Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 10.4s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. flip-- N/A

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

   2. lift-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-+.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lift-+.f64 N/A

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

   10. flip-- N/A

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

   11. lift--.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*l/ N/A

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

   14. lower-*.f64 N/A

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

4. Applied egg-rr 99.9%

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

Alternative 2: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + -1\right) \cdot 6\\ t_1 := 4 \cdot \sqrt{x}\\ \mathbf{if}\;\frac{t\_0}{\left(x + 1\right) + t\_1} \leq -1:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 6, -6\right)}{x + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (+ x -1.0) 6.0)) (t_1 (* 4.0 (sqrt x))))
   (if (<= (/ t_0 (+ (+ x 1.0) t_1)) -1.0)
     (/ t_0 (fma 4.0 (sqrt x) 1.0))
     (/ (fma x 6.0 -6.0) (+ x t_1)))))

C

double code(double x) {
	double t_0 = (x + -1.0) * 6.0;
	double t_1 = 4.0 * sqrt(x);
	double tmp;
	if ((t_0 / ((x + 1.0) + t_1)) <= -1.0) {
		tmp = t_0 / fma(4.0, sqrt(x), 1.0);
	} else {
		tmp = fma(x, 6.0, -6.0) / (x + t_1);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x + -1.0) * 6.0)
	t_1 = Float64(4.0 * sqrt(x))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(x + 1.0) + t_1)) <= -1.0)
		tmp = Float64(t_0 / fma(4.0, sqrt(x), 1.0));
	else
		tmp = Float64(fma(x, 6.0, -6.0) / Float64(x + t_1));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(x + 1.0), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], -1.0], N[(t$95$0 / N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * 6.0 + -6.0), $MachinePrecision] / N[(x + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 96.9

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

   5. Simplified 96.9%

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

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

   1. Initial program 98.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 98.8

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

   4. Applied egg-rr 98.8%

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 98.9

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 96.9

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

   9. Simplified 96.9%

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

4. Final simplification 96.9%

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

Alternative 3: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(4, \sqrt{x}, 1\right)\\ t_1 := \left(x + -1\right) \cdot 6\\ \mathbf{if}\;\frac{t\_1}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -1:\\ \;\;\;\;\frac{t\_1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{x + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 4.0 (sqrt x) 1.0)) (t_1 (* (+ x -1.0) 6.0)))
   (if (<= (/ t_1 (+ (+ x 1.0) (* 4.0 (sqrt x)))) -1.0)
     (/ t_1 t_0)
     (/ (* x 6.0) (+ x t_0)))))

C

double code(double x) {
	double t_0 = fma(4.0, sqrt(x), 1.0);
	double t_1 = (x + -1.0) * 6.0;
	double tmp;
	if ((t_1 / ((x + 1.0) + (4.0 * sqrt(x)))) <= -1.0) {
		tmp = t_1 / t_0;
	} else {
		tmp = (x * 6.0) / (x + t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(4.0, sqrt(x), 1.0)
	t_1 = Float64(Float64(x + -1.0) * 6.0)
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -1.0)
		tmp = Float64(t_1 / t_0);
	else
		tmp = Float64(Float64(x * 6.0) / Float64(x + t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(t$95$1 / t$95$0), $MachinePrecision], N[(N[(x * 6.0), $MachinePrecision] / N[(x + t$95$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)))) < -1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 96.9

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

   5. Simplified 96.9%

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

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

   1. Initial program 98.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 98.8

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.9

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

   7. Simplified 96.9%

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

4. Final simplification 96.9%

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

Alternative 4: 11.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* (+ x -1.0) 6.0) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -1.0)
   (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)))) <= -1.0) {
		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)))) <= -1.0)
		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], -1.0], 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)))) < -1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 96.9

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

   5. Simplified 96.9%

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

   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. lower-*.f64 2.3

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

   8. Simplified 2.3%

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

   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. lower-fma.f64 N/A

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

       7. lower-sqrt.f64 15.7

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

   11. Simplified 15.7%

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

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

   1. Initial program 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 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. lower-fma.f64 N/A

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

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

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

      3. lower-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 11.1%

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

Alternative 5: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. clear-num N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. associate-/r/ N/A

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

   11. clear-num N/A

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

   12. lower-*.f64 N/A

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. flip-- N/A

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

   2. lift-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-+.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lift-+.f64 N/A

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

   10. flip-- N/A

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

   11. lift--.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*l/ N/A

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

   14. lower-*.f64 N/A

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

4. Applied egg-rr 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. frac-2neg N/A

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

   7. associate-*l/ N/A

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

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

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

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

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

   10. metadata-eval N/A

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

   11. lift-+.f64 N/A

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

   12. lift-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

   17. neg-mul-1 N/A

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

   18. metadata-eval N/A

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

   19. distribute-neg-in N/A

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

   20. lift-fma.f64 N/A

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.7

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

9. Simplified 99.7%

   \[\leadsto \frac{x + -1}{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(-4, \sqrt{x}, -1\right), x \cdot 0.16666666666666666\right)}} \]
10. Step-by-step derivation

    1. lift-sqrt.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. +-commutative N/A

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

    5. lift-*.f64 N/A

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

    6. lower-fma.f64 N/A

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

    7. lift-fma.f64 N/A

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

    8. distribute-rgt-in N/A

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

    9. *-commutative N/A

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

    10. associate-*l* N/A

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

    11. metadata-eval N/A

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

    12. metadata-eval N/A

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

    13. metadata-eval N/A

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

    14. lower-fma.f64 N/A

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

    15. metadata-eval 99.7

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

11. Applied egg-rr 99.7%

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

Alternative 7: 52.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

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

   3. lower-sqrt.f64 49.1

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

5. Simplified 49.1%

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

6. Final simplification 49.1%

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

Alternative 8: 52.5% 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 99.3%

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

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

   3. lower-sqrt.f64 49.1

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

5. Simplified 49.1%

   \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]
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. lower-fma.f64 49.1

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

7. Applied egg-rr 49.1%

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

Alternative 9: 7.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		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) 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) {
		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:
		tmp = math.sqrt(x) * -1.5
	else:
		tmp = math.sqrt(x) * 1.5
	return tmp

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], 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 x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 96.9

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

   5. Simplified 96.9%

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

   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. lower-*.f64 N/A

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

      3. lower-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 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 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. lower-fma.f64 N/A

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

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

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

      3. lower-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. Add Preprocessing

Alternative 10: 52.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Sqrt[x], $MachinePrecision] * 24.0 + -6.0), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

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

   13. lower-sqrt.f64 N/A

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

   14. metadata-eval 46.4

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

5. Simplified 46.4%

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

   1. lift-sqrt.f64 N/A

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

   2. flip-+ N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

7. Applied egg-rr 46.4%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

      \[\leadsto \sqrt{x} \cdot \color{blue}{24} + 1 \cdot -6 \]

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. metadata-eval 48.8

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

10. Simplified 48.8%

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

Alternative 11: 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 99.3%

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

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

   3. lower-sqrt.f64 49.1

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

5. Simplified 49.1%

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

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. lower-*.f64 4.8

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

8. Simplified 4.8%

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

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. lower-fma.f64 N/A

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

    5. lower-sqrt.f64 10.9

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

11. Simplified 10.9%

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

Alternative 12: 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 99.3%

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

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

   3. lower-sqrt.f64 49.1

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

5. Simplified 49.1%

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

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. lower-*.f64 N/A

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

   3. lower-sqrt.f64 4.0

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

8. Simplified 4.0%

   \[\leadsto \color{blue}{\sqrt{x} \cdot -1.5} \]
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 2024207 
(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)))))