Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 7.0s

Alternatives: 8

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   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. metadata-eval 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/r/ 100.0%

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

   4. fma-udef 100.0%

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

   5. +-commutative 100.0%

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

   6. +-commutative 100.0%

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

   7. fma-udef 100.0%

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

   8. sub-neg 100.0%

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

   9. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. div-inv 99.9%

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

   2. frac-2neg 99.9%

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

   3. metadata-eval 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-neg-in 99.9%

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

   6. metadata-eval 99.9%

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

   7. *-rgt-identity 99.9%

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

   8. sub-neg 99.9%

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

   9. *-rgt-identity 99.9%

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

8. Applied egg-rr 99.9%

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

   1. associate-*r/ 100.0%

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

   2. *-commutative 100.0%

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

   3. neg-mul-1 100.0%

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

   4. neg-sub0 100.0%

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

   5. fma-udef 100.0%

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

   6. associate--r+ 100.0%

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

   7. neg-sub0 100.0%

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

   8. *-commutative 100.0%

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

   9. distribute-rgt-neg-in 100.0%

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

   10. metadata-eval 100.0%

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

   11. +-commutative 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 95.2% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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. metadata-eval 99.9%

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

   2. sub-neg 99.9%

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

   3. *-commutative 99.9%

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

   4. clear-num 99.7%

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

   5. un-div-inv 100.0%

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

   6. sub-neg 100.0%

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

   7. metadata-eval 100.0%

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

   8. div-inv 99.7%

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

   9. metadata-eval 99.7%

      \[\leadsto \frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right) \cdot \color{blue}{0.16666666666666666}} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right) \cdot 0.16666666666666666}} \]
7. Step-by-step derivation

   1. fma-udef 99.7%

      \[\leadsto \frac{x + -1}{\color{blue}{\left(4 \cdot \sqrt{x} + \left(x + 1\right)\right)} \cdot 0.16666666666666666} \]

   2. +-commutative 99.7%

      \[\leadsto \frac{x + -1}{\color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \cdot 0.16666666666666666} \]

   3. *-commutative 99.7%

      \[\leadsto \frac{x + -1}{\color{blue}{0.16666666666666666 \cdot \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}} \]

   4. associate-+l+ 99.7%

      \[\leadsto \frac{x + -1}{0.16666666666666666 \cdot \color{blue}{\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

   5. distribute-rgt-in 99.7%

      \[\leadsto \frac{x + -1}{\color{blue}{x \cdot 0.16666666666666666 + \left(1 + 4 \cdot \sqrt{x}\right) \cdot 0.16666666666666666}} \]

8. Applied egg-rr 99.7%

   \[\leadsto \frac{x + -1}{\color{blue}{x \cdot 0.16666666666666666 + \left(1 + 4 \cdot \sqrt{x}\right) \cdot 0.16666666666666666}} \]

9. Taylor expanded in x around 0 95.5%

   \[\leadsto \frac{x + -1}{x \cdot 0.16666666666666666 + \color{blue}{0.16666666666666666}} \]
10. Step-by-step derivation

    1. *-un-lft-identity 95.5%

       \[\leadsto \frac{\color{blue}{1 \cdot \left(x + -1\right)}}{x \cdot 0.16666666666666666 + 0.16666666666666666} \]

    2. distribute-lft1-in 95.5%

       \[\leadsto \frac{1 \cdot \left(x + -1\right)}{\color{blue}{\left(x + 1\right) \cdot 0.16666666666666666}} \]

    3. times-frac 95.5%

       \[\leadsto \color{blue}{\frac{1}{x + 1} \cdot \frac{x + -1}{0.16666666666666666}} \]

    4. div-inv 95.5%

       \[\leadsto \frac{1}{x + 1} \cdot \color{blue}{\left(\left(x + -1\right) \cdot \frac{1}{0.16666666666666666}\right)} \]

    5. metadata-eval 95.5%

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

11. Applied egg-rr 95.5%

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

12. Final simplification 95.5%

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

Alternative 3: 95.5% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 96.9%

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

   if 0.5 < x

   1. Initial program 99.6%

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

      1. associate-*l/ 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-def 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 94.2%

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

   6. Taylor expanded in x around 0 94.3%

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

      1. associate-*r/ 94.3%

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

      2. metadata-eval 94.3%

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

   8. Simplified 94.3%

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

4. Final simplification 95.7%

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

Alternative 4: 95.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) -6.0 (- 6.0 (/ 12.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0;
	} else {
		tmp = 6.0 - (12.0 / 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
    else
        tmp = 6.0d0 - (12.0d0 / x)
    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 - (12.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0
	else:
		tmp = 6.0 - (12.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = -6.0;
	else
		tmp = Float64(6.0 - Float64(12.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0;
	else
		tmp = 6.0 - (12.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], -6.0, N[(6.0 - N[(12.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 96.9%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. associate-*l/ 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-def 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. metadata-eval 99.8%

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

      2. sub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. clear-num 99.4%

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

      5. un-div-inv 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. div-inv 99.4%

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

      9. metadata-eval 99.4%

         \[\leadsto \frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right) \cdot \color{blue}{0.16666666666666666}} \]

   6. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right) \cdot 0.16666666666666666}} \]
   7. Step-by-step derivation

      1. fma-udef 99.4%

         \[\leadsto \frac{x + -1}{\color{blue}{\left(4 \cdot \sqrt{x} + \left(x + 1\right)\right)} \cdot 0.16666666666666666} \]

      2. +-commutative 99.4%

         \[\leadsto \frac{x + -1}{\color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \cdot 0.16666666666666666} \]

      3. *-commutative 99.4%

         \[\leadsto \frac{x + -1}{\color{blue}{0.16666666666666666 \cdot \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}} \]

      4. associate-+l+ 99.4%

         \[\leadsto \frac{x + -1}{0.16666666666666666 \cdot \color{blue}{\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      5. distribute-rgt-in 99.4%

         \[\leadsto \frac{x + -1}{\color{blue}{x \cdot 0.16666666666666666 + \left(1 + 4 \cdot \sqrt{x}\right) \cdot 0.16666666666666666}} \]

   8. Applied egg-rr 99.4%

      \[\leadsto \frac{x + -1}{\color{blue}{x \cdot 0.16666666666666666 + \left(1 + 4 \cdot \sqrt{x}\right) \cdot 0.16666666666666666}} \]

   9. Taylor expanded in x around 0 93.9%

      \[\leadsto \frac{x + -1}{x \cdot 0.16666666666666666 + \color{blue}{0.16666666666666666}} \]

   10. Taylor expanded in x around inf 94.3%

       \[\leadsto \color{blue}{6 - 12 \cdot \frac{1}{x}} \]
   11. Step-by-step derivation

       1. associate-*r/ 94.3%

          \[\leadsto 6 - \color{blue}{\frac{12 \cdot 1}{x}} \]

       2. metadata-eval 94.3%

          \[\leadsto 6 - \frac{\color{blue}{12}}{x} \]

   12. Simplified 94.3%

       \[\leadsto \color{blue}{6 - \frac{12}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;x \cdot 12 - 6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.0) (- (* x 12.0) 6.0) (- 6.0 (/ 12.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = (x * 12.0) - 6.0;
	} else {
		tmp = 6.0 - (12.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = (x * 12.0d0) - 6.0d0
    else
        tmp = 6.0d0 - (12.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = (x * 12.0) - 6.0;
	} else {
		tmp = 6.0 - (12.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = (x * 12.0) - 6.0
	else:
		tmp = 6.0 - (12.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(x * 12.0) - 6.0);
	else
		tmp = Float64(6.0 - Float64(12.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (x * 12.0) - 6.0;
	else
		tmp = 6.0 - (12.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[(N[(x * 12.0), $MachinePrecision] - 6.0), $MachinePrecision], N[(6.0 - N[(12.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. metadata-eval 100.0%

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

      2. sub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. clear-num 100.0%

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

      5. un-div-inv 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. div-inv 100.0%

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

      9. metadata-eval 100.0%

         \[\leadsto \frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right) \cdot \color{blue}{0.16666666666666666}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right) \cdot 0.16666666666666666}} \]
   7. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \frac{x + -1}{\color{blue}{\left(4 \cdot \sqrt{x} + \left(x + 1\right)\right)} \cdot 0.16666666666666666} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{x + -1}{\color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \cdot 0.16666666666666666} \]

      3. *-commutative 100.0%

         \[\leadsto \frac{x + -1}{\color{blue}{0.16666666666666666 \cdot \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}} \]

      4. associate-+l+ 100.0%

         \[\leadsto \frac{x + -1}{0.16666666666666666 \cdot \color{blue}{\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      5. distribute-rgt-in 100.0%

         \[\leadsto \frac{x + -1}{\color{blue}{x \cdot 0.16666666666666666 + \left(1 + 4 \cdot \sqrt{x}\right) \cdot 0.16666666666666666}} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \frac{x + -1}{\color{blue}{x \cdot 0.16666666666666666 + \left(1 + 4 \cdot \sqrt{x}\right) \cdot 0.16666666666666666}} \]

   9. Taylor expanded in x around 0 97.0%

      \[\leadsto \frac{x + -1}{x \cdot 0.16666666666666666 + \color{blue}{0.16666666666666666}} \]

   10. Taylor expanded in x around 0 97.0%

       \[\leadsto \color{blue}{12 \cdot x - 6} \]

   if 2 < x

   1. Initial program 99.6%

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

      1. associate-*l/ 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-def 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. metadata-eval 99.8%

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

      2. sub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. clear-num 99.4%

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

      5. un-div-inv 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. div-inv 99.4%

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

      9. metadata-eval 99.4%

         \[\leadsto \frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right) \cdot \color{blue}{0.16666666666666666}} \]

   6. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right) \cdot 0.16666666666666666}} \]
   7. Step-by-step derivation

      1. fma-udef 99.4%

         \[\leadsto \frac{x + -1}{\color{blue}{\left(4 \cdot \sqrt{x} + \left(x + 1\right)\right)} \cdot 0.16666666666666666} \]

      2. +-commutative 99.4%

         \[\leadsto \frac{x + -1}{\color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \cdot 0.16666666666666666} \]

      3. *-commutative 99.4%

         \[\leadsto \frac{x + -1}{\color{blue}{0.16666666666666666 \cdot \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}} \]

      4. associate-+l+ 99.4%

         \[\leadsto \frac{x + -1}{0.16666666666666666 \cdot \color{blue}{\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      5. distribute-rgt-in 99.4%

         \[\leadsto \frac{x + -1}{\color{blue}{x \cdot 0.16666666666666666 + \left(1 + 4 \cdot \sqrt{x}\right) \cdot 0.16666666666666666}} \]

   8. Applied egg-rr 99.4%

      \[\leadsto \frac{x + -1}{\color{blue}{x \cdot 0.16666666666666666 + \left(1 + 4 \cdot \sqrt{x}\right) \cdot 0.16666666666666666}} \]

   9. Taylor expanded in x around 0 93.9%

      \[\leadsto \frac{x + -1}{x \cdot 0.16666666666666666 + \color{blue}{0.16666666666666666}} \]

   10. Taylor expanded in x around inf 94.3%

       \[\leadsto \color{blue}{6 - 12 \cdot \frac{1}{x}} \]
   11. Step-by-step derivation

       1. associate-*r/ 94.3%

          \[\leadsto 6 - \color{blue}{\frac{12 \cdot 1}{x}} \]

       2. metadata-eval 94.3%

          \[\leadsto 6 - \frac{\color{blue}{12}}{x} \]

   12. Simplified 94.3%

       \[\leadsto \color{blue}{6 - \frac{12}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;x \cdot 12 - 6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.3% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \frac{x + -1}{0.16666666666666666 + x \cdot 0.16666666666666666} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ x -1.0) (+ 0.16666666666666666 (* x 0.16666666666666666))))

C

double code(double x) {
	return (x + -1.0) / (0.16666666666666666 + (x * 0.16666666666666666));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + (-1.0d0)) / (0.16666666666666666d0 + (x * 0.16666666666666666d0))
end function

Java

public static double code(double x) {
	return (x + -1.0) / (0.16666666666666666 + (x * 0.16666666666666666));
}

Python

def code(x):
	return (x + -1.0) / (0.16666666666666666 + (x * 0.16666666666666666))

Julia

function code(x)
	return Float64(Float64(x + -1.0) / Float64(0.16666666666666666 + Float64(x * 0.16666666666666666)))
end

MATLAB

function tmp = code(x)
	tmp = (x + -1.0) / (0.16666666666666666 + (x * 0.16666666666666666));
end

Wolfram

code[x_] := N[(N[(x + -1.0), $MachinePrecision] / N[(0.16666666666666666 + N[(x * 0.16666666666666666), $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%

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

   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. metadata-eval 99.9%

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

   2. sub-neg 99.9%

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

   3. *-commutative 99.9%

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

   4. clear-num 99.7%

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

   5. un-div-inv 100.0%

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

   6. sub-neg 100.0%

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

   7. metadata-eval 100.0%

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

   8. div-inv 99.7%

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

   9. metadata-eval 99.7%

      \[\leadsto \frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right) \cdot \color{blue}{0.16666666666666666}} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right) \cdot 0.16666666666666666}} \]
7. Step-by-step derivation

   1. fma-udef 99.7%

      \[\leadsto \frac{x + -1}{\color{blue}{\left(4 \cdot \sqrt{x} + \left(x + 1\right)\right)} \cdot 0.16666666666666666} \]

   2. +-commutative 99.7%

      \[\leadsto \frac{x + -1}{\color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \cdot 0.16666666666666666} \]

   3. *-commutative 99.7%

      \[\leadsto \frac{x + -1}{\color{blue}{0.16666666666666666 \cdot \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}} \]

   4. associate-+l+ 99.7%

      \[\leadsto \frac{x + -1}{0.16666666666666666 \cdot \color{blue}{\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

   5. distribute-rgt-in 99.7%

      \[\leadsto \frac{x + -1}{\color{blue}{x \cdot 0.16666666666666666 + \left(1 + 4 \cdot \sqrt{x}\right) \cdot 0.16666666666666666}} \]

8. Applied egg-rr 99.7%

   \[\leadsto \frac{x + -1}{\color{blue}{x \cdot 0.16666666666666666 + \left(1 + 4 \cdot \sqrt{x}\right) \cdot 0.16666666666666666}} \]

9. Taylor expanded in x around 0 95.5%

   \[\leadsto \frac{x + -1}{x \cdot 0.16666666666666666 + \color{blue}{0.16666666666666666}} \]

10. Final simplification 95.5%

    \[\leadsto \frac{x + -1}{0.16666666666666666 + x \cdot 0.16666666666666666} \]
11. Add Preprocessing

Alternative 7: 95.5% 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 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 96.9%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. associate-*l/ 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-def 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 94.3%

      \[\leadsto \color{blue}{6} \]
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 8: 48.2% 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%

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

   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 52.6%

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

6. Final simplification 52.6%

   \[\leadsto -6 \]
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 2024021 
(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)))))