Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 6.3s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(x - 1.0), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(1.0 + x), $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. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 99.9

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 6.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -2:\\ \;\;\;\;\frac{-1.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot \sqrt{{x}^{-1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -2.0)
   (/ -1.5 (sqrt x))
   (* 1.5 (sqrt (pow x -1.0)))))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -2.0) {
		tmp = -1.5 / sqrt(x);
	} else {
		tmp = 1.5 * sqrt(pow(x, -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((6.0d0 * (x - 1.0d0)) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))) <= (-2.0d0)) then
        tmp = (-1.5d0) / sqrt(x)
    else
        tmp = 1.5d0 * sqrt((x ** (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * Math.sqrt(x)))) <= -2.0) {
		tmp = -1.5 / Math.sqrt(x);
	} else {
		tmp = 1.5 * Math.sqrt(Math.pow(x, -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * math.sqrt(x)))) <= -2.0:
		tmp = -1.5 / math.sqrt(x)
	else:
		tmp = 1.5 * math.sqrt(math.pow(x, -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -2.0)
		tmp = Float64(-1.5 / sqrt(x));
	else
		tmp = Float64(1.5 * sqrt((x ^ -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -2.0)
		tmp = -1.5 / sqrt(x);
	else
		tmp = 1.5 * sqrt((x ^ -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[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], -2.0], N[(-1.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.5 * N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 6.8%

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

   10. Applied rewrites 6.8%

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

   if -2 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 1.8

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

   5. Applied rewrites 1.8%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \frac{3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 7.2%

      \[\leadsto 1.5 \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -2:\\ \;\;\;\;\frac{-1.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot \sqrt{{x}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -2:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot x}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -2.0)
   (/ -6.0 (fma 4.0 (sqrt x) (+ 1.0 x)))
   (/ (* 6.0 x) (fma (sqrt x) 4.0 1.0))))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -2.0) {
		tmp = -6.0 / fma(4.0, sqrt(x), (1.0 + x));
	} else {
		tmp = (6.0 * x) / fma(sqrt(x), 4.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -2.0)
		tmp = Float64(-6.0 / fma(4.0, sqrt(x), Float64(1.0 + x)));
	else
		tmp = Float64(Float64(6.0 * x) / fma(sqrt(x), 4.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[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], -2.0], N[(-6.0 / N[(4.0 * N[Sqrt[x], $MachinePrecision] + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 * x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lft-mult-inverse N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.6%

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

   if -2 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 7.3

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 7.3

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

   8. Applied rewrites 7.3%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -2:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot x}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 6.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -2:\\ \;\;\;\;\frac{-1.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \sqrt{x}, 0.375\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -2.0)
   (/ -1.5 (sqrt x))
   (/ (fma 1.5 (sqrt x) 0.375) x)))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -2.0) {
		tmp = -1.5 / sqrt(x);
	} else {
		tmp = fma(1.5, sqrt(x), 0.375) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -2.0)
		tmp = Float64(-1.5 / sqrt(x));
	else
		tmp = Float64(fma(1.5, sqrt(x), 0.375) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[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], -2.0], N[(-1.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(1.5 * N[Sqrt[x], $MachinePrecision] + 0.375), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 6.8%

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

   10. Applied rewrites 6.8%

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

   if -2 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 1.8

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

   5. Applied rewrites 1.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 1.8%

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

   9. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{-3}{2} \cdot \sqrt{x} + \frac{3}{8} \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 7.2%

       \[\leadsto \frac{\mathsf{fma}\left(1.5, \sqrt{x}, 0.375\right)}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 51.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \sqrt{x}, 0.375\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (fma 4.0 (sqrt x) (+ 1.0 x)))
   (/ (fma 1.5 (sqrt x) 0.375) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / fma(4.0, sqrt(x), (1.0 + x));
	} else {
		tmp = fma(1.5, sqrt(x), 0.375) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / fma(4.0, sqrt(x), Float64(1.0 + x)));
	else
		tmp = Float64(fma(1.5, sqrt(x), 0.375) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(4.0 * N[Sqrt[x], $MachinePrecision] + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.5 * N[Sqrt[x], $MachinePrecision] + 0.375), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lft-mult-inverse N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.6%

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

   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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 1.8

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

   5. Applied rewrites 1.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 1.8%

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

   9. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{-3}{2} \cdot \sqrt{x} + \frac{3}{8} \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 7.2%

       \[\leadsto \frac{\mathsf{fma}\left(1.5, \sqrt{x}, 0.375\right)}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf

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

   1. fp-cancel-sub-sign-inv N/A

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

   2. distribute-rgt-in N/A

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

   3. metadata-eval N/A

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

   4. associate-*l* N/A

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

   5. lft-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 99.9

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

5. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lower-fma.f64 99.9

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

7. Applied rewrites 99.9%

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

Alternative 7: 51.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \sqrt{x}, 0.375\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (fma (sqrt x) 4.0 1.0))
   (/ (fma 1.5 (sqrt x) 0.375) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / fma(sqrt(x), 4.0, 1.0);
	} else {
		tmp = fma(1.5, sqrt(x), 0.375) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / fma(sqrt(x), 4.0, 1.0));
	else
		tmp = Float64(fma(1.5, sqrt(x), 0.375) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.5 * N[Sqrt[x], $MachinePrecision] + 0.375), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 97.5

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

   5. Applied rewrites 97.5%

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

   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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 1.8

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

   5. Applied rewrites 1.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 1.8%

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

   9. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{-3}{2} \cdot \sqrt{x} + \frac{3}{8} \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 7.2%

       \[\leadsto \frac{\mathsf{fma}\left(1.5, \sqrt{x}, 0.375\right)}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 51.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf

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

   1. fp-cancel-sub-sign-inv N/A

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

   2. distribute-rgt-in N/A

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

   3. metadata-eval N/A

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

   4. associate-*l* N/A

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

   5. lft-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 99.9

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

5. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lower-fma.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-sqrt.f64 53.2

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

10. Applied rewrites 53.2%

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

Alternative 9: 4.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1.5}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -1.5 (sqrt x)))

C

double code(double x) {
	return -1.5 / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.5d0) / sqrt(x)
end function

Java

public static double code(double x) {
	return -1.5 / Math.sqrt(x);
}

Python

def code(x):
	return -1.5 / math.sqrt(x)

Julia

function code(x)
	return Float64(-1.5 / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = -1.5 / sqrt(x);
end

Wolfram

code[x_] := N[(-1.5 / N[Sqrt[x], $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

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sqrt.f64 50.4

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

5. Applied rewrites 50.4%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{-3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
7. Step-by-step derivation

8. Applied rewrites 4.4%

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

10. Applied rewrites 4.4%

    \[\leadsto \color{blue}{\frac{-1.5}{\sqrt{x}}} \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× 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 2024326 
(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)))))