Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 7.4s

Alternatives: 9

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / 6.0), $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/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

   4. sub-neg 99.8%

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

   5. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. associate-*l/ 99.8%

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

   2. fma-udef 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

   5. associate-/l* 99.9%

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

   6. +-commutative 99.9%

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

   7. fma-udef 99.9%

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

5. Applied egg-rr 99.9%

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

   1. fma-udef 99.9%

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

   2. add-sqr-sqrt 99.9%

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

   3. sqrt-unprod 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. swap-sqr 99.9%

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

   7. add-sqr-sqrt 99.9%

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

   8. metadata-eval 99.9%

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

7. Applied egg-rr 99.9%

   \[\leadsto \frac{6}{\frac{x + \color{blue}{\left(\sqrt{x \cdot 16} + 1\right)}}{x + -1}} \]
8. Step-by-step derivation

   1. div-inv 99.9%

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

   2. sqrt-prod 99.9%

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

   3. metadata-eval 99.9%

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

   4. *-commutative 99.9%

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

9. Applied egg-rr 99.9%

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

    1. un-div-inv 99.9%

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

    2. clear-num 99.9%

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

    3. fma-def 99.9%

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

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

Alternative 2: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;\left(x + -1\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{x + \left(1 + 4 \cdot \sqrt{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.15)
   (* (+ x -1.0) 6.0)
   (/ (* x 6.0) (+ x (+ 1.0 (* 4.0 (sqrt x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.15) {
		tmp = (x + -1.0) * 6.0;
	} else {
		tmp = (x * 6.0) / (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 <= 1.15d0) then
        tmp = (x + (-1.0d0)) * 6.0d0
    else
        tmp = (x * 6.0d0) / (x + (1.0d0 + (4.0d0 * sqrt(x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.15) {
		tmp = (x + -1.0) * 6.0;
	} else {
		tmp = (x * 6.0) / (x + (1.0 + (4.0 * Math.sqrt(x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.15:
		tmp = (x + -1.0) * 6.0
	else:
		tmp = (x * 6.0) / (x + (1.0 + (4.0 * math.sqrt(x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.15)
		tmp = Float64(Float64(x + -1.0) * 6.0);
	else
		tmp = Float64(Float64(x * 6.0) / Float64(x + Float64(1.0 + Float64(4.0 * sqrt(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.15)
		tmp = (x + -1.0) * 6.0;
	else
		tmp = (x * 6.0) / (x + (1.0 + (4.0 * sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.15], N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision], N[(N[(x * 6.0), $MachinePrecision] / N[(x + N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   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. +-commutative 99.9%

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

      3. fma-def 99.9%

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

      4. sub-neg 99.9%

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

      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. Taylor expanded in x around 0 95.7%

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

   5. Taylor expanded in x around 0 95.7%

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

      1. sub-neg 95.7%

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

      2. metadata-eval 95.7%

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

      3. metadata-eval 95.7%

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

      4. distribute-lft-in 95.7%

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

      5. *-commutative 95.7%

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

   7. Applied egg-rr 95.7%

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

   if 1.1499999999999999 < x

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      3. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 97.1%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;\left(x + -1\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{x + \left(1 + 4 \cdot \sqrt{x}\right)}\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

   4. sub-neg 99.8%

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

   5. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. fma-udef 99.8%

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

   2. +-commutative 99.8%

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

   3. add-sqr-sqrt 99.8%

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

   4. sqrt-unprod 99.8%

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

   5. *-commutative 99.8%

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

   6. *-commutative 99.8%

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

   7. swap-sqr 99.8%

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

   8. add-sqr-sqrt 99.8%

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

   9. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-+l+ 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

   4. sub-neg 99.8%

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

   5. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. associate-*l/ 99.8%

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

   2. fma-udef 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

   5. associate-/l* 99.9%

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

   6. +-commutative 99.9%

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

   7. fma-udef 99.9%

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

5. Applied egg-rr 99.9%

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

   1. fma-udef 99.9%

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

   2. add-sqr-sqrt 99.9%

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

   3. sqrt-unprod 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. swap-sqr 99.9%

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

   7. add-sqr-sqrt 99.9%

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

   8. metadata-eval 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 5: 95.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left(x + -1\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\frac{36 - \frac{36}{x \cdot x}}{6 + \frac{6}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (* (+ x -1.0) 6.0)
   (/ (- 36.0 (/ 36.0 (* x x))) (+ 6.0 (/ 6.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x + -1.0) * 6.0;
	} else {
		tmp = (36.0 - (36.0 / (x * x))) / (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 = (x + (-1.0d0)) * 6.0d0
    else
        tmp = (36.0d0 - (36.0d0 / (x * x))) / (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 = (x + -1.0) * 6.0;
	} else {
		tmp = (36.0 - (36.0 / (x * x))) / (6.0 + (6.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = (x + -1.0) * 6.0
	else:
		tmp = (36.0 - (36.0 / (x * x))) / (6.0 + (6.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(x + -1.0) * 6.0);
	else
		tmp = Float64(Float64(36.0 - Float64(36.0 / Float64(x * x))) / 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 = (x + -1.0) * 6.0;
	else
		tmp = (36.0 - (36.0 / (x * x))) / (6.0 + (6.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision], N[(N[(36.0 - N[(36.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(6.0 / 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. +-commutative 99.9%

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

      3. fma-def 99.9%

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

      4. sub-neg 99.9%

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

      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. Taylor expanded in x around 0 95.7%

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

   5. Taylor expanded in x around 0 95.7%

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

      1. sub-neg 95.7%

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

      2. metadata-eval 95.7%

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

      3. metadata-eval 95.7%

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

      4. distribute-lft-in 95.7%

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

      5. *-commutative 95.7%

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

   7. Applied egg-rr 95.7%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

      4. sub-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*l/ 99.6%

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

      2. fma-udef 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-/l* 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-udef 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 94.8%

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

   7. Taylor expanded in x around 0 94.8%

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

      1. associate-*r/ 94.8%

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

      2. metadata-eval 94.8%

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

   9. Simplified 94.8%

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

       1. flip-- 94.8%

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

       2. metadata-eval 94.8%

          \[\leadsto \frac{\color{blue}{36} - \frac{6}{x} \cdot \frac{6}{x}}{6 + \frac{6}{x}} \]

       3. frac-times 94.8%

          \[\leadsto \frac{36 - \color{blue}{\frac{6 \cdot 6}{x \cdot x}}}{6 + \frac{6}{x}} \]

       4. metadata-eval 94.8%

          \[\leadsto \frac{36 - \frac{\color{blue}{36}}{x \cdot x}}{6 + \frac{6}{x}} \]

   11. Applied egg-rr 94.8%

       \[\leadsto \color{blue}{\frac{36 - \frac{36}{x \cdot x}}{6 + \frac{6}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left(x + -1\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\frac{36 - \frac{36}{x \cdot x}}{6 + \frac{6}{x}}\\ \end{array} \]

Alternative 6: 95.5% accurate, 16.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = (x + -1.0) * 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 = (x + (-1.0d0)) * 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 = (x + -1.0) * 6.0;
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[(N[(x + -1.0), $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. +-commutative 99.9%

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

      3. fma-def 99.9%

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

      4. sub-neg 99.9%

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

      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. Taylor expanded in x around 0 95.7%

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

   5. Taylor expanded in x around 0 95.7%

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

      1. sub-neg 95.7%

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

      2. metadata-eval 95.7%

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

      3. metadata-eval 95.7%

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

      4. distribute-lft-in 95.7%

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

      5. *-commutative 95.7%

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

   7. Applied egg-rr 95.7%

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

   if 2 < x

   1. Initial program 99.6%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

      4. sub-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 94.8%

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

4. Final simplification 95.2%

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

Alternative 7: 95.5% accurate, 16.0× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = (x + -1.0) * 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(x + -1.0) * 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 = (x + -1.0) * 6.0;
	else
		tmp = 6.0 - (6.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[(x + -1.0), $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. +-commutative 99.9%

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

      3. fma-def 99.9%

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

      4. sub-neg 99.9%

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

      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. Taylor expanded in x around 0 95.7%

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

   5. Taylor expanded in x around 0 95.7%

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

      1. sub-neg 95.7%

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

      2. metadata-eval 95.7%

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

      3. metadata-eval 95.7%

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

      4. distribute-lft-in 95.7%

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

      5. *-commutative 95.7%

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

   7. Applied egg-rr 95.7%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

      4. sub-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*l/ 99.6%

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

      2. fma-udef 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-/l* 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-udef 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 94.8%

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

   7. Taylor expanded in x around 0 94.8%

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

      1. associate-*r/ 94.8%

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

      2. metadata-eval 94.8%

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

   9. Simplified 94.8%

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

4. Final simplification 95.2%

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

Alternative 8: 95.5% accurate, 36.9× 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. +-commutative 99.9%

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

      3. fma-def 99.9%

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

      4. sub-neg 99.9%

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

      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. Taylor expanded in x around 0 95.7%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

      4. sub-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 94.8%

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

4. Final simplification 95.2%

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

Alternative 9: 48.3% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ -6 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -6.0)

C

double code(double x) {
	return -6.0;
}

Fortran

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

Java

public static double code(double x) {
	return -6.0;
}

Python

def code(x):
	return -6.0

Julia

function code(x)
	return -6.0
end

MATLAB

function tmp = code(x)
	tmp = -6.0;
end

Wolfram

code[x_] := -6.0

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

   4. sub-neg 99.8%

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

   5. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 42.7%

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

5. Final simplification 42.7%

   \[\leadsto -6 \]

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 2023196 
(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)))))