Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.7%

Time: 8.6s

Alternatives: 10

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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x + -1}{1 + \left(x + \sqrt{x \cdot 16}\right)}}{0.16666666666666666} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (+ x -1.0) (+ 1.0 (+ x (sqrt (* x 16.0))))) 0.16666666666666666))

C

double code(double x) {
	return ((x + -1.0) / (1.0 + (x + sqrt((x * 16.0))))) / 0.16666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + (-1.0d0)) / (1.0d0 + (x + sqrt((x * 16.0d0))))) / 0.16666666666666666d0
end function

Java

public static double code(double x) {
	return ((x + -1.0) / (1.0 + (x + Math.sqrt((x * 16.0))))) / 0.16666666666666666;
}

Python

def code(x):
	return ((x + -1.0) / (1.0 + (x + math.sqrt((x * 16.0))))) / 0.16666666666666666

Julia

function code(x)
	return Float64(Float64(Float64(x + -1.0) / Float64(1.0 + Float64(x + sqrt(Float64(x * 16.0))))) / 0.16666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = ((x + -1.0) / (1.0 + (x + sqrt((x * 16.0))))) / 0.16666666666666666;
end

Wolfram

code[x_] := N[(N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + N[(x + N[Sqrt[N[(x * 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.16666666666666666), $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. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. +-commutative 99.9%

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

   4. associate-+l+ 99.9%

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

6. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. clear-num 99.7%

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

   3. un-div-inv 99.9%

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

   4. associate-+r+ 99.9%

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

   5. +-commutative 99.9%

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

   6. div-inv 99.7%

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

   7. metadata-eval 99.7%

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

   8. associate-/r* 99.9%

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

   9. +-commutative 99.9%

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

   10. associate-+r+ 99.9%

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

   11. +-commutative 99.9%

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

   12. fma-def 99.9%

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

8. Applied egg-rr 99.9%

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

   1. fma-udef 99.9%

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

   2. *-commutative 99.9%

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

   3. metadata-eval 99.9%

      \[\leadsto \frac{\frac{x + -1}{1 + \left(\sqrt{x} \cdot \color{blue}{\sqrt{16}} + x\right)}}{0.16666666666666666} \]

   4. sqrt-prod 99.9%

      \[\leadsto \frac{\frac{x + -1}{1 + \left(\color{blue}{\sqrt{x \cdot 16}} + x\right)}}{0.16666666666666666} \]

10. Applied egg-rr 99.9%

    \[\leadsto \frac{\frac{x + -1}{1 + \color{blue}{\left(\sqrt{x \cdot 16} + x\right)}}}{0.16666666666666666} \]

11. Final simplification 99.9%

    \[\leadsto \frac{\frac{x + -1}{1 + \left(x + \sqrt{x \cdot 16}\right)}}{0.16666666666666666} \]
12. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   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. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. +-commutative 99.9%

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

   4. associate-+l+ 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 95.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.98:\\ \;\;\;\;\left(x + -1\right) \cdot \left(6 + x \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{6}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.98) (* (+ x -1.0) (+ 6.0 (* x -6.0))) (- 6.0 (/ 6.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.98) {
		tmp = (x + -1.0) * (6.0 + (x * -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 <= 0.98d0) then
        tmp = (x + (-1.0d0)) * (6.0d0 + (x * (-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 <= 0.98) {
		tmp = (x + -1.0) * (6.0 + (x * -6.0));
	} else {
		tmp = 6.0 - (6.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.98:
		tmp = (x + -1.0) * (6.0 + (x * -6.0))
	else:
		tmp = 6.0 - (6.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.98)
		tmp = Float64(Float64(x + -1.0) * Float64(6.0 + Float64(x * -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 <= 0.98)
		tmp = (x + -1.0) * (6.0 + (x * -6.0));
	else
		tmp = 6.0 - (6.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.98], N[(N[(x + -1.0), $MachinePrecision] * N[(6.0 + N[(x * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 - N[(6.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.97999999999999998

   1. Initial program 99.9%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

      2. +-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 94.2%

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

   8. Taylor expanded in x around 0 94.2%

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

   if 0.97999999999999998 < 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/ 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 94.8%

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

   6. Taylor expanded in x around 0 94.9%

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

      1. associate-*r/ 94.9%

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

      2. metadata-eval 94.9%

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

   8. Simplified 94.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.98:\\ \;\;\;\;\left(x + -1\right) \cdot \left(6 + x \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{6}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. 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 94.8%

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

   if 0.5 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. 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 94.5%

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

Alternative 5: 95.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;x \cdot 6 - 6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.05) (- (* x 6.0) 6.0) (- 6.0 (/ 12.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.05) {
		tmp = (x * 6.0) - 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 <= 2.05d0) then
        tmp = (x * 6.0d0) - 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 <= 2.05) {
		tmp = (x * 6.0) - 6.0;
	} else {
		tmp = 6.0 - (12.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.05:
		tmp = (x * 6.0) - 6.0
	else:
		tmp = 6.0 - (12.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.05)
		tmp = Float64(Float64(x * 6.0) - 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 <= 2.05)
		tmp = (x * 6.0) - 6.0;
	else
		tmp = 6.0 - (12.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.05], N[(N[(x * 6.0), $MachinePrecision] - 6.0), $MachinePrecision], N[(6.0 - N[(12.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0499999999999998

   1. Initial program 99.9%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 93.6%

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

   6. Taylor expanded in x around 0 93.6%

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

   if 2.0499999999999998 < 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/ 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. fma-udef 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 95.4%

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

   8. Taylor expanded in x around inf 95.5%

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

      1. associate-*r/ 95.5%

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

      2. metadata-eval 95.5%

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

   10. Simplified 95.5%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;x \cdot 6 - 6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;\left(x + -1\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.05) (* (+ x -1.0) 6.0) (- 6.0 (/ 12.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.05) {
		tmp = (x + -1.0) * 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 <= 2.05d0) then
        tmp = (x + (-1.0d0)) * 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 <= 2.05) {
		tmp = (x + -1.0) * 6.0;
	} else {
		tmp = 6.0 - (12.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.05:
		tmp = (x + -1.0) * 6.0
	else:
		tmp = 6.0 - (12.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.05)
		tmp = Float64(Float64(x + -1.0) * 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 <= 2.05)
		tmp = (x + -1.0) * 6.0;
	else
		tmp = 6.0 - (12.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.05], N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision], N[(6.0 - N[(12.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0499999999999998

   1. Initial program 99.9%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 93.6%

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

   if 2.0499999999999998 < 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/ 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. fma-udef 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 95.4%

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

   8. Taylor expanded in x around inf 95.5%

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

      1. associate-*r/ 95.5%

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

      2. metadata-eval 95.5%

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

   10. Simplified 95.5%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;\left(x + -1\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.5% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x + -1.0), $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. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. +-commutative 99.9%

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

   4. associate-+l+ 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf 94.5%

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

8. Final simplification 94.5%

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

Alternative 8: 95.5% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / 0.16666666666666666), $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. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. +-commutative 99.9%

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

   4. associate-+l+ 99.9%

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

6. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. clear-num 99.7%

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

   3. un-div-inv 99.9%

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

   4. associate-+r+ 99.9%

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

   5. +-commutative 99.9%

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

   6. div-inv 99.7%

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

   7. metadata-eval 99.7%

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

   8. associate-/r* 99.9%

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

   9. +-commutative 99.9%

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

   10. associate-+r+ 99.9%

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

   11. +-commutative 99.9%

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

   12. fma-def 99.9%

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

8. Applied egg-rr 99.9%

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

9. Taylor expanded in x around inf 94.5%

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

10. Final simplification 94.5%

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

Alternative 9: 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 99.9%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 94.2%

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

   if 1 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. 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 94.9%

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

4. Final simplification 94.5%

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

Alternative 10: 48.3% 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 47.1%

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

6. Final simplification 47.1%

   \[\leadsto -6 \]
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 2024031 
(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)))))