Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 8.3s

Alternatives: 7

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: 99.7% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.9%

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

   2. *-commutative 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

   5. associate-+l+ 99.9%

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

   6. +-commutative 99.9%

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

   7. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.038:\\ \;\;\;\;-6 + \sqrt{x} \cdot 24\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \frac{6}{x + 4 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.038)
   (+ -6.0 (* (sqrt x) 24.0))
   (* (+ x -1.0) (/ 6.0 (+ x (* 4.0 (sqrt x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.038) {
		tmp = -6.0 + (sqrt(x) * 24.0);
	} else {
		tmp = (x + -1.0) * (6.0 / (x + (4.0 * sqrt(x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.038d0) then
        tmp = (-6.0d0) + (sqrt(x) * 24.0d0)
    else
        tmp = (x + (-1.0d0)) * (6.0d0 / (x + (4.0d0 * sqrt(x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.038) {
		tmp = -6.0 + (Math.sqrt(x) * 24.0);
	} else {
		tmp = (x + -1.0) * (6.0 / (x + (4.0 * Math.sqrt(x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.038:
		tmp = -6.0 + (math.sqrt(x) * 24.0)
	else:
		tmp = (x + -1.0) * (6.0 / (x + (4.0 * math.sqrt(x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.038)
		tmp = Float64(-6.0 + Float64(sqrt(x) * 24.0));
	else
		tmp = Float64(Float64(x + -1.0) * Float64(6.0 / Float64(x + Float64(4.0 * sqrt(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.038)
		tmp = -6.0 + (sqrt(x) * 24.0);
	else
		tmp = (x + -1.0) * (6.0 / (x + (4.0 * sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.038], N[(-6.0 + N[(N[Sqrt[x], $MachinePrecision] * 24.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + -1.0), $MachinePrecision] * N[(6.0 / N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0379999999999999991

   1. Initial program 99.9%

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

      1. flip-+ 99.9%

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

      2. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. fma-define 100.0%

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

      3. *-commutative 100.0%

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

      4. fma-define 100.0%

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

      5. associate-+l+ 100.0%

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

      6. *-commutative 100.0%

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

      7. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto -6 + \sqrt{x} \cdot \color{blue}{24} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{-6 + \sqrt{x} \cdot 24} \]

   if 0.0379999999999999991 < x

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/l* 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

      5. associate-+l+ 99.8%

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

      6. +-commutative 99.8%

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

      7. fma-define 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 95.4%

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

      1. *-commutative 95.4%

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

   7. Simplified 95.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.038:\\ \;\;\;\;-6 + \sqrt{x} \cdot 24\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \frac{6}{x + 4 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x + -1.0), $MachinePrecision] * N[(6.0 / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 99.8%

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

   2. associate-/l* 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

   5. associate-+l+ 99.9%

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

   6. +-commutative 99.9%

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

   7. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

   1. associate-+r+ 99.9%

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

   2. *-commutative 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.07:\\ \;\;\;\;-6 + \sqrt{x} \cdot 24\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 + \frac{4}{\sqrt{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.07) (+ -6.0 (* (sqrt x) 24.0)) (/ 6.0 (+ 1.0 (/ 4.0 (sqrt x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.07) {
		tmp = -6.0 + (sqrt(x) * 24.0);
	} else {
		tmp = 6.0 / (1.0 + (4.0 / sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.07d0) then
        tmp = (-6.0d0) + (sqrt(x) * 24.0d0)
    else
        tmp = 6.0d0 / (1.0d0 + (4.0d0 / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.07) {
		tmp = -6.0 + (Math.sqrt(x) * 24.0);
	} else {
		tmp = 6.0 / (1.0 + (4.0 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.07:
		tmp = -6.0 + (math.sqrt(x) * 24.0)
	else:
		tmp = 6.0 / (1.0 + (4.0 / math.sqrt(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.07)
		tmp = Float64(-6.0 + Float64(sqrt(x) * 24.0));
	else
		tmp = Float64(6.0 / Float64(1.0 + Float64(4.0 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.07)
		tmp = -6.0 + (sqrt(x) * 24.0);
	else
		tmp = 6.0 / (1.0 + (4.0 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.07], N[(-6.0 + N[(N[Sqrt[x], $MachinePrecision] * 24.0), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 + N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.070000000000000007

   1. Initial program 99.9%

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

      1. flip-+ 99.9%

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

      2. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. fma-define 100.0%

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

      3. *-commutative 100.0%

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

      4. fma-define 100.0%

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

      5. associate-+l+ 100.0%

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

      6. *-commutative 100.0%

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

      7. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto -6 + \sqrt{x} \cdot \color{blue}{24} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{-6 + \sqrt{x} \cdot 24} \]

   if 0.070000000000000007 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 95.3%

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

      1. *-commutative 95.3%

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

   5. Simplified 95.3%

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

   6. Taylor expanded in x around inf 95.4%

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

      1. *-commutative 95.4%

         \[\leadsto \frac{6}{1 + \color{blue}{\sqrt{\frac{1}{x}} \cdot 4}} \]

   8. Simplified 95.4%

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

      1. add-exp-log 95.3%

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

      2. log-div 95.4%

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

      3. log1p-define 95.4%

         \[\leadsto e^{\log 6 - \color{blue}{\mathsf{log1p}\left(\sqrt{\frac{1}{x}} \cdot 4\right)}} \]

      4. *-commutative 95.4%

         \[\leadsto e^{\log 6 - \mathsf{log1p}\left(\color{blue}{4 \cdot \sqrt{\frac{1}{x}}}\right)} \]

      5. sqrt-div 95.4%

         \[\leadsto e^{\log 6 - \mathsf{log1p}\left(4 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)} \]

      6. metadata-eval 95.4%

         \[\leadsto e^{\log 6 - \mathsf{log1p}\left(4 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)} \]

      7. un-div-inv 95.4%

         \[\leadsto e^{\log 6 - \mathsf{log1p}\left(\color{blue}{\frac{4}{\sqrt{x}}}\right)} \]

   10. Applied egg-rr 95.4%

       \[\leadsto \color{blue}{e^{\log 6 - \mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right)}} \]
   11. Step-by-step derivation

       1. exp-diff 95.4%

          \[\leadsto \color{blue}{\frac{e^{\log 6}}{e^{\mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right)}}} \]

       2. rem-exp-log 95.4%

          \[\leadsto \frac{\color{blue}{6}}{e^{\mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right)}} \]

       3. log1p-undefine 95.4%

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

       4. rem-exp-log 95.4%

          \[\leadsto \frac{6}{\color{blue}{1 + \frac{4}{\sqrt{x}}}} \]

   12. Simplified 95.4%

       \[\leadsto \color{blue}{\frac{6}{1 + \frac{4}{\sqrt{x}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.06:\\ \;\;\;\;-6 + \sqrt{x} \cdot 24\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-24}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.06) (+ -6.0 (* (sqrt x) 24.0)) (+ 6.0 (/ -24.0 (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.06) {
		tmp = -6.0 + (sqrt(x) * 24.0);
	} else {
		tmp = 6.0 + (-24.0 / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.06d0) then
        tmp = (-6.0d0) + (sqrt(x) * 24.0d0)
    else
        tmp = 6.0d0 + ((-24.0d0) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.06) {
		tmp = -6.0 + (Math.sqrt(x) * 24.0);
	} else {
		tmp = 6.0 + (-24.0 / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.06:
		tmp = -6.0 + (math.sqrt(x) * 24.0)
	else:
		tmp = 6.0 + (-24.0 / math.sqrt(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.06)
		tmp = Float64(-6.0 + Float64(sqrt(x) * 24.0));
	else
		tmp = Float64(6.0 + Float64(-24.0 / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.06)
		tmp = -6.0 + (sqrt(x) * 24.0);
	else
		tmp = 6.0 + (-24.0 / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.06], N[(-6.0 + N[(N[Sqrt[x], $MachinePrecision] * 24.0), $MachinePrecision]), $MachinePrecision], N[(6.0 + N[(-24.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.059999999999999998

   1. Initial program 99.9%

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

      1. flip-+ 99.9%

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

      2. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. fma-define 100.0%

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

      3. *-commutative 100.0%

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

      4. fma-define 100.0%

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

      5. associate-+l+ 100.0%

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

      6. *-commutative 100.0%

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

      7. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto -6 + \sqrt{x} \cdot \color{blue}{24} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{-6 + \sqrt{x} \cdot 24} \]

   if 0.059999999999999998 < x

   1. Initial program 99.6%

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

      1. flip-+ 58.4%

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

      2. associate-/r/ 58.4%

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

   4. Applied egg-rr 58.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 57.0%

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

      2. fma-define 57.0%

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

      3. *-commutative 57.0%

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

      4. fma-define 57.0%

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

      5. associate-+l+ 57.0%

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

      6. *-commutative 57.0%

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

      7. +-commutative 57.0%

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

   6. Simplified 57.0%

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

   7. Taylor expanded in x around inf 95.0%

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

      1. distribute-rgt-in 95.0%

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

      2. metadata-eval 95.0%

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

      3. *-commutative 95.0%

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

      4. associate-*l* 95.0%

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

      5. metadata-eval 95.0%

         \[\leadsto 6 + \sqrt{\frac{1}{x}} \cdot \color{blue}{-24} \]

   9. Simplified 95.0%

      \[\leadsto \color{blue}{6 + \sqrt{\frac{1}{x}} \cdot -24} \]
   10. Step-by-step derivation

       1. *-commutative 95.0%

          \[\leadsto 6 + \color{blue}{-24 \cdot \sqrt{\frac{1}{x}}} \]

       2. sqrt-div 95.0%

          \[\leadsto 6 + -24 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}} \]

       3. metadata-eval 95.0%

          \[\leadsto 6 + -24 \cdot \frac{\color{blue}{1}}{\sqrt{x}} \]

       4. un-div-inv 95.0%

          \[\leadsto 6 + \color{blue}{\frac{-24}{\sqrt{x}}} \]

   11. Applied egg-rr 95.0%

       \[\leadsto 6 + \color{blue}{\frac{-24}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 52.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ -6 + \sqrt{x} \cdot 24 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ -6.0 (* (sqrt x) 24.0)))

C

double code(double x) {
	return -6.0 + (sqrt(x) * 24.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-6.0d0) + (sqrt(x) * 24.0d0)
end function

Java

public static double code(double x) {
	return -6.0 + (Math.sqrt(x) * 24.0);
}

Python

def code(x):
	return -6.0 + (math.sqrt(x) * 24.0)

Julia

function code(x)
	return Float64(-6.0 + Float64(sqrt(x) * 24.0))
end

MATLAB

function tmp = code(x)
	tmp = -6.0 + (sqrt(x) * 24.0);
end

Wolfram

code[x_] := N[(-6.0 + N[(N[Sqrt[x], $MachinePrecision] * 24.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. flip-+ 80.1%

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

   2. associate-/r/ 80.1%

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

4. Applied egg-rr 80.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]
5. Step-by-step derivation

   1. associate-*l/ 79.5%

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

   2. fma-define 79.5%

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

   3. *-commutative 79.5%

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

   4. fma-define 79.5%

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

   5. associate-+l+ 79.5%

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

   6. *-commutative 79.5%

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

   7. +-commutative 79.5%

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

6. Simplified 79.5%

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

7. Taylor expanded in x around 0 55.8%

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

   1. distribute-rgt-in 55.8%

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

   2. metadata-eval 55.8%

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

   3. *-commutative 55.8%

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

   4. associate-*l* 55.8%

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

   5. metadata-eval 55.8%

      \[\leadsto -6 + \sqrt{x} \cdot \color{blue}{24} \]

9. Simplified 55.8%

   \[\leadsto \color{blue}{-6 + \sqrt{x} \cdot 24} \]
10. Add Preprocessing

Alternative 7: 4.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x \cdot 2.25} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* x 2.25)))

C

double code(double x) {
	return sqrt((x * 2.25));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x * 2.25d0))
end function

Java

public static double code(double x) {
	return Math.sqrt((x * 2.25));
}

Python

def code(x):
	return math.sqrt((x * 2.25))

Julia

function code(x)
	return sqrt(Float64(x * 2.25))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x * 2.25));
end

Wolfram

code[x_] := N[Sqrt[N[(x * 2.25), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf 46.6%

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

   1. *-commutative 46.6%

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

5. Simplified 46.6%

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

6. Taylor expanded in x around inf 46.4%

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

   1. *-commutative 46.4%

      \[\leadsto \frac{6}{1 + \color{blue}{\sqrt{\frac{1}{x}} \cdot 4}} \]

8. Simplified 46.4%

   \[\leadsto \color{blue}{\frac{6}{1 + \sqrt{\frac{1}{x}} \cdot 4}} \]

9. Taylor expanded in x around 0 4.5%

   \[\leadsto \color{blue}{1.5 \cdot \sqrt{x}} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 4.5%

       \[\leadsto \color{blue}{\sqrt{1.5 \cdot \sqrt{x}} \cdot \sqrt{1.5 \cdot \sqrt{x}}} \]

    2. sqrt-unprod 4.5%

       \[\leadsto \color{blue}{\sqrt{\left(1.5 \cdot \sqrt{x}\right) \cdot \left(1.5 \cdot \sqrt{x}\right)}} \]

    3. *-commutative 4.5%

       \[\leadsto \sqrt{\color{blue}{\left(\sqrt{x} \cdot 1.5\right)} \cdot \left(1.5 \cdot \sqrt{x}\right)} \]

    4. *-commutative 4.5%

       \[\leadsto \sqrt{\left(\sqrt{x} \cdot 1.5\right) \cdot \color{blue}{\left(\sqrt{x} \cdot 1.5\right)}} \]

    5. swap-sqr 4.5%

       \[\leadsto \sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(1.5 \cdot 1.5\right)}} \]

    6. add-sqr-sqrt 4.5%

       \[\leadsto \sqrt{\color{blue}{x} \cdot \left(1.5 \cdot 1.5\right)} \]

    7. metadata-eval 4.5%

       \[\leadsto \sqrt{x \cdot \color{blue}{2.25}} \]

11. Applied egg-rr 4.5%

    \[\leadsto \color{blue}{\sqrt{x \cdot 2.25}} \]
12. 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 2024111 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

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

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