Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 8.2s

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, 1.0× speedup

Math

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

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(6.0 * Float64(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[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / 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. 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. sub-neg 99.9%

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

   3. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 0.07:\\ \;\;\;\;-6 \cdot \left(\left(x + 1\right) - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x} \cdot \left(x + \left(1 - t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= x 0.07)
     (* -6.0 (- (+ x 1.0) t_0))
     (* (/ 6.0 x) (+ x (- 1.0 t_0))))))

C

double code(double x) {
	double t_0 = 4.0 * sqrt(x);
	double tmp;
	if (x <= 0.07) {
		tmp = -6.0 * ((x + 1.0) - t_0);
	} else {
		tmp = (6.0 / x) * (x + (1.0 - t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * sqrt(x)
    if (x <= 0.07d0) then
        tmp = (-6.0d0) * ((x + 1.0d0) - t_0)
    else
        tmp = (6.0d0 / x) * (x + (1.0d0 - t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 4.0 * Math.sqrt(x);
	double tmp;
	if (x <= 0.07) {
		tmp = -6.0 * ((x + 1.0) - t_0);
	} else {
		tmp = (6.0 / x) * (x + (1.0 - t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 4.0 * math.sqrt(x)
	tmp = 0
	if x <= 0.07:
		tmp = -6.0 * ((x + 1.0) - t_0)
	else:
		tmp = (6.0 / x) * (x + (1.0 - t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(4.0 * sqrt(x))
	tmp = 0.0
	if (x <= 0.07)
		tmp = Float64(-6.0 * Float64(Float64(x + 1.0) - t_0));
	else
		tmp = Float64(Float64(6.0 / x) * Float64(x + Float64(1.0 - t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 4.0 * sqrt(x);
	tmp = 0.0;
	if (x <= 0.07)
		tmp = -6.0 * ((x + 1.0) - t_0);
	else
		tmp = (6.0 / x) * (x + (1.0 - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.07], N[(-6.0 * N[(N[(x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 / x), $MachinePrecision] * N[(x + N[(1.0 - t$95$0), $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/ 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. pow2 99.9%

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

      9. *-commutative 99.9%

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

      10. *-commutative 99.9%

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

      11. swap-sqr 99.9%

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

      12. add-sqr-sqrt 99.9%

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

      13. metadata-eval 99.9%

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

      14. associate--l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 97.6%

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

      1. associate-+r- 97.6%

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

   7. Applied egg-rr 97.6%

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

   if 0.070000000000000007 < x

   1. Initial program 99.7%

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

      1. flip-+ 49.5%

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

      2. associate-/r/ 49.5%

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

      3. sub-neg 49.5%

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

      4. metadata-eval 49.5%

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

      5. +-commutative 49.5%

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

      6. distribute-lft-in 49.4%

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

      7. metadata-eval 49.4%

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

      8. pow2 49.4%

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

      9. *-commutative 49.4%

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

      10. *-commutative 49.4%

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

      11. swap-sqr 49.4%

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

      12. add-sqr-sqrt 49.4%

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

      13. metadata-eval 49.4%

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

      14. associate--l+ 49.4%

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

   4. Applied egg-rr 49.4%

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

   5. Taylor expanded in x around inf 95.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.07:\\ \;\;\;\;-6 \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x} \cdot \left(x + \left(1 - 4 \cdot \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 0.076:\\ \;\;\;\;-6 \cdot \left(\left(x + 1\right) - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot x}{\left(x + 1\right) + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= x 0.076)
     (* -6.0 (- (+ x 1.0) t_0))
     (/ (* 6.0 x) (+ (+ x 1.0) t_0)))))

C

double code(double x) {
	double t_0 = 4.0 * sqrt(x);
	double tmp;
	if (x <= 0.076) {
		tmp = -6.0 * ((x + 1.0) - t_0);
	} else {
		tmp = (6.0 * x) / ((x + 1.0) + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * sqrt(x)
    if (x <= 0.076d0) then
        tmp = (-6.0d0) * ((x + 1.0d0) - t_0)
    else
        tmp = (6.0d0 * x) / ((x + 1.0d0) + t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 4.0 * Math.sqrt(x);
	double tmp;
	if (x <= 0.076) {
		tmp = -6.0 * ((x + 1.0) - t_0);
	} else {
		tmp = (6.0 * x) / ((x + 1.0) + t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 4.0 * math.sqrt(x)
	tmp = 0
	if x <= 0.076:
		tmp = -6.0 * ((x + 1.0) - t_0)
	else:
		tmp = (6.0 * x) / ((x + 1.0) + t_0)
	return tmp

Julia

function code(x)
	t_0 = Float64(4.0 * sqrt(x))
	tmp = 0.0
	if (x <= 0.076)
		tmp = Float64(-6.0 * Float64(Float64(x + 1.0) - t_0));
	else
		tmp = Float64(Float64(6.0 * x) / Float64(Float64(x + 1.0) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 4.0 * sqrt(x);
	tmp = 0.0;
	if (x <= 0.076)
		tmp = -6.0 * ((x + 1.0) - t_0);
	else
		tmp = (6.0 * x) / ((x + 1.0) + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.076], N[(-6.0 * N[(N[(x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 * x), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0759999999999999981

   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/ 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. pow2 99.9%

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

      9. *-commutative 99.9%

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

      10. *-commutative 99.9%

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

      11. swap-sqr 99.9%

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

      12. add-sqr-sqrt 99.9%

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

      13. metadata-eval 99.9%

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

      14. associate--l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 97.6%

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

      1. associate-+r- 97.6%

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

   7. Applied egg-rr 97.6%

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

   if 0.0759999999999999981 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 95.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.076:\\ \;\;\;\;-6 \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot x}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.34:\\ \;\;\;\;-6 \cdot \left(x + \left(1 - 4 \cdot \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.34) (* -6.0 (+ x (- 1.0 (* 4.0 (sqrt x))))) 6.0))

C

double code(double x) {
	double tmp;
	if (x <= 0.34) {
		tmp = -6.0 * (x + (1.0 - (4.0 * sqrt(x))));
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.34d0) then
        tmp = (-6.0d0) * (x + (1.0d0 - (4.0d0 * sqrt(x))))
    else
        tmp = 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.34) {
		tmp = -6.0 * (x + (1.0 - (4.0 * Math.sqrt(x))));
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.34:
		tmp = -6.0 * (x + (1.0 - (4.0 * math.sqrt(x))))
	else:
		tmp = 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.34)
		tmp = Float64(-6.0 * Float64(x + Float64(1.0 - Float64(4.0 * sqrt(x)))));
	else
		tmp = 6.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.34)
		tmp = -6.0 * (x + (1.0 - (4.0 * sqrt(x))));
	else
		tmp = 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.34], N[(-6.0 * N[(x + N[(1.0 - N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 6.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.340000000000000024

   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/ 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. pow2 99.9%

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

      9. *-commutative 99.9%

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

      10. *-commutative 99.9%

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

      11. swap-sqr 99.9%

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

      12. add-sqr-sqrt 99.9%

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

      13. metadata-eval 99.9%

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

      14. associate--l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 96.9%

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

   if 0.340000000000000024 < x

   1. Initial program 99.7%

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

      1. associate-/l* 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 94.8%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.34:\\ \;\;\;\;-6 \cdot \left(x + \left(1 - 4 \cdot \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.34:\\ \;\;\;\;-6 \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.34) (* -6.0 (- (+ x 1.0) (* 4.0 (sqrt x)))) 6.0))

C

double code(double x) {
	double tmp;
	if (x <= 0.34) {
		tmp = -6.0 * ((x + 1.0) - (4.0 * sqrt(x)));
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.34d0) then
        tmp = (-6.0d0) * ((x + 1.0d0) - (4.0d0 * sqrt(x)))
    else
        tmp = 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.34) {
		tmp = -6.0 * ((x + 1.0) - (4.0 * Math.sqrt(x)));
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.34:
		tmp = -6.0 * ((x + 1.0) - (4.0 * math.sqrt(x)))
	else:
		tmp = 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.34)
		tmp = Float64(-6.0 * Float64(Float64(x + 1.0) - Float64(4.0 * sqrt(x))));
	else
		tmp = 6.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.34)
		tmp = -6.0 * ((x + 1.0) - (4.0 * sqrt(x)));
	else
		tmp = 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.34], N[(-6.0 * N[(N[(x + 1.0), $MachinePrecision] - N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 6.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.340000000000000024

   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/ 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. pow2 99.9%

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

      9. *-commutative 99.9%

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

      10. *-commutative 99.9%

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

      11. swap-sqr 99.9%

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

      12. add-sqr-sqrt 99.9%

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

      13. metadata-eval 99.9%

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

      14. associate--l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 96.9%

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

      1. associate-+r- 96.9%

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

   7. Applied egg-rr 96.9%

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

   if 0.340000000000000024 < x

   1. Initial program 99.7%

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

      1. associate-/l* 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 94.8%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.34:\\ \;\;\;\;-6 \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.7% 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%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 94.0%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. associate-/l* 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 95.5%

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

4. Final simplification 94.7%

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

Alternative 7: 48.1% 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. 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. sub-neg 99.9%

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

   3. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 46.7%

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

6. Final simplification 46.7%

   \[\leadsto -6 \]
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 2024046 
(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)))))