Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.6% → 99.9%

Time: 10.4s

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.9%

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

   2. *-commutative 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. associate-+r+ 99.9%

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

   7. fma-undefine 99.9%

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

   8. +-commutative 99.9%

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

4. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

   2. add-sqr-sqrt 99.9%

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

   3. sqrt-unprod 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. swap-sqr 99.9%

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

   7. add-sqr-sqrt 99.9%

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

   8. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf 99.9%

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

   1. sqrt-div 99.9%

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

   2. metadata-eval 99.9%

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

   3. un-div-inv 99.9%

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

9. Applied egg-rr 99.9%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;\frac{\left(x + -1\right) \cdot 6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{1 + \left(x + \sqrt{x \cdot 16}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.4)
   (/ (* (+ x -1.0) 6.0) (+ 1.0 (* 4.0 (sqrt x))))
   (* 6.0 (/ x (+ 1.0 (+ x (sqrt (* x 16.0))))))))

C

double code(double x) {
	double tmp;
	if (x <= 3.4) {
		tmp = ((x + -1.0) * 6.0) / (1.0 + (4.0 * sqrt(x)));
	} else {
		tmp = 6.0 * (x / (1.0 + (x + sqrt((x * 16.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.4d0) then
        tmp = ((x + (-1.0d0)) * 6.0d0) / (1.0d0 + (4.0d0 * sqrt(x)))
    else
        tmp = 6.0d0 * (x / (1.0d0 + (x + sqrt((x * 16.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.4) {
		tmp = ((x + -1.0) * 6.0) / (1.0 + (4.0 * Math.sqrt(x)));
	} else {
		tmp = 6.0 * (x / (1.0 + (x + Math.sqrt((x * 16.0)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.4:
		tmp = ((x + -1.0) * 6.0) / (1.0 + (4.0 * math.sqrt(x)))
	else:
		tmp = 6.0 * (x / (1.0 + (x + math.sqrt((x * 16.0)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.4)
		tmp = Float64(Float64(Float64(x + -1.0) * 6.0) / Float64(1.0 + Float64(4.0 * sqrt(x))));
	else
		tmp = Float64(6.0 * Float64(x / Float64(1.0 + Float64(x + sqrt(Float64(x * 16.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.4)
		tmp = ((x + -1.0) * 6.0) / (1.0 + (4.0 * sqrt(x)));
	else
		tmp = 6.0 * (x / (1.0 + (x + sqrt((x * 16.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.4], N[(N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(x / N[(1.0 + N[(x + N[Sqrt[N[(x * 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999991

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 97.2%

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

   if 3.39999999999999991 < x

   1. Initial program 99.7%

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

      1. associate-/l* 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

      7. fma-undefine 99.9%

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

      8. +-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. fma-undefine 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. sqrt-unprod 99.9%

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

      4. *-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. swap-sqr 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 98.7%

      \[\leadsto \frac{\color{blue}{x}}{1 + \left(\sqrt{x \cdot 16} + x\right)} \cdot 6 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;\frac{\left(x + -1\right) \cdot 6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{1 + \left(x + \sqrt{x \cdot 16}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;\left(x + -1\right) \cdot \frac{6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{1 + \left(x + \sqrt{x \cdot 16}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.4)
   (* (+ x -1.0) (/ 6.0 (+ 1.0 (* 4.0 (sqrt x)))))
   (* 6.0 (/ x (+ 1.0 (+ x (sqrt (* x 16.0))))))))

C

double code(double x) {
	double tmp;
	if (x <= 3.4) {
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * sqrt(x))));
	} else {
		tmp = 6.0 * (x / (1.0 + (x + sqrt((x * 16.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.4d0) then
        tmp = (x + (-1.0d0)) * (6.0d0 / (1.0d0 + (4.0d0 * sqrt(x))))
    else
        tmp = 6.0d0 * (x / (1.0d0 + (x + sqrt((x * 16.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.4) {
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * Math.sqrt(x))));
	} else {
		tmp = 6.0 * (x / (1.0 + (x + Math.sqrt((x * 16.0)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.4:
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * math.sqrt(x))))
	else:
		tmp = 6.0 * (x / (1.0 + (x + math.sqrt((x * 16.0)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.4)
		tmp = Float64(Float64(x + -1.0) * Float64(6.0 / Float64(1.0 + Float64(4.0 * sqrt(x)))));
	else
		tmp = Float64(6.0 * Float64(x / Float64(1.0 + Float64(x + sqrt(Float64(x * 16.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.4)
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * sqrt(x))));
	else
		tmp = 6.0 * (x / (1.0 + (x + sqrt((x * 16.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.4], N[(N[(x + -1.0), $MachinePrecision] * N[(6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(x / N[(1.0 + N[(x + N[Sqrt[N[(x * 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999991

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

      7. fma-undefine 99.9%

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

      8. +-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 97.2%

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

   if 3.39999999999999991 < x

   1. Initial program 99.7%

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

      1. associate-/l* 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

      7. fma-undefine 99.9%

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

      8. +-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. fma-undefine 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. sqrt-unprod 99.9%

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

      4. *-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. swap-sqr 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 98.7%

      \[\leadsto \frac{\color{blue}{x}}{1 + \left(\sqrt{x \cdot 16} + x\right)} \cdot 6 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;\left(x + -1\right) \cdot \frac{6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{1 + \left(x + \sqrt{x \cdot 16}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\left(x + -1\right) \cdot \frac{6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-6}{-1 + {x}^{-0.5} \cdot -4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.0)
   (* (+ x -1.0) (/ 6.0 (+ 1.0 (* 4.0 (sqrt x)))))
   (/ -6.0 (+ -1.0 (* (pow x -0.5) -4.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * sqrt(x))));
	} else {
		tmp = -6.0 / (-1.0 + (pow(x, -0.5) * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.0d0) then
        tmp = (x + (-1.0d0)) * (6.0d0 / (1.0d0 + (4.0d0 * sqrt(x))))
    else
        tmp = (-6.0d0) / ((-1.0d0) + ((x ** (-0.5d0)) * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * Math.sqrt(x))));
	} else {
		tmp = -6.0 / (-1.0 + (Math.pow(x, -0.5) * -4.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.0:
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * math.sqrt(x))))
	else:
		tmp = -6.0 / (-1.0 + (math.pow(x, -0.5) * -4.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(Float64(x + -1.0) * Float64(6.0 / Float64(1.0 + Float64(4.0 * sqrt(x)))));
	else
		tmp = Float64(-6.0 / Float64(-1.0 + Float64((x ^ -0.5) * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.0)
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * sqrt(x))));
	else
		tmp = -6.0 / (-1.0 + ((x ^ -0.5) * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.0], N[(N[(x + -1.0), $MachinePrecision] * N[(6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-6.0 / N[(-1.0 + N[(N[Power[x, -0.5], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

      7. fma-undefine 99.9%

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

      8. +-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 97.2%

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

   if 4 < 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 99.7%

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

   4. Taylor expanded in x around -inf 0.0%

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

      1. sub-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow-1 0.0%

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

      4. metadata-eval 0.0%

         \[\leadsto \frac{-6}{\left(\sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      5. pow-sqr 0.0%

         \[\leadsto \frac{-6}{\left(\sqrt{\color{blue}{{x}^{-0.5} \cdot {x}^{-0.5}}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      6. rem-sqrt-square 0.0%

         \[\leadsto \frac{-6}{\left(\color{blue}{\left|{x}^{-0.5}\right|} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      7. rem-square-sqrt 0.0%

         \[\leadsto \frac{-6}{\left(\left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right| \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      8. fabs-sqr 0.0%

         \[\leadsto \frac{-6}{\left(\color{blue}{\left(\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}\right)} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      9. rem-square-sqrt 0.0%

         \[\leadsto \frac{-6}{\left(\color{blue}{{x}^{-0.5}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      10. unpow2 0.0%

          \[\leadsto \frac{-6}{\left({x}^{-0.5} \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot 4 + \left(-1\right)} \]

      11. rem-square-sqrt 98.6%

          \[\leadsto \frac{-6}{\left({x}^{-0.5} \cdot \color{blue}{-1}\right) \cdot 4 + \left(-1\right)} \]

      12. associate-*l* 98.6%

          \[\leadsto \frac{-6}{\color{blue}{{x}^{-0.5} \cdot \left(-1 \cdot 4\right)} + \left(-1\right)} \]

      13. metadata-eval 98.6%

          \[\leadsto \frac{-6}{{x}^{-0.5} \cdot \color{blue}{-4} + \left(-1\right)} \]

      14. metadata-eval 98.6%

          \[\leadsto \frac{-6}{{x}^{-0.5} \cdot -4 + \color{blue}{-1}} \]

   6. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{-6}{{x}^{-0.5} \cdot -4 + -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\left(x + -1\right) \cdot \frac{6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-6}{-1 + {x}^{-0.5} \cdot -4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{4 \cdot \sqrt{x} + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-6}{-1 + {x}^{-0.5} \cdot -4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (+ (* 4.0 (sqrt x)) (+ x 1.0)))
   (/ -6.0 (+ -1.0 (* (pow x -0.5) -4.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / ((4.0 * sqrt(x)) + (x + 1.0));
	} else {
		tmp = -6.0 / (-1.0 + (pow(x, -0.5) * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / ((4.0d0 * sqrt(x)) + (x + 1.0d0))
    else
        tmp = (-6.0d0) / ((-1.0d0) + ((x ** (-0.5d0)) * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / ((4.0 * Math.sqrt(x)) + (x + 1.0));
	} else {
		tmp = -6.0 / (-1.0 + (Math.pow(x, -0.5) * -4.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / ((4.0 * math.sqrt(x)) + (x + 1.0))
	else:
		tmp = -6.0 / (-1.0 + (math.pow(x, -0.5) * -4.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(Float64(4.0 * sqrt(x)) + Float64(x + 1.0)));
	else
		tmp = Float64(-6.0 / Float64(-1.0 + Float64((x ^ -0.5) * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / ((4.0 * sqrt(x)) + (x + 1.0));
	else
		tmp = -6.0 / (-1.0 + ((x ^ -0.5) * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-6.0 / N[(-1.0 + N[(N[Power[x, -0.5], $MachinePrecision] * -4.0), $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. Add Preprocessing

   3. Taylor expanded in x around 0 97.2%

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

   if 1 < 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 99.7%

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

   4. Taylor expanded in x around -inf 0.0%

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

      1. sub-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow-1 0.0%

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

      4. metadata-eval 0.0%

         \[\leadsto \frac{-6}{\left(\sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      5. pow-sqr 0.0%

         \[\leadsto \frac{-6}{\left(\sqrt{\color{blue}{{x}^{-0.5} \cdot {x}^{-0.5}}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      6. rem-sqrt-square 0.0%

         \[\leadsto \frac{-6}{\left(\color{blue}{\left|{x}^{-0.5}\right|} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      7. rem-square-sqrt 0.0%

         \[\leadsto \frac{-6}{\left(\left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right| \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      8. fabs-sqr 0.0%

         \[\leadsto \frac{-6}{\left(\color{blue}{\left(\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}\right)} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      9. rem-square-sqrt 0.0%

         \[\leadsto \frac{-6}{\left(\color{blue}{{x}^{-0.5}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      10. unpow2 0.0%

          \[\leadsto \frac{-6}{\left({x}^{-0.5} \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot 4 + \left(-1\right)} \]

      11. rem-square-sqrt 98.6%

          \[\leadsto \frac{-6}{\left({x}^{-0.5} \cdot \color{blue}{-1}\right) \cdot 4 + \left(-1\right)} \]

      12. associate-*l* 98.6%

          \[\leadsto \frac{-6}{\color{blue}{{x}^{-0.5} \cdot \left(-1 \cdot 4\right)} + \left(-1\right)} \]

      13. metadata-eval 98.6%

          \[\leadsto \frac{-6}{{x}^{-0.5} \cdot \color{blue}{-4} + \left(-1\right)} \]

      14. metadata-eval 98.6%

          \[\leadsto \frac{-6}{{x}^{-0.5} \cdot -4 + \color{blue}{-1}} \]

   6. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{-6}{{x}^{-0.5} \cdot -4 + -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{4 \cdot \sqrt{x} + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-6}{-1 + {x}^{-0.5} \cdot -4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-6}{-1 + {x}^{-0.5} \cdot -4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (+ 1.0 (* 4.0 (sqrt x))))
   (/ -6.0 (+ -1.0 (* (pow x -0.5) -4.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	} else {
		tmp = -6.0 / (-1.0 + (pow(x, -0.5) * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / (1.0d0 + (4.0d0 * sqrt(x)))
    else
        tmp = (-6.0d0) / ((-1.0d0) + ((x ** (-0.5d0)) * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (4.0 * Math.sqrt(x)));
	} else {
		tmp = -6.0 / (-1.0 + (Math.pow(x, -0.5) * -4.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / (1.0 + (4.0 * math.sqrt(x)))
	else:
		tmp = -6.0 / (-1.0 + (math.pow(x, -0.5) * -4.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(1.0 + Float64(4.0 * sqrt(x))));
	else
		tmp = Float64(-6.0 / Float64(-1.0 + Float64((x ^ -0.5) * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	else
		tmp = -6.0 / (-1.0 + ((x ^ -0.5) * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-6.0 / N[(-1.0 + N[(N[Power[x, -0.5], $MachinePrecision] * -4.0), $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. Add Preprocessing

   3. Taylor expanded in x around 0 97.1%

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

   if 1 < 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 99.7%

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

   4. Taylor expanded in x around -inf 0.0%

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

      1. sub-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow-1 0.0%

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

      4. metadata-eval 0.0%

         \[\leadsto \frac{-6}{\left(\sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      5. pow-sqr 0.0%

         \[\leadsto \frac{-6}{\left(\sqrt{\color{blue}{{x}^{-0.5} \cdot {x}^{-0.5}}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      6. rem-sqrt-square 0.0%

         \[\leadsto \frac{-6}{\left(\color{blue}{\left|{x}^{-0.5}\right|} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      7. rem-square-sqrt 0.0%

         \[\leadsto \frac{-6}{\left(\left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right| \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      8. fabs-sqr 0.0%

         \[\leadsto \frac{-6}{\left(\color{blue}{\left(\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}\right)} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      9. rem-square-sqrt 0.0%

         \[\leadsto \frac{-6}{\left(\color{blue}{{x}^{-0.5}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 4 + \left(-1\right)} \]

      10. unpow2 0.0%

          \[\leadsto \frac{-6}{\left({x}^{-0.5} \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot 4 + \left(-1\right)} \]

      11. rem-square-sqrt 98.6%

          \[\leadsto \frac{-6}{\left({x}^{-0.5} \cdot \color{blue}{-1}\right) \cdot 4 + \left(-1\right)} \]

      12. associate-*l* 98.6%

          \[\leadsto \frac{-6}{\color{blue}{{x}^{-0.5} \cdot \left(-1 \cdot 4\right)} + \left(-1\right)} \]

      13. metadata-eval 98.6%

          \[\leadsto \frac{-6}{{x}^{-0.5} \cdot \color{blue}{-4} + \left(-1\right)} \]

      14. metadata-eval 98.6%

          \[\leadsto \frac{-6}{{x}^{-0.5} \cdot -4 + \color{blue}{-1}} \]

   6. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{-6}{{x}^{-0.5} \cdot -4 + -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-6}{-1 + {x}^{-0.5} \cdot -4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 6 \cdot \frac{x + -1}{1 + \left(x + \sqrt{x \cdot 16}\right)} \end{array} \]

FPCore

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

C

double code(double x) {
	return 6.0 * ((x + -1.0) / (1.0 + (x + sqrt((x * 16.0)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0 * ((x + (-1.0d0)) / (1.0d0 + (x + sqrt((x * 16.0d0)))))
end function

Java

public static double code(double x) {
	return 6.0 * ((x + -1.0) / (1.0 + (x + Math.sqrt((x * 16.0)))));
}

Python

def code(x):
	return 6.0 * ((x + -1.0) / (1.0 + (x + math.sqrt((x * 16.0)))))

Julia

function code(x)
	return Float64(6.0 * Float64(Float64(x + -1.0) / Float64(1.0 + Float64(x + sqrt(Float64(x * 16.0))))))
end

MATLAB

function tmp = code(x)
	tmp = 6.0 * ((x + -1.0) / (1.0 + (x + sqrt((x * 16.0)))));
end

Wolfram

code[x_] := N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + N[(x + N[Sqrt[N[(x * 16.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. associate-/l* 99.9%

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

   2. *-commutative 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. associate-+r+ 99.9%

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

   7. fma-undefine 99.9%

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

   8. +-commutative 99.9%

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

4. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

   2. add-sqr-sqrt 99.9%

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

   3. sqrt-unprod 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. swap-sqr 99.9%

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

   7. add-sqr-sqrt 99.9%

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

   8. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto 6 \cdot \frac{x + -1}{1 + \left(x + \sqrt{x \cdot 16}\right)} \]
8. Add Preprocessing

Alternative 8: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + -1\right) \cdot \frac{6}{1 + \left(x + \sqrt{x \cdot 16}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (+ x -1.0) (/ 6.0 (+ 1.0 (+ x (sqrt (* x 16.0)))))))

C

double code(double x) {
	return (x + -1.0) * (6.0 / (1.0 + (x + sqrt((x * 16.0)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + (-1.0d0)) * (6.0d0 / (1.0d0 + (x + sqrt((x * 16.0d0)))))
end function

Java

public static double code(double x) {
	return (x + -1.0) * (6.0 / (1.0 + (x + Math.sqrt((x * 16.0)))));
}

Python

def code(x):
	return (x + -1.0) * (6.0 / (1.0 + (x + math.sqrt((x * 16.0)))))

Julia

function code(x)
	return Float64(Float64(x + -1.0) * Float64(6.0 / Float64(1.0 + Float64(x + sqrt(Float64(x * 16.0))))))
end

MATLAB

function tmp = code(x)
	tmp = (x + -1.0) * (6.0 / (1.0 + (x + sqrt((x * 16.0)))));
end

Wolfram

code[x_] := N[(N[(x + -1.0), $MachinePrecision] * N[(6.0 / N[(1.0 + N[(x + N[Sqrt[N[(x * 16.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 99.8%

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

   2. associate-/l* 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. associate-+r+ 99.9%

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

   7. fma-undefine 99.9%

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

   8. +-commutative 99.9%

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

4. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

   2. add-sqr-sqrt 99.9%

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

   3. sqrt-unprod 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. swap-sqr 99.9%

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

   7. add-sqr-sqrt 99.9%

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

   8. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \left(x + -1\right) \cdot \frac{6}{1 + \left(x + \sqrt{x \cdot 16}\right)} \]
8. Add Preprocessing

Alternative 9: 51.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (/ -6.0 (+ 1.0 (* 4.0 (sqrt x)))) (sqrt (* x 2.25))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	} else {
		tmp = sqrt((x * 2.25));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / (1.0d0 + (4.0d0 * sqrt(x)))
    else
        tmp = sqrt((x * 2.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (4.0 * Math.sqrt(x)));
	} else {
		tmp = Math.sqrt((x * 2.25));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / (1.0 + (4.0 * math.sqrt(x)))
	else:
		tmp = math.sqrt((x * 2.25))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(1.0 + Float64(4.0 * sqrt(x))));
	else
		tmp = sqrt(Float64(x * 2.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	else
		tmp = sqrt((x * 2.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(x * 2.25), $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. Add Preprocessing

   3. Taylor expanded in x around 0 97.1%

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

   if 1 < 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 0 7.0%

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

   4. Taylor expanded in x around inf 7.0%

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

      1. *-commutative 7.0%

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

   6. Simplified 7.0%

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

      1. pow1 7.0%

         \[\leadsto \color{blue}{{\left(\sqrt{x} \cdot 1.5\right)}^{1}} \]

      2. add-sqr-sqrt 7.0%

         \[\leadsto {\color{blue}{\left(\sqrt{\sqrt{x} \cdot 1.5} \cdot \sqrt{\sqrt{x} \cdot 1.5}\right)}}^{1} \]

      3. sqrt-unprod 7.0%

         \[\leadsto {\color{blue}{\left(\sqrt{\left(\sqrt{x} \cdot 1.5\right) \cdot \left(\sqrt{x} \cdot 1.5\right)}\right)}}^{1} \]

      4. swap-sqr 7.0%

         \[\leadsto {\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(1.5 \cdot 1.5\right)}}\right)}^{1} \]

      5. add-sqr-sqrt 7.0%

         \[\leadsto {\left(\sqrt{\color{blue}{x} \cdot \left(1.5 \cdot 1.5\right)}\right)}^{1} \]

      6. metadata-eval 7.0%

         \[\leadsto {\left(\sqrt{x \cdot \color{blue}{2.25}}\right)}^{1} \]

   8. Applied egg-rr 7.0%

      \[\leadsto \color{blue}{{\left(\sqrt{x \cdot 2.25}\right)}^{1}} \]
   9. Step-by-step derivation

      1. unpow1 7.0%

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

   10. Simplified 7.0%

       \[\leadsto \color{blue}{\sqrt{x \cdot 2.25}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-1.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (/ -1.5 (sqrt x)) (sqrt (* x 2.25))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -1.5 / sqrt(x);
	} else {
		tmp = sqrt((x * 2.25));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-1.5d0) / sqrt(x)
    else
        tmp = sqrt((x * 2.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -1.5 / Math.sqrt(x);
	} else {
		tmp = Math.sqrt((x * 2.25));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -1.5 / math.sqrt(x)
	else:
		tmp = math.sqrt((x * 2.25))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-1.5 / sqrt(x));
	else
		tmp = sqrt(Float64(x * 2.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -1.5 / sqrt(x);
	else
		tmp = sqrt((x * 2.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-1.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(x * 2.25), $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. Add Preprocessing

   3. Taylor expanded in x around 0 97.1%

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

   4. Taylor expanded in x around inf 7.0%

      \[\leadsto \color{blue}{-1.5 \cdot \sqrt{\frac{1}{x}}} \]
   5. Step-by-step derivation

      1. *-commutative 7.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot -1.5} \]

   6. Simplified 7.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot -1.5} \]
   7. Step-by-step derivation

      1. *-commutative 7.0%

         \[\leadsto \color{blue}{-1.5 \cdot \sqrt{\frac{1}{x}}} \]

      2. sqrt-div 7.0%

         \[\leadsto -1.5 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}} \]

      3. metadata-eval 7.0%

         \[\leadsto -1.5 \cdot \frac{\color{blue}{1}}{\sqrt{x}} \]

      4. un-div-inv 7.0%

         \[\leadsto \color{blue}{\frac{-1.5}{\sqrt{x}}} \]

   8. Applied egg-rr 7.0%

      \[\leadsto \color{blue}{\frac{-1.5}{\sqrt{x}}} \]

   if 1 < 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 0 7.0%

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

   4. Taylor expanded in x around inf 7.0%

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

      1. *-commutative 7.0%

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

   6. Simplified 7.0%

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

      1. pow1 7.0%

         \[\leadsto \color{blue}{{\left(\sqrt{x} \cdot 1.5\right)}^{1}} \]

      2. add-sqr-sqrt 7.0%

         \[\leadsto {\color{blue}{\left(\sqrt{\sqrt{x} \cdot 1.5} \cdot \sqrt{\sqrt{x} \cdot 1.5}\right)}}^{1} \]

      3. sqrt-unprod 7.0%

         \[\leadsto {\color{blue}{\left(\sqrt{\left(\sqrt{x} \cdot 1.5\right) \cdot \left(\sqrt{x} \cdot 1.5\right)}\right)}}^{1} \]

      4. swap-sqr 7.0%

         \[\leadsto {\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(1.5 \cdot 1.5\right)}}\right)}^{1} \]

      5. add-sqr-sqrt 7.0%

         \[\leadsto {\left(\sqrt{\color{blue}{x} \cdot \left(1.5 \cdot 1.5\right)}\right)}^{1} \]

      6. metadata-eval 7.0%

         \[\leadsto {\left(\sqrt{x \cdot \color{blue}{2.25}}\right)}^{1} \]

   8. Applied egg-rr 7.0%

      \[\leadsto \color{blue}{{\left(\sqrt{x \cdot 2.25}\right)}^{1}} \]
   9. Step-by-step derivation

      1. unpow1 7.0%

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

   10. Simplified 7.0%

       \[\leadsto \color{blue}{\sqrt{x \cdot 2.25}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 4.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 54.6%

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

4. Taylor expanded in x around inf 4.3%

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

   1. *-commutative 4.3%

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

6. Simplified 4.3%

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

   1. pow1 4.3%

      \[\leadsto \color{blue}{{\left(\sqrt{x} \cdot 1.5\right)}^{1}} \]

   2. add-sqr-sqrt 4.3%

      \[\leadsto {\color{blue}{\left(\sqrt{\sqrt{x} \cdot 1.5} \cdot \sqrt{\sqrt{x} \cdot 1.5}\right)}}^{1} \]

   3. sqrt-unprod 4.3%

      \[\leadsto {\color{blue}{\left(\sqrt{\left(\sqrt{x} \cdot 1.5\right) \cdot \left(\sqrt{x} \cdot 1.5\right)}\right)}}^{1} \]

   4. swap-sqr 4.3%

      \[\leadsto {\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(1.5 \cdot 1.5\right)}}\right)}^{1} \]

   5. add-sqr-sqrt 4.3%

      \[\leadsto {\left(\sqrt{\color{blue}{x} \cdot \left(1.5 \cdot 1.5\right)}\right)}^{1} \]

   6. metadata-eval 4.3%

      \[\leadsto {\left(\sqrt{x \cdot \color{blue}{2.25}}\right)}^{1} \]

8. Applied egg-rr 4.3%

   \[\leadsto \color{blue}{{\left(\sqrt{x \cdot 2.25}\right)}^{1}} \]
9. Step-by-step derivation

   1. unpow1 4.3%

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

10. Simplified 4.3%

    \[\leadsto \color{blue}{\sqrt{x \cdot 2.25}} \]
11. Add Preprocessing

Developer Target 1: 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 2024123 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 6 (/ (+ (+ x 1) (* 4 (sqrt x))) (- x 1))))

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