Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 8.4s

Alternatives: 7

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Applied rewrites 99.9%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. lower-+.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 97.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x - -1.0) + (4.0 * sqrt(x)))) <= -1.0) {
		tmp = fma(x, 6.0, -6.0) / fma(sqrt(x), 4.0, 1.0);
	} else {
		tmp = (6.0 * x) / fma(sqrt(x), 4.0, (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)))) <= -1.0)
		tmp = Float64(fma(x, 6.0, -6.0) / fma(sqrt(x), 4.0, 1.0));
	else
		tmp = Float64(Float64(6.0 * x) / fma(sqrt(x), 4.0, Float64(x - -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], -1.0], N[(N[(x * 6.0 + -6.0), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 * x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(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)))) < -1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
   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 97.7

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

   5. Applied rewrites 97.7%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}} \]
   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 97.7

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

   7. Applied rewrites 97.7%

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

   if -1 < (/.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.6

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.6

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

      17. lift-+.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. sub-neg N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 97.6

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

   7. Applied rewrites 97.6%

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

4. Final simplification 97.6%

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

Alternative 3: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Applied rewrites 99.9%

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

Alternative 4: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. distribute-rgt-in N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval 99.8

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 99.8

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

   17. lift-+.f64 N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. sub-neg N/A

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

   21. lower--.f64 N/A

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

   22. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. lift-fma.f64 N/A

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

   5. lower--.f64 99.8

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

   6. lift-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 5: 52.6% 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 57.3

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

5. Applied rewrites 57.3%

   \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}} \]
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 57.3

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

7. Applied rewrites 57.3%

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

Alternative 6: 52.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(6.0 * N[(4.0 * N[Sqrt[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. 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 55.0

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

5. Applied rewrites 55.0%

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

7. Applied rewrites 55.0%

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

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 57.0%

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

Alternative 7: 4.5% 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 55.0

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

5. Applied rewrites 55.0%

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

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 4.8%

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

10. Applied rewrites 4.8%

    \[\leadsto \color{blue}{\frac{-1.5}{\sqrt{x}}} \]
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 2024318 
(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)))))