Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.6% → 99.9%

Time: 10.5s

Alternatives: 14

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sub-neg N/A

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

   6. +-lowering-+.f64 N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. sqrt-lowering-sqrt.f64 N/A

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

   11. +-lowering-+.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 97.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 97.9

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

   5. Simplified 97.9%

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval 97.9

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

   7. Applied egg-rr 97.9%

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

   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 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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sub-neg N/A

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

      6. +-lowering-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. +-lowering-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 98.7%

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

4. Final simplification 98.3%

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

Alternative 3: 11.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], N[(N[Sqrt[x], $MachinePrecision] * -1.5 + -0.375), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * 1.5), $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. accelerator-lowering-fma.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 97.9

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 2.2

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

   8. Simplified 2.2%

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

   9. Taylor expanded in x around -inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{x} \cdot \frac{-3}{2}} + \frac{3}{8} \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} \]

       2. unpow2 N/A

          \[\leadsto \sqrt{x} \cdot \frac{-3}{2} + \frac{3}{8} \cdot \frac{1}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}} \]

       3. rem-square-sqrt N/A

          \[\leadsto \sqrt{x} \cdot \frac{-3}{2} + \frac{3}{8} \cdot \frac{1}{\color{blue}{-1}} \]

       4. metadata-eval N/A

          \[\leadsto \sqrt{x} \cdot \frac{-3}{2} + \frac{3}{8} \cdot \color{blue}{-1} \]

       5. metadata-eval N/A

          \[\leadsto \sqrt{x} \cdot \frac{-3}{2} + \color{blue}{\frac{-3}{8}} \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \frac{-3}{2}, \frac{-3}{8}\right)} \]

       7. sqrt-lowering-sqrt.f64 15.6

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x}}, -1.5, -0.375\right) \]

   11. Simplified 15.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, -1.5, -0.375\right)} \]

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 7.0

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

   5. Simplified 7.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{x} \cdot \frac{3}{2}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{x} \cdot \frac{3}{2}} \]

      3. sqrt-lowering-sqrt.f64 7.0

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

   8. Simplified 7.0%

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

4. Final simplification 11.3%

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

Alternative 4: 6.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], N[(N[Sqrt[x], $MachinePrecision] * -1.5), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * 1.5), $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. accelerator-lowering-fma.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 97.9

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 7.0

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

   8. Simplified 7.0%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 7.0

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

   5. Simplified 7.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{x} \cdot \frac{3}{2}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{x} \cdot \frac{3}{2}} \]

      3. sqrt-lowering-sqrt.f64 7.0

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

   8. Simplified 7.0%

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

4. Final simplification 7.0%

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

Alternative 5: 52.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * 1.5 + -0.375), $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. Simplified 97.9%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 7.0

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

   5. Simplified 7.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 7.0

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

   8. Simplified 7.0%

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

   9. Taylor expanded in x around inf

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

       1. sub-neg N/A

          \[\leadsto \color{blue}{\frac{3}{2} \cdot \sqrt{x} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{x} \cdot \frac{3}{2}} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \sqrt{x} \cdot \frac{3}{2} + \color{blue}{\frac{-3}{8}} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \frac{3}{2}, \frac{-3}{8}\right)} \]

       5. sqrt-lowering-sqrt.f64 7.0

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x}}, 1.5, -0.375\right) \]

   11. Simplified 7.0%

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

Alternative 6: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. clear-num N/A

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

   2. associate-/r* N/A

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

   3. associate-/r/ N/A

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

   4. clear-num N/A

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

   5. *-lowering-*.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. +-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. sqrt-lowering-sqrt.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. sub-neg N/A

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

   12. +-lowering-+.f64 N/A

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

   13. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 7: 52.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 97.9%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 7.0

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

   5. Simplified 7.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 7.0

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

   8. Simplified 7.0%

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

   9. Taylor expanded in x around inf

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

       1. sub-neg N/A

          \[\leadsto \color{blue}{\frac{3}{2} \cdot \sqrt{x} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{x} \cdot \frac{3}{2}} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \sqrt{x} \cdot \frac{3}{2} + \color{blue}{\frac{-3}{8}} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \frac{3}{2}, \frac{-3}{8}\right)} \]

       5. sqrt-lowering-sqrt.f64 7.0

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x}}, 1.5, -0.375\right) \]

   11. Simplified 7.0%

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

4. Final simplification 52.5%

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

Alternative 8: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-+r+ N/A

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

   3. +-lowering-+.f64 N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 99.8

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

4. Applied egg-rr 99.8%

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. metadata-eval 99.8

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 9: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. frac-2neg N/A

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

   2. /-lowering-/.f64 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. sub-neg N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. metadata-eval N/A

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

   11. neg-sub0 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, -6, 6\right)}{\color{blue}{0 - \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}} \]

   12. associate-+l+ N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, -6, 6\right)}{0 - \color{blue}{\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

   13. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, -6, 6\right)}{0 - \color{blue}{\left(\left(1 + 4 \cdot \sqrt{x}\right) + x\right)}} \]

   14. associate--r+ N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, -6, 6\right)}{\color{blue}{\left(0 - \left(1 + 4 \cdot \sqrt{x}\right)\right) - x}} \]

   15. neg-sub0 N/A

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

   16. --lowering--.f64 N/A

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

4. Applied egg-rr 99.8%

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

Alternative 10: 52.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Wolfram

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

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

      2. distribute-neg-frac N/A

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

      3. distribute-neg-frac2 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. sqrt-lowering-sqrt.f64 N/A

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

      14. metadata-eval 97.9

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

   5. Simplified 97.9%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 7.0

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

   5. Simplified 7.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 7.0

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

   8. Simplified 7.0%

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

   9. Taylor expanded in x around inf

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

       1. sub-neg N/A

          \[\leadsto \color{blue}{\frac{3}{2} \cdot \sqrt{x} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{x} \cdot \frac{3}{2}} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \sqrt{x} \cdot \frac{3}{2} + \color{blue}{\frac{-3}{8}} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \frac{3}{2}, \frac{-3}{8}\right)} \]

       5. sqrt-lowering-sqrt.f64 7.0

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x}}, 1.5, -0.375\right) \]

   11. Simplified 7.0%

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

Alternative 11: 52.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt{x}, -0.6666666666666666, -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, 1.5, -0.375\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ 1.0 (fma (sqrt x) -0.6666666666666666 -0.16666666666666666))
   (fma (sqrt x) 1.5 -0.375)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 / fma(sqrt(x), -0.6666666666666666, -0.16666666666666666);
	} else {
		tmp = fma(sqrt(x), 1.5, -0.375);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 / fma(sqrt(x), -0.6666666666666666, -0.16666666666666666));
	else
		tmp = fma(sqrt(x), 1.5, -0.375);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] * -0.6666666666666666 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * 1.5 + -0.375), $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

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 97.9

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

   5. Simplified 97.9%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. metadata-eval 98.0

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

   7. Applied egg-rr 98.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 97.8

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

   10. Simplified 97.8%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 7.0

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

   5. Simplified 7.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 7.0

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

   8. Simplified 7.0%

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

   9. Taylor expanded in x around inf

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

       1. sub-neg N/A

          \[\leadsto \color{blue}{\frac{3}{2} \cdot \sqrt{x} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{x} \cdot \frac{3}{2}} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \sqrt{x} \cdot \frac{3}{2} + \color{blue}{\frac{-3}{8}} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \frac{3}{2}, \frac{-3}{8}\right)} \]

       5. sqrt-lowering-sqrt.f64 7.0

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x}}, 1.5, -0.375\right) \]

   11. Simplified 7.0%

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

Alternative 12: 52.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
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. accelerator-lowering-fma.f64 N/A

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

   3. sqrt-lowering-sqrt.f64 52.5

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

5. Simplified 52.5%

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. metadata-eval 52.5

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

7. Applied egg-rr 52.5%

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

Alternative 13: 11.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{x}, 1.5, -0.375\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (sqrt x) 1.5 -0.375))

C

double code(double x) {
	return fma(sqrt(x), 1.5, -0.375);
}

Julia

function code(x)
	return fma(sqrt(x), 1.5, -0.375)
end

Wolfram

code[x_] := N[(N[Sqrt[x], $MachinePrecision] * 1.5 + -0.375), $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. accelerator-lowering-fma.f64 N/A

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

   3. sqrt-lowering-sqrt.f64 52.5

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

5. Simplified 52.5%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 4.6

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

8. Simplified 4.6%

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

9. Taylor expanded in x around inf

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

    1. sub-neg N/A

       \[\leadsto \color{blue}{\frac{3}{2} \cdot \sqrt{x} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \]

    2. *-commutative N/A

       \[\leadsto \color{blue}{\sqrt{x} \cdot \frac{3}{2}} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \sqrt{x} \cdot \frac{3}{2} + \color{blue}{\frac{-3}{8}} \]

    4. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \frac{3}{2}, \frac{-3}{8}\right)} \]

    5. sqrt-lowering-sqrt.f64 11.3

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x}}, 1.5, -0.375\right) \]

11. Simplified 11.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, 1.5, -0.375\right)} \]
12. Add Preprocessing

Alternative 14: 4.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[x], $MachinePrecision] * -1.5), $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. accelerator-lowering-fma.f64 N/A

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

   3. sqrt-lowering-sqrt.f64 52.5

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

5. Simplified 52.5%

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

6. Taylor expanded in x around -inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. sqrt-lowering-sqrt.f64 4.2

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

8. Simplified 4.2%

   \[\leadsto \color{blue}{\sqrt{x} \cdot -1.5} \]
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 2024205 
(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)))))