadsto \frac{6}{{\left(\frac{x + -1}{\color{blue}{x + \left(4 \cdot \sqrt{x} + 1\right)}}\right)}^{-1}} \]

   9. fma-udef 99.9%

      \[\leadsto \frac{6}{{\left(\frac{x + -1}{x + \color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}}\right)}^{-1}} \]

7. Applied egg-rr 99.9%

   \[\leadsto \frac{6}{\color{blue}{{\left(\frac{x + -1}{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)}\right)}^{-1}}} \]

8. Final simplification 99.9%

   \[\leadsto \frac{6}{{\left(\frac{x + -1}{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)}\right)}^{-1}} \]

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{1 + \left(x + 4 \cdot \sqrt{x}\right)}{x + -1}}{6}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (/ (/ (+ 1.0 (+ x (* 4.0 (sqrt x)))) (+ x -1.0)) 6.0)))

C

double code(double x) {
	return 1.0 / (((1.0 + (x + (4.0 * sqrt(x)))) / (x + -1.0)) / 6.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (((1.0d0 + (x + (4.0d0 * sqrt(x)))) / (x + (-1.0d0))) / 6.0d0)
end function

Java

public static double code(double x) {
	return 1.0 / (((1.0 + (x + (4.0 * Math.sqrt(x)))) / (x + -1.0)) / 6.0);
}

Python

def code(x):
	return 1.0 / (((1.0 + (x + (4.0 * math.sqrt(x)))) / (x + -1.0)) / 6.0)

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(1.0 + Float64(x + Float64(4.0 * sqrt(x)))) / Float64(x + -1.0)) / 6.0))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (((1.0 + (x + (4.0 * sqrt(x)))) / (x + -1.0)) / 6.0);
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(1.0 + N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / 6.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. div-inv 99.7%

      \[\leadsto \color{blue}{\left(6 \cdot \left(x - 1\right)\right) \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

   2. associate-*l* 99.8%

      \[\leadsto \color{blue}{6 \cdot \left(\left(x - 1\right) \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right)} \]

   3. sub-neg 99.8%

      \[\leadsto 6 \cdot \left(\color{blue}{\left(x + \left(-1\right)\right)} \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right) \]

   4. metadata-eval 99.8%

      \[\leadsto 6 \cdot \left(\left(x + \color{blue}{-1}\right) \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right) \]

   5. +-commutative 99.8%

      \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}\right) \]

   6. fma-udef 99.8%

      \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right) \]

3. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}\right)} \]
4. Step-by-step derivation

   1. un-div-inv 99.9%

      \[\leadsto 6 \cdot \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \]

   2. clear-num 99.9%

      \[\leadsto 6 \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}{x + -1}}} \]

   3. fma-udef 99.9%

      \[\leadsto 6 \cdot \frac{1}{\frac{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}{x + -1}} \]

   4. +-commutative 99.9%

      \[\leadsto 6 \cdot \frac{1}{\frac{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}}{x + -1}} \]

   5. flip-+ 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\color{blue}{\frac{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}}{x + -1}} \]

   6. unpow2 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\frac{\color{blue}{{\left(x + 1\right)}^{2}} - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   7. swap-sqr 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - \color{blue}{\left(4 \cdot 4\right) \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   8. metadata-eval 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - \color{blue}{16} \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   9. add-sqr-sqrt 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - 16 \cdot \color{blue}{x}}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   10. *-commutative 76.4%

       \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - \color{blue}{x \cdot 16}}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   11. +-commutative 76.4%

       \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - x \cdot 16}{\color{blue}{\left(1 + x\right)} - 4 \cdot \sqrt{x}}}{x + -1}} \]

   12. associate-+r- 76.4%

       \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - x \cdot 16}{\color{blue}{1 + \left(x - 4 \cdot \sqrt{x}\right)}}}{x + -1}} \]

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}{x + -1}}{6}}} \]
6. Step-by-step derivation

   1. fma-udef 99.8%

      \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}\right) \]

   2. associate-+r+ 99.8%

      \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) + 1}}\right) \]

7. Applied egg-rr 99.9%

   \[\leadsto \frac{1}{\frac{\frac{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) + 1}}{x + -1}}{6}} \]

8. Final simplification 99.9%

   \[\leadsto \frac{1}{\frac{\frac{1 + \left(x + 4 \cdot \sqrt{x}\right)}{x + -1}}{6}} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{6}{\frac{1 + \left(x + 4 \cdot \sqrt{x}\right)}{x + -1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 6.0 (/ (+ 1.0 (+ x (* 4.0 (sqrt x)))) (+ x -1.0))))

C

double code(double x) {
	return 6.0 / ((1.0 + (x + (4.0 * sqrt(x)))) / (x + -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0 / ((1.0d0 + (x + (4.0d0 * sqrt(x)))) / (x + (-1.0d0)))
end function

Java

public static double code(double x) {
	return 6.0 / ((1.0 + (x + (4.0 * Math.sqrt(x)))) / (x + -1.0));
}

Python

def code(x):
	return 6.0 / ((1.0 + (x + (4.0 * math.sqrt(x)))) / (x + -1.0))

Julia

function code(x)
	return Float64(6.0 / Float64(Float64(1.0 + Float64(x + Float64(4.0 * sqrt(x)))) / Float64(x + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 6.0 / ((1.0 + (x + (4.0 * sqrt(x)))) / (x + -1.0));
end

Wolfram

code[x_] := N[(6.0 / N[(N[(1.0 + N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. div-inv 99.7%

      \[\leadsto \color{blue}{\left(6 \cdot \left(x - 1\right)\right) \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

   2. associate-*l* 99.8%

      \[\leadsto \color{blue}{6 \cdot \left(\left(x - 1\right) \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right)} \]

   3. sub-neg 99.8%

      \[\leadsto 6 \cdot \left(\color{blue}{\left(x + \left(-1\right)\right)} \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right) \]

   4. metadata-eval 99.8%

      \[\leadsto 6 \cdot \left(\left(x + \color{blue}{-1}\right) \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right) \]

   5. +-commutative 99.8%

      \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}\right) \]

   6. fma-udef 99.8%

      \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right) \]

3. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}\right)} \]
4. Step-by-step derivation

   1. un-div-inv 99.9%

      \[\leadsto 6 \cdot \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \]

   2. clear-num 99.9%

      \[\leadsto 6 \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}{x + -1}}} \]

   3. fma-udef 99.9%

      \[\leadsto 6 \cdot \frac{1}{\frac{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}{x + -1}} \]

   4. +-commutative 99.9%

      \[\leadsto 6 \cdot \frac{1}{\frac{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}}{x + -1}} \]

   5. flip-+ 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\color{blue}{\frac{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}}{x + -1}} \]

   6. unpow2 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\frac{\color{blue}{{\left(x + 1\right)}^{2}} - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   7. swap-sqr 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - \color{blue}{\left(4 \cdot 4\right) \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   8. metadata-eval 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - \color{blue}{16} \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   9. add-sqr-sqrt 76.4%

      \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - 16 \cdot \color{blue}{x}}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   10. *-commutative 76.4%

       \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - \color{blue}{x \cdot 16}}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}{x + -1}} \]

   11. +-commutative 76.4%

       \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - x \cdot 16}{\color{blue}{\left(1 + x\right)} - 4 \cdot \sqrt{x}}}{x + -1}} \]

   12. associate-+r- 76.4%

       \[\leadsto 6 \cdot \frac{1}{\frac{\frac{{\left(x + 1\right)}^{2} - x \cdot 16}{\color{blue}{1 + \left(x - 4 \cdot \sqrt{x}\right)}}}{x + -1}} \]

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{6}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}{x + -1}}} \]
6. Step-by-step derivation

   1. fma-udef 99.8%

      \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}\right) \]

   2. associate-+r+ 99.8%

      \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) + 1}}\right) \]

7. Applied egg-rr 99.9%

   \[\leadsto \frac{6}{\frac{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) + 1}}{x + -1}} \]

8. Final simplification 99.9%

   \[\leadsto \frac{6}{\frac{1 + \left(x + 4 \cdot \sqrt{x}\right)}{x + -1}} \]

Alternative 4: 95.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;6 \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{1}{\frac{x}{x + -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (* 6.0 (+ x -1.0)) (* 6.0 (/ 1.0 (/ x (+ x -1.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 6.0 * (x + -1.0);
	} else {
		tmp = 6.0 * (1.0 / (x / (x + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 6.0d0 * (x + (-1.0d0))
    else
        tmp = 6.0d0 * (1.0d0 / (x / (x + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 6.0 * (x + -1.0);
	} else {
		tmp = 6.0 * (1.0 / (x / (x + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 6.0 * (x + -1.0)
	else:
		tmp = 6.0 * (1.0 / (x / (x + -1.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(6.0 * Float64(x + -1.0));
	else
		tmp = Float64(6.0 * Float64(1.0 / Float64(x / Float64(x + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 6.0 * (x + -1.0);
	else
		tmp = 6.0 * (1.0 / (x / (x + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(6.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(1.0 / N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0 97.2%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1}} \]

   if 1 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf 94.6%

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

      1. sub-neg 94.6%

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(-1\right)\right)}}{x} \]

      2. metadata-eval 94.6%

         \[\leadsto \frac{6 \cdot \left(x + \color{blue}{-1}\right)}{x} \]

      3. associate-/l* 94.8%

         \[\leadsto \color{blue}{\frac{6}{\frac{x}{x + -1}}} \]

      4. div-inv 94.8%

         \[\leadsto \color{blue}{6 \cdot \frac{1}{\frac{x}{x + -1}}} \]

   4. Applied egg-rr 94.8%

      \[\leadsto \color{blue}{6 \cdot \frac{1}{\frac{x}{x + -1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;6 \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{1}{\frac{x}{x + -1}}\\ \end{array} \]

Alternative 5: 95.3% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{x + 1}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 6.0 (* (+ x -1.0) (/ 1.0 (+ x 1.0)))))

C

double code(double x) {
	return 6.0 * ((x + -1.0) * (1.0 / (x + 1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0 * ((x + (-1.0d0)) * (1.0d0 / (x + 1.0d0)))
end function

Java

public static double code(double x) {
	return 6.0 * ((x + -1.0) * (1.0 / (x + 1.0)));
}

Python

def code(x):
	return 6.0 * ((x + -1.0) * (1.0 / (x + 1.0)))

Julia

function code(x)
	return Float64(6.0 * Float64(Float64(x + -1.0) * Float64(1.0 / Float64(x + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = 6.0 * ((x + -1.0) * (1.0 / (x + 1.0)));
end

Wolfram

code[x_] := N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. div-inv 99.7%

      \[\leadsto \color{blue}{\left(6 \cdot \left(x - 1\right)\right) \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

   2. associate-*l* 99.8%

      \[\leadsto \color{blue}{6 \cdot \left(\left(x - 1\right) \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right)} \]

   3. sub-neg 99.8%

      \[\leadsto 6 \cdot \left(\color{blue}{\left(x + \left(-1\right)\right)} \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right) \]

   4. metadata-eval 99.8%

      \[\leadsto 6 \cdot \left(\left(x + \color{blue}{-1}\right) \cdot \frac{1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right) \]

   5. +-commutative 99.8%

      \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}\right) \]

   6. fma-udef 99.8%

      \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right) \]

3. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.8%

      \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}\right) \]

   2. associate-+r+ 99.8%

      \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) + 1}}\right) \]

5. Applied egg-rr 99.8%

   \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) + 1}}\right) \]

6. Taylor expanded in x around inf 95.9%

   \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{\color{blue}{x} + 1}\right) \]

7. Final simplification 95.9%

   \[\leadsto 6 \cdot \left(\left(x + -1\right) \cdot \frac{1}{x + 1}\right) \]

Alternative 6: 95.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x + -1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.5) -6.0 (* 6.0 (/ (+ x -1.0) x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = -6.0;
	} else {
		tmp = 6.0 * ((x + -1.0) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.5d0) then
        tmp = -6.0d0
    else
        tmp = 6.0d0 * ((x + (-1.0d0)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = -6.0;
	} else {
		tmp = 6.0 * ((x + -1.0) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.5:
		tmp = -6.0
	else:
		tmp = 6.0 * ((x + -1.0) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = -6.0;
	else
		tmp = Float64(6.0 * Float64(Float64(x + -1.0) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.5)
		tmp = -6.0;
	else
		tmp = 6.0 * ((x + -1.0) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], -6.0, N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0 97.2%

      \[\leadsto \color{blue}{-6} \]

   if 0.5 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf 94.6%

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

      1. sub-neg 94.6%

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(-1\right)\right)}}{x} \]

      2. metadata-eval 94.6%

         \[\leadsto \frac{6 \cdot \left(x + \color{blue}{-1}\right)}{x} \]

      3. *-un-lft-identity 94.6%

         \[\leadsto \frac{6 \cdot \left(x + -1\right)}{\color{blue}{1 \cdot x}} \]

      4. times-frac 94.8%

         \[\leadsto \color{blue}{\frac{6}{1} \cdot \frac{x + -1}{x}} \]

      5. metadata-eval 94.8%

         \[\leadsto \color{blue}{6} \cdot \frac{x + -1}{x} \]

   4. Applied egg-rr 94.8%

      \[\leadsto \color{blue}{6 \cdot \frac{x + -1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x + -1}{x}\\ \end{array} \]

Alternative 7: 95.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{x}{x + -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.5) -6.0 (/ 6.0 (/ x (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = -6.0;
	} else {
		tmp = 6.0 / (x / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.5d0) then
        tmp = -6.0d0
    else
        tmp = 6.0d0 / (x / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = -6.0;
	} else {
		tmp = 6.0 / (x / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.5:
		tmp = -6.0
	else:
		tmp = 6.0 / (x / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = -6.0;
	else
		tmp = Float64(6.0 / Float64(x / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.5)
		tmp = -6.0;
	else
		tmp = 6.0 / (x / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], -6.0, N[(6.0 / N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0 97.2%

      \[\leadsto \color{blue}{-6} \]

   if 0.5 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf 94.6%

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

      1. sub-neg 94.6%

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(-1\right)\right)}}{x} \]

      2. metadata-eval 94.6%

         \[\leadsto \frac{6 \cdot \left(x + \color{blue}{-1}\right)}{x} \]

      3. associate-/l* 94.8%

         \[\leadsto \color{blue}{\frac{6}{\frac{x}{x + -1}}} \]

      4. div-inv 94.8%

         \[\leadsto \color{blue}{6 \cdot \frac{1}{\frac{x}{x + -1}}} \]

   4. Applied egg-rr 94.8%

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

      1. un-div-inv 94.8%

         \[\leadsto \color{blue}{\frac{6}{\frac{x}{x + -1}}} \]

   6. Applied egg-rr 94.8%

      \[\leadsto \color{blue}{\frac{6}{\frac{x}{x + -1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{x}{x + -1}}\\ \end{array} \]

Alternative 8: 95.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;6 \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{x}{x + -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (* 6.0 (+ x -1.0)) (/ 6.0 (/ x (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 6.0 * (x + -1.0);
	} else {
		tmp = 6.0 / (x / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 6.0d0 * (x + (-1.0d0))
    else
        tmp = 6.0d0 / (x / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 6.0 * (x + -1.0);
	} else {
		tmp = 6.0 / (x / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 6.0 * (x + -1.0)
	else:
		tmp = 6.0 / (x / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(6.0 * Float64(x + -1.0));
	else
		tmp = Float64(6.0 / Float64(x / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 6.0 * (x + -1.0);
	else
		tmp = 6.0 / (x / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(6.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0 97.2%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1}} \]

   if 1 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf 94.6%

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

      1. sub-neg 94.6%

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(-1\right)\right)}}{x} \]

      2. metadata-eval 94.6%

         \[\leadsto \frac{6 \cdot \left(x + \color{blue}{-1}\right)}{x} \]

      3. associate-/l* 94.8%

         \[\leadsto \color{blue}{\frac{6}{\frac{x}{x + -1}}} \]

      4. div-inv 94.8%

         \[\leadsto \color{blue}{6 \cdot \frac{1}{\frac{x}{x + -1}}} \]

   4. Applied egg-rr 94.8%

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

      1. un-div-inv 94.8%

         \[\leadsto \color{blue}{\frac{6}{\frac{x}{x + -1}}} \]

   6. Applied egg-rr 94.8%

      \[\leadsto \color{blue}{\frac{6}{\frac{x}{x + -1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;6 \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{x}{x + -1}}\\ \end{array} \]

Alternative 9: 95.4% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{6}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.5) -6.0 (- 6.0 (/ 6.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = -6.0;
	} else {
		tmp = 6.0 - (6.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.5d0) then
        tmp = -6.0d0
    else
        tmp = 6.0d0 - (6.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = -6.0;
	} else {
		tmp = 6.0 - (6.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.5:
		tmp = -6.0
	else:
		tmp = 6.0 - (6.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = -6.0;
	else
		tmp = Float64(6.0 - Float64(6.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.5)
		tmp = -6.0;
	else
		tmp = 6.0 - (6.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], -6.0, N[(6.0 - N[(6.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0 97.2%

      \[\leadsto \color{blue}{-6} \]

   if 0.5 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf 94.6%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{x}} \]

   3. Taylor expanded in x around 0 94.8%

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

      1. associate-*r/ 94.8%

         \[\leadsto 6 - \color{blue}{\frac{6 \cdot 1}{x}} \]

      2. metadata-eval 94.8%

         \[\leadsto 6 - \frac{\color{blue}{6}}{x} \]

   5. Simplified 94.8%

      \[\leadsto \color{blue}{6 - \frac{6}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{6}{x}\\ \end{array} \]

Alternative 10: 95.4% accurate, 36.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) -6.0 6.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0;
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = -6.0d0
    else
        tmp = 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0;
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0
	else:
		tmp = 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = -6.0;
	else
		tmp = 6.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0;
	else
		tmp = 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], -6.0, 6.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0 97.2%

      \[\leadsto \color{blue}{-6} \]

   if 1 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf 94.8%

      \[\leadsto \color{blue}{6} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

Alternative 11: 49.1% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ -6 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -6.0)

C

double code(double x) {
	return -6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -6.0d0
end function

Java

public static double code(double x) {
	return -6.0;
}

Python

def code(x):
	return -6.0

Julia

function code(x)
	return -6.0
end

MATLAB

function tmp = code(x)
	tmp = -6.0;
end

Wolfram

code[x_] := -6.0

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

2. Taylor expanded in x around 0 48.6%

   \[\leadsto \color{blue}{-6} \]

3. Final simplification 48.6%

   \[\leadsto -6 \]

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0))))

C

double code(double x) {
	return 6.0 / (((x + 1.0) + (4.0 * sqrt(x))) / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0 / (((x + 1.0d0) + (4.0d0 * sqrt(x))) / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return 6.0 / (((x + 1.0) + (4.0 * Math.sqrt(x))) / (x - 1.0));
}

Python

def code(x):
	return 6.0 / (((x + 1.0) + (4.0 * math.sqrt(x))) / (x - 1.0))

Julia

function code(x)
	return Float64(6.0 / Float64(Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))) / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 6.0 / (((x + 1.0) + (4.0 * sqrt(x))) / (x - 1.0));
end

Wolfram

code[x_] := N[(6.0 / N[(N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023332 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))