Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.8% → 99.9%

Time: 8.6s

Alternatives: 6

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: 99.8% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. times-frac 99.9%

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

   3. metadata-eval 99.9%

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

   4. sub-neg 99.9%

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

   5. metadata-eval 99.9%

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

   6. associate-+l+ 99.9%

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

   7. +-commutative 99.9%

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

   8. fma-def 99.9%

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

4. Applied egg-rr 99.9%

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

   1. fma-udef 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;6 \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-6}{\sqrt{x} \cdot -4 + \left(-1 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.2)
   (* 6.0 (+ x -1.0))
   (* x (/ -6.0 (+ (* (sqrt x) -4.0) (- -1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = 6.0 * (x + -1.0);
	} else {
		tmp = x * (-6.0 / ((sqrt(x) * -4.0) + (-1.0 - x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.2d0) then
        tmp = 6.0d0 * (x + (-1.0d0))
    else
        tmp = x * ((-6.0d0) / ((sqrt(x) * (-4.0d0)) + ((-1.0d0) - x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = 6.0 * (x + -1.0);
	} else {
		tmp = x * (-6.0 / ((Math.sqrt(x) * -4.0) + (-1.0 - x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.2:
		tmp = 6.0 * (x + -1.0)
	else:
		tmp = x * (-6.0 / ((math.sqrt(x) * -4.0) + (-1.0 - x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.2)
		tmp = Float64(6.0 * Float64(x + -1.0));
	else
		tmp = Float64(x * Float64(-6.0 / Float64(Float64(sqrt(x) * -4.0) + Float64(-1.0 - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.2)
		tmp = 6.0 * (x + -1.0);
	else
		tmp = x * (-6.0 / ((sqrt(x) * -4.0) + (-1.0 - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.2], N[(6.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(-6.0 / N[(N[(N[Sqrt[x], $MachinePrecision] * -4.0), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 99.9%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 97.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \cdot \sqrt[3]{\frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right) \cdot \sqrt[3]{\frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right)} \cdot \left(x + -1\right) \]

      2. pow3 97.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right)}^{3}} \cdot \left(x + -1\right) \]

      3. fma-udef 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

      4. +-commutative 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}}}\right)}^{3} \cdot \left(x + -1\right) \]

      5. associate-+l+ 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{\color{blue}{x + \left(1 + 4 \cdot \sqrt{x}\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

      6. +-commutative 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{x + \color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

      7. fma-def 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{x + \color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

   6. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{6}{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)}}\right)}^{3}} \cdot \left(x + -1\right) \]
   7. Step-by-step derivation

      1. fma-udef 99.9%

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

   8. Applied egg-rr 97.6%

      \[\leadsto {\left(\sqrt[3]{\frac{6}{x + \color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

   9. Taylor expanded in x around 0 96.8%

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

   if 1.19999999999999996 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf 96.7%

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

      1. frac-2neg 96.7%

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

      2. associate-+l+ 96.7%

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

      3. +-commutative 96.7%

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

      4. div-inv 96.3%

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

      5. *-commutative 96.3%

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

      6. distribute-rgt-neg-in 96.3%

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

      7. metadata-eval 96.3%

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

      8. +-commutative 96.3%

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

      9. associate-+l+ 96.3%

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

      10. +-commutative 96.3%

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

      11. fma-def 96.3%

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

   5. Applied egg-rr 96.3%

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

      1. associate-*l* 97.1%

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

      2. associate-*r/ 97.5%

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

      3. metadata-eval 97.5%

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

      4. neg-sub0 97.5%

         \[\leadsto x \cdot \frac{-6}{\color{blue}{0 - \mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \]

      5. fma-udef 97.5%

         \[\leadsto x \cdot \frac{-6}{0 - \color{blue}{\left(4 \cdot \sqrt{x} + \left(x + 1\right)\right)}} \]

      6. associate--r+ 97.5%

         \[\leadsto x \cdot \frac{-6}{\color{blue}{\left(0 - 4 \cdot \sqrt{x}\right) - \left(x + 1\right)}} \]

      7. neg-sub0 97.5%

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

      8. *-commutative 97.5%

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

      9. distribute-rgt-neg-in 97.5%

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

      10. metadata-eval 97.5%

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

      11. +-commutative 97.5%

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

   7. Simplified 97.5%

      \[\leadsto \color{blue}{x \cdot \frac{-6}{\sqrt{x} \cdot -4 - \left(1 + x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;6 \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-6}{\sqrt{x} \cdot -4 + \left(-1 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;6 \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.0) (* 6.0 (+ x -1.0)) 6.0))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 6.0 * (x + -1.0);
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = 6.0d0 * (x + (-1.0d0))
    else
        tmp = 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 6.0 * (x + -1.0);
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = 6.0 * (x + -1.0)
	else:
		tmp = 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(6.0 * Float64(x + -1.0));
	else
		tmp = 6.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = 6.0 * (x + -1.0);
	else
		tmp = 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[(6.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], 6.0]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 99.9%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 97.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \cdot \sqrt[3]{\frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right) \cdot \sqrt[3]{\frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right)} \cdot \left(x + -1\right) \]

      2. pow3 97.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right)}^{3}} \cdot \left(x + -1\right) \]

      3. fma-udef 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

      4. +-commutative 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}}}\right)}^{3} \cdot \left(x + -1\right) \]

      5. associate-+l+ 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{\color{blue}{x + \left(1 + 4 \cdot \sqrt{x}\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

      6. +-commutative 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{x + \color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

      7. fma-def 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{x + \color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

   6. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{6}{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)}}\right)}^{3}} \cdot \left(x + -1\right) \]
   7. Step-by-step derivation

      1. fma-udef 99.9%

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

   8. Applied egg-rr 97.6%

      \[\leadsto {\left(\sqrt[3]{\frac{6}{x + \color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

   9. Taylor expanded in x around 0 96.8%

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

   if 2 < x

   1. Initial program 99.1%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 94.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;6 \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.3% accurate, 11.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 6.0 * (x + -1.0);
	} 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 <= 1.0d0) then
        tmp = 6.0d0 * (x + (-1.0d0))
    else
        tmp = 6.0d0 - (6.0d0 / x)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 6.0 * (x + -1.0)
	else:
		tmp = 6.0 - (6.0 / x)
	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(6.0 / x));
	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 - (6.0 / x);
	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[(6.0 / x), $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. Step-by-step derivation

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 97.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \cdot \sqrt[3]{\frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right) \cdot \sqrt[3]{\frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right)} \cdot \left(x + -1\right) \]

      2. pow3 97.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right)}^{3}} \cdot \left(x + -1\right) \]

      3. fma-udef 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

      4. +-commutative 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}}}\right)}^{3} \cdot \left(x + -1\right) \]

      5. associate-+l+ 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{\color{blue}{x + \left(1 + 4 \cdot \sqrt{x}\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

      6. +-commutative 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{x + \color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

      7. fma-def 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{6}{x + \color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

   6. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{6}{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)}}\right)}^{3}} \cdot \left(x + -1\right) \]
   7. Step-by-step derivation

      1. fma-udef 99.9%

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

   8. Applied egg-rr 97.6%

      \[\leadsto {\left(\sqrt[3]{\frac{6}{x + \color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}}}\right)}^{3} \cdot \left(x + -1\right) \]

   9. Taylor expanded in x around 0 96.8%

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

   if 1 < x

   1. Initial program 99.1%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 94.7%

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

   6. Taylor expanded in x around 0 94.7%

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

      1. associate-*r/ 94.7%

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

      2. metadata-eval 94.7%

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

   8. Simplified 94.7%

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

4. Final simplification 95.7%

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

Alternative 5: 95.3% accurate, 18.8× 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. Step-by-step derivation

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 96.8%

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

   if 1 < x

   1. Initial program 99.1%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 94.7%

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

4. Final simplification 95.7%

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

Alternative 6: 48.6% 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 99.5%

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

   1. associate-*l/ 99.9%

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

   2. sub-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. fma-def 99.9%

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

   5. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 46.6%

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

6. Final simplification 46.6%

   \[\leadsto -6 \]
7. Add Preprocessing

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 2024027 
(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)))))