Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 8.0s

Alternatives: 10

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 99.4%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
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: 97.9% accurate, 0.5× speedup

Math

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

FPCore

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

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
   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 97.6

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

   5. Applied rewrites 97.6%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}} \]
   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 97.6

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

   7. Applied rewrites 97.6%

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

   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 98.8%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   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 97.1

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

   5. Applied rewrites 97.1%

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

   7. Applied rewrites 97.1%

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

4. Final simplification 97.4%

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

Alternative 3: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
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. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-/r/ N/A

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

   6. clear-num N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 4: 52.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   5. Applied rewrites 97.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 97.6

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

      5. lift-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lift--.f64 97.6

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

   7. Applied rewrites 97.6%

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

   if 1 < x

   1. Initial program 98.8%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   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 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 6.8%

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

Alternative 5: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{x - 1}{\mathsf{fma}\left(\mathsf{fma}\left(4, \sqrt{x}, x\right), 0.16666666666666666, 0.16666666666666666\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (- x 1.0)
  (fma (fma 4.0 (sqrt x) x) 0.16666666666666666 0.16666666666666666)))

C

double code(double x) {
	return (x - 1.0) / fma(fma(4.0, sqrt(x), x), 0.16666666666666666, 0.16666666666666666);
}

Julia

function code(x)
	return Float64(Float64(x - 1.0) / fma(fma(4.0, sqrt(x), x), 0.16666666666666666, 0.16666666666666666))
end

Wolfram

code[x_] := N[(N[(x - 1.0), $MachinePrecision] / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision] * 0.16666666666666666 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
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. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-/r/ N/A

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

   6. clear-num N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. associate-+r- N/A

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

   11. div-sub N/A

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

   12. sub-neg N/A

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

   13. div-inv N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-fma.f64 N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval 99.6

       \[\leadsto \frac{x - 1}{\mathsf{fma}\left(\mathsf{fma}\left(4, \sqrt{x}, x\right), 0.16666666666666666, \color{blue}{0.16666666666666666}\right)} \]

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{x - 1}{\mathsf{fma}\left(\mathsf{fma}\left(4, \sqrt{x}, x\right), 0.16666666666666666, 0.16666666666666666\right)}} \]
7. Add Preprocessing

Alternative 6: 51.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / fma(sqrt(x), 4.0, 1.0);
	} else {
		tmp = 1.5 * sqrt(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(1.5 * sqrt(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[(1.5 * N[Sqrt[x], $MachinePrecision]), $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 98.8%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   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 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 6.8%

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

Alternative 7: 52.0% 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 99.4%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
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 52.6

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

5. Applied rewrites 52.6%

   \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}} \]
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 52.6

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

7. Applied rewrites 52.6%

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

Alternative 8: 52.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{x - 1}{\mathsf{fma}\left(0.6666666666666666, \sqrt{x}, 0.16666666666666666\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- x 1.0) (fma 0.6666666666666666 (sqrt x) 0.16666666666666666)))

C

double code(double x) {
	return (x - 1.0) / fma(0.6666666666666666, sqrt(x), 0.16666666666666666);
}

Julia

function code(x)
	return Float64(Float64(x - 1.0) / fma(0.6666666666666666, sqrt(x), 0.16666666666666666))
end

Wolfram

code[x_] := N[(N[(x - 1.0), $MachinePrecision] / N[(0.6666666666666666 * N[Sqrt[x], $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
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. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-/r/ N/A

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

   6. clear-num N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. associate-+r- N/A

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

   11. div-sub N/A

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

   12. sub-neg N/A

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

   13. div-inv N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-fma.f64 N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval 99.6

       \[\leadsto \frac{x - 1}{\mathsf{fma}\left(\mathsf{fma}\left(4, \sqrt{x}, x\right), 0.16666666666666666, \color{blue}{0.16666666666666666}\right)} \]

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{x - 1}{\mathsf{fma}\left(\mathsf{fma}\left(4, \sqrt{x}, x\right), 0.16666666666666666, 0.16666666666666666\right)}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{x - 1}{\color{blue}{\frac{1}{6} + \frac{2}{3} \cdot \sqrt{x}}} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{x - 1}{\color{blue}{\frac{2}{3} \cdot \sqrt{x} + \frac{1}{6}}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{x - 1}{\color{blue}{\mathsf{fma}\left(\frac{2}{3}, \sqrt{x}, \frac{1}{6}\right)}} \]

   3. lower-sqrt.f64 52.5

      \[\leadsto \frac{x - 1}{\mathsf{fma}\left(0.6666666666666666, \color{blue}{\sqrt{x}}, 0.16666666666666666\right)} \]

9. Applied rewrites 52.5%

   \[\leadsto \frac{x - 1}{\color{blue}{\mathsf{fma}\left(0.6666666666666666, \sqrt{x}, 0.16666666666666666\right)}} \]
10. Add Preprocessing

Alternative 9: 7.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   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 6.9%

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

   10. Applied rewrites 6.9%

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

   if 1 < x

   1. Initial program 98.8%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   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 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 6.8%

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

Alternative 10: 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 99.4%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
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 49.1

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

5. Applied rewrites 49.1%

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

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 2024288 
(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)))))