Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 8.2s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 97.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* (- x 1.0) 6.0) (+ (* 4.0 (sqrt x)) (+ x 1.0))) -5.0)
   (* (/ (- x 1.0) (fma 4.0 (sqrt x) 1.0)) 6.0)
   (/ 6.0 (+ (/ 4.0 (sqrt x)) 1.0))))

C

double code(double x) {
	double tmp;
	if ((((x - 1.0) * 6.0) / ((4.0 * sqrt(x)) + (x + 1.0))) <= -5.0) {
		tmp = ((x - 1.0) / fma(4.0, sqrt(x), 1.0)) * 6.0;
	} else {
		tmp = 6.0 / ((4.0 / sqrt(x)) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 1.0) * 6.0) / Float64(Float64(4.0 * sqrt(x)) + Float64(x + 1.0))) <= -5.0)
		tmp = Float64(Float64(Float64(x - 1.0) / fma(4.0, sqrt(x), 1.0)) * 6.0);
	else
		tmp = Float64(6.0 / Float64(Float64(4.0 / sqrt(x)) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(x - 1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5.0], N[(N[(N[(x - 1.0), $MachinePrecision] / N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision], N[(6.0 / N[(N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 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)))) < -5

   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

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.0

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

   7. Applied rewrites 98.0%

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

   if -5 < (/.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 inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 98.9%

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

Alternative 3: 97.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* (- x 1.0) 6.0) (+ (* 4.0 (sqrt x)) (+ x 1.0))) -5.0)
   (* (/ 6.0 (fma 4.0 (sqrt x) 1.0)) (- x 1.0))
   (/ 6.0 (+ (/ 4.0 (sqrt x)) 1.0))))

C

double code(double x) {
	double tmp;
	if ((((x - 1.0) * 6.0) / ((4.0 * sqrt(x)) + (x + 1.0))) <= -5.0) {
		tmp = (6.0 / fma(4.0, sqrt(x), 1.0)) * (x - 1.0);
	} else {
		tmp = 6.0 / ((4.0 / sqrt(x)) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 1.0) * 6.0) / Float64(Float64(4.0 * sqrt(x)) + Float64(x + 1.0))) <= -5.0)
		tmp = Float64(Float64(6.0 / fma(4.0, sqrt(x), 1.0)) * Float64(x - 1.0));
	else
		tmp = Float64(6.0 / Float64(Float64(4.0 / sqrt(x)) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(x - 1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5.0], N[(N[(6.0 / N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 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)))) < -5

   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

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.0

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

   7. Applied rewrites 98.0%

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

   if -5 < (/.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 inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 98.9%

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

Alternative 4: 97.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* (- x 1.0) 6.0) (+ (* 4.0 (sqrt x)) (+ x 1.0))) -5.0)
   (/ (fma x 6.0 -6.0) (fma (sqrt x) 4.0 1.0))
   (/ 6.0 (+ (/ 4.0 (sqrt x)) 1.0))))

C

double code(double x) {
	double tmp;
	if ((((x - 1.0) * 6.0) / ((4.0 * sqrt(x)) + (x + 1.0))) <= -5.0) {
		tmp = fma(x, 6.0, -6.0) / fma(sqrt(x), 4.0, 1.0);
	} else {
		tmp = 6.0 / ((4.0 / sqrt(x)) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 1.0) * 6.0) / Float64(Float64(4.0 * sqrt(x)) + Float64(x + 1.0))) <= -5.0)
		tmp = Float64(fma(x, 6.0, -6.0) / fma(sqrt(x), 4.0, 1.0));
	else
		tmp = Float64(6.0 / Float64(Float64(4.0 / sqrt(x)) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(x - 1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5.0], N[(N[(x * 6.0 + -6.0), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 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)))) < -5

   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

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 98.0

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

   7. Applied rewrites 98.0%

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

   if -5 < (/.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 inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 98.9%

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

Alternative 5: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * 6.0 + -6.0), $MachinePrecision] / N[(N[(N[Sqrt[x], $MachinePrecision] * 4.0 + x), $MachinePrecision] + 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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+r+ N/A

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

   5. lower-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lift-fma.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 6: 52.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * 6.0 + -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 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 52.0

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

5. Applied rewrites 52.0%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 52.0

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

7. Applied rewrites 52.0%

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

Alternative 7: 51.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(24, \sqrt{x}, -6\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma 24.0 (sqrt x) -6.0))

C

double code(double x) {
	return fma(24.0, sqrt(x), -6.0);
}

Julia

function code(x)
	return fma(24.0, sqrt(x), -6.0)
end

Wolfram

code[x_] := N[(24.0 * N[Sqrt[x], $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. 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 49.5

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

5. Applied rewrites 49.5%

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

7. Applied rewrites 49.5%

   \[\leadsto \frac{-6}{\mathsf{fma}\left(16, x, -1\right)} \cdot \color{blue}{\mathsf{fma}\left(4, \sqrt{x}, -1\right)} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 51.8%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 51.8%

    \[\leadsto \mathsf{fma}\left(24, \color{blue}{\sqrt{x}}, -6\right) \]
14. Add Preprocessing

Alternative 8: 4.5% accurate, 2.6× speedup

Math

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

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 inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 51.2

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

5. Applied rewrites 51.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 4.3%

   \[\leadsto 1.5 \cdot \color{blue}{\sqrt{x}} \]
9. 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 2024268 
(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)))))