Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 10.8s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.9%

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

   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 99.8%

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

   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 99.7%

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

   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. distribute-rgt-in 99.7%

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

8. Applied egg-rr 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. distribute-rgt-out 99.7%

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

   3. times-frac 99.9%

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

   4. metadata-eval 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.45:\\ \;\;\;\;\frac{x + -1}{0.16666666666666666 + \sqrt{x} \cdot 0.6666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot x}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.45)
   (/ (+ x -1.0) (+ 0.16666666666666666 (* (sqrt x) 0.6666666666666666)))
   (/ (* 6.0 x) (+ (+ x 1.0) (* 4.0 (sqrt x))))))

C

double code(double x) {
	double tmp;
	if (x <= 3.45) {
		tmp = (x + -1.0) / (0.16666666666666666 + (sqrt(x) * 0.6666666666666666));
	} else {
		tmp = (6.0 * x) / ((x + 1.0) + (4.0 * sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.45d0) then
        tmp = (x + (-1.0d0)) / (0.16666666666666666d0 + (sqrt(x) * 0.6666666666666666d0))
    else
        tmp = (6.0d0 * x) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.45) {
		tmp = (x + -1.0) / (0.16666666666666666 + (Math.sqrt(x) * 0.6666666666666666));
	} else {
		tmp = (6.0 * x) / ((x + 1.0) + (4.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.45:
		tmp = (x + -1.0) / (0.16666666666666666 + (math.sqrt(x) * 0.6666666666666666))
	else:
		tmp = (6.0 * x) / ((x + 1.0) + (4.0 * math.sqrt(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.45)
		tmp = Float64(Float64(x + -1.0) / Float64(0.16666666666666666 + Float64(sqrt(x) * 0.6666666666666666)));
	else
		tmp = Float64(Float64(6.0 * x) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.45)
		tmp = (x + -1.0) / (0.16666666666666666 + (sqrt(x) * 0.6666666666666666));
	else
		tmp = (6.0 * x) / ((x + 1.0) + (4.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.45], N[(N[(x + -1.0), $MachinePrecision] / N[(0.16666666666666666 + N[(N[Sqrt[x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 * x), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4500000000000002

   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. 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 99.9%

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

      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 99.9%

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

      9. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. distribute-rgt-in 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \frac{x + -1}{\left(x + 1\right) \cdot 0.16666666666666666 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(4 \cdot \sqrt{x}\right) \cdot 0.16666666666666666\right)\right)}} \]

      2. expm1-udef 99.7%

         \[\leadsto \frac{x + -1}{\left(x + 1\right) \cdot 0.16666666666666666 + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(4 \cdot \sqrt{x}\right) \cdot 0.16666666666666666\right)} - 1\right)}} \]

      3. associate-*l* 99.7%

         \[\leadsto \frac{x + -1}{\left(x + 1\right) \cdot 0.16666666666666666 + \left(e^{\mathsf{log1p}\left(\color{blue}{4 \cdot \left(\sqrt{x} \cdot 0.16666666666666666\right)}\right)} - 1\right)} \]

   10. Applied egg-rr 99.7%

       \[\leadsto \frac{x + -1}{\left(x + 1\right) \cdot 0.16666666666666666 + \color{blue}{\left(e^{\mathsf{log1p}\left(4 \cdot \left(\sqrt{x} \cdot 0.16666666666666666\right)\right)} - 1\right)}} \]
   11. Step-by-step derivation

       1. expm1-def 99.9%

          \[\leadsto \frac{x + -1}{\left(x + 1\right) \cdot 0.16666666666666666 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(4 \cdot \left(\sqrt{x} \cdot 0.16666666666666666\right)\right)\right)}} \]

       2. expm1-log1p 99.9%

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

       3. *-commutative 99.9%

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

       4. associate-*l* 99.9%

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

       5. metadata-eval 99.9%

          \[\leadsto \frac{x + -1}{\left(x + 1\right) \cdot 0.16666666666666666 + \sqrt{x} \cdot \color{blue}{0.6666666666666666}} \]

   12. Simplified 99.9%

       \[\leadsto \frac{x + -1}{\left(x + 1\right) \cdot 0.16666666666666666 + \color{blue}{\sqrt{x} \cdot 0.6666666666666666}} \]

   13. Taylor expanded in x around 0 98.0%

       \[\leadsto \frac{x + -1}{\color{blue}{0.16666666666666666} + \sqrt{x} \cdot 0.6666666666666666} \]

   if 3.4500000000000002 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 98.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.45:\\ \;\;\;\;\frac{x + -1}{0.16666666666666666 + \sqrt{x} \cdot 0.6666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot x}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 18:\\ \;\;\;\;\frac{x + -1}{0.16666666666666666 + \sqrt{x} \cdot 0.6666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{1}{\frac{x + -1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 18.0)
   (/ (+ x -1.0) (+ 0.16666666666666666 (* (sqrt x) 0.6666666666666666)))
   (/ 6.0 (/ 1.0 (/ (+ x -1.0) x)))))

C

double code(double x) {
	double tmp;
	if (x <= 18.0) {
		tmp = (x + -1.0) / (0.16666666666666666 + (sqrt(x) * 0.6666666666666666));
	} else {
		tmp = 6.0 / (1.0 / ((x + -1.0) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 18.0d0) then
        tmp = (x + (-1.0d0)) / (0.16666666666666666d0 + (sqrt(x) * 0.6666666666666666d0))
    else
        tmp = 6.0d0 / (1.0d0 / ((x + (-1.0d0)) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 18.0) {
		tmp = (x + -1.0) / (0.16666666666666666 + (Math.sqrt(x) * 0.6666666666666666));
	} else {
		tmp = 6.0 / (1.0 / ((x + -1.0) / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 18.0:
		tmp = (x + -1.0) / (0.16666666666666666 + (math.sqrt(x) * 0.6666666666666666))
	else:
		tmp = 6.0 / (1.0 / ((x + -1.0) / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 18.0)
		tmp = Float64(Float64(x + -1.0) / Float64(0.16666666666666666 + Float64(sqrt(x) * 0.6666666666666666)));
	else
		tmp = Float64(6.0 / Float64(1.0 / Float64(Float64(x + -1.0) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 18.0)
		tmp = (x + -1.0) / (0.16666666666666666 + (sqrt(x) * 0.6666666666666666));
	else
		tmp = 6.0 / (1.0 / ((x + -1.0) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 18.0], N[(N[(x + -1.0), $MachinePrecision] / N[(0.16666666666666666 + N[(N[Sqrt[x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 / N[(N[(x + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 18

   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. 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 99.9%

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

      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 99.9%

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

      9. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. distribute-rgt-in 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \frac{x + -1}{\left(x + 1\right) \cdot 0.16666666666666666 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(4 \cdot \sqrt{x}\right) \cdot 0.16666666666666666\right)\right)}} \]

      2. expm1-udef 99.7%

         \[\leadsto \frac{x + -1}{\left(x + 1\right) \cdot 0.16666666666666666 + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(4 \cdot \sqrt{x}\right) \cdot 0.16666666666666666\right)} - 1\right)}} \]

      3. associate-*l* 99.7%

         \[\leadsto \frac{x + -1}{\left(x + 1\right) \cdot 0.16666666666666666 + \left(e^{\mathsf{log1p}\left(\color{blue}{4 \cdot \left(\sqrt{x} \cdot 0.16666666666666666\right)}\right)} - 1\right)} \]

   10. Applied egg-rr 99.7%

       \[\leadsto \frac{x + -1}{\left(x + 1\right) \cdot 0.16666666666666666 + \color{blue}{\left(e^{\mathsf{log1p}\left(4 \cdot \left(\sqrt{x} \cdot 0.16666666666666666\right)\right)} - 1\right)}} \]
   11. Step-by-step derivation

       1. expm1-def 99.9%

          \[\leadsto \frac{x + -1}{\left(x + 1\right) \cdot 0.16666666666666666 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(4 \cdot \left(\sqrt{x} \cdot 0.16666666666666666\right)\right)\right)}} \]

       2. expm1-log1p 99.9%

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

       3. *-commutative 99.9%

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

       4. associate-*l* 99.9%

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

       5. metadata-eval 99.9%

          \[\leadsto \frac{x + -1}{\left(x + 1\right) \cdot 0.16666666666666666 + \sqrt{x} \cdot \color{blue}{0.6666666666666666}} \]

   12. Simplified 99.9%

       \[\leadsto \frac{x + -1}{\left(x + 1\right) \cdot 0.16666666666666666 + \color{blue}{\sqrt{x} \cdot 0.6666666666666666}} \]

   13. Taylor expanded in x around 0 98.0%

       \[\leadsto \frac{x + -1}{\color{blue}{0.16666666666666666} + \sqrt{x} \cdot 0.6666666666666666} \]

   if 18 < 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/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 96.0%

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

      1. associate-*l/ 95.8%

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

      2. associate-/l* 96.1%

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

   7. Applied egg-rr 96.1%

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

      1. clear-num 96.1%

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

      2. inv-pow 96.1%

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

   9. Applied egg-rr 96.1%

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

       1. unpow-1 96.1%

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

   11. Simplified 96.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 18:\\ \;\;\;\;\frac{x + -1}{0.16666666666666666 + \sqrt{x} \cdot 0.6666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{1}{\frac{x + -1}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. 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 99.9%

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

      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 99.9%

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

      9. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 95.4%

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

   8. Taylor expanded in x around 0 95.4%

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

   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/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 96.0%

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

      1. associate-*l/ 95.8%

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

      2. associate-/l* 96.1%

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

   7. Applied egg-rr 96.1%

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

      1. clear-num 96.1%

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

      2. inv-pow 96.1%

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

   9. Applied egg-rr 96.1%

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

       1. unpow-1 96.1%

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

   11. Simplified 96.1%

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

4. Final simplification 95.7%

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

Alternative 5: 95.5% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. 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 99.9%

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

      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 99.9%

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

      9. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 95.4%

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

   8. Taylor expanded in x around 0 95.4%

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

   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/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 96.0%

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

      1. associate-*l/ 95.8%

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

      2. associate-/l* 96.1%

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

   7. Applied egg-rr 96.1%

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

      1. div-inv 96.1%

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

      2. clear-num 96.1%

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

   9. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{6 \cdot \frac{x + -1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

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

Alternative 6: 95.5% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. 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 99.9%

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

      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 99.9%

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

      9. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 95.4%

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

   8. Taylor expanded in x around 0 95.4%

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

   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/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 96.0%

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

      1. associate-*l/ 95.8%

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

      2. associate-/l* 96.1%

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

   7. Applied egg-rr 96.1%

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

4. Final simplification 95.7%

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

Alternative 7: 95.5% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 99.9%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. 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 99.9%

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

      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 99.9%

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

      9. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 95.4%

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

      1. frac-2neg 95.4%

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

      2. div-inv 95.4%

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

      3. +-commutative 95.4%

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

      4. distribute-neg-in 95.4%

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

      5. metadata-eval 95.4%

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

      6. *-rgt-identity 95.4%

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

      7. sub-neg 95.4%

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

      8. *-rgt-identity 95.4%

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

      9. metadata-eval 95.4%

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

      10. metadata-eval 95.4%

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

   9. Applied egg-rr 95.4%

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

   if 2 < 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/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 96.1%

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

4. Final simplification 95.7%

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

Alternative 8: 95.5% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. 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 99.9%

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

      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 99.9%

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

      9. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 95.4%

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

      1. frac-2neg 95.4%

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

      2. div-inv 95.4%

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

      3. +-commutative 95.4%

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

      4. distribute-neg-in 95.4%

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

      5. metadata-eval 95.4%

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

      6. *-rgt-identity 95.4%

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

      7. sub-neg 95.4%

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

      8. *-rgt-identity 95.4%

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

      9. metadata-eval 95.4%

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

      10. metadata-eval 95.4%

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

   9. Applied egg-rr 95.4%

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

   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/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 96.0%

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

   6. Taylor expanded in x around 0 96.1%

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

      1. associate-*r/ 96.1%

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

      2. metadata-eval 96.1%

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

   8. Simplified 96.1%

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

4. Final simplification 95.7%

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

Alternative 9: 95.5% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. associate-*l/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. 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 99.9%

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

      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 99.9%

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

      9. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 95.4%

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

   8. Taylor expanded in x around 0 95.4%

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

   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/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 96.0%

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

   6. Taylor expanded in x around 0 96.1%

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

      1. associate-*r/ 96.1%

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

      2. metadata-eval 96.1%

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

   8. Simplified 96.1%

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

4. Final simplification 95.7%

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

Alternative 10: 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 95.4%

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

   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/ 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 96.1%

      \[\leadsto \color{blue}{6} \]
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 11: 48.8% 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 47.0%

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

6. Final simplification 47.0%

   \[\leadsto -6 \]
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 2024017 
(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)))))