Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 8.9s

Alternatives: 15

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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 99.9%

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

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - 1\right) \cdot 6\\ \mathbf{if}\;\frac{t\_0}{4 \cdot \sqrt{x} + \left(1 + x\right)} \leq 2:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\sqrt{x} \cdot -4 - x} \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (- x 1.0) 6.0)))
   (if (<= (/ t_0 (+ (* 4.0 (sqrt x)) (+ 1.0 x))) 2.0)
     (/ t_0 (fma (sqrt x) 4.0 1.0))
     (* (/ (- x) (- (* (sqrt x) -4.0) x)) 6.0))))

C

double code(double x) {
	double t_0 = (x - 1.0) * 6.0;
	double tmp;
	if ((t_0 / ((4.0 * sqrt(x)) + (1.0 + x))) <= 2.0) {
		tmp = t_0 / fma(sqrt(x), 4.0, 1.0);
	} else {
		tmp = (-x / ((sqrt(x) * -4.0) - x)) * 6.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x - 1.0) * 6.0)
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(4.0 * sqrt(x)) + Float64(1.0 + x))) <= 2.0)
		tmp = Float64(t_0 / fma(sqrt(x), 4.0, 1.0));
	else
		tmp = Float64(Float64(Float64(-x) / Float64(Float64(sqrt(x) * -4.0) - x)) * 6.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x - 1.0), $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / N[(N[(N[Sqrt[x], $MachinePrecision] * -4.0), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] * 6.0), $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 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 96.3

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

   5. Applied rewrites 96.3%

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

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

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 98.9

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.9

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

   10. Applied rewrites 98.9%

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

4. Final simplification 97.5%

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

Alternative 3: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - 1\right) \cdot 6\\ \mathbf{if}\;\frac{t\_0}{4 \cdot \sqrt{x} + \left(1 + x\right)} \leq 2:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (- x 1.0) 6.0)))
   (if (<= (/ t_0 (+ (* 4.0 (sqrt x)) (+ 1.0 x))) 2.0)
     (/ t_0 (fma (sqrt x) 4.0 1.0))
     (/ (* x 6.0) (fma (sqrt x) 4.0 (- x -1.0))))))

C

double code(double x) {
	double t_0 = (x - 1.0) * 6.0;
	double tmp;
	if ((t_0 / ((4.0 * sqrt(x)) + (1.0 + x))) <= 2.0) {
		tmp = t_0 / fma(sqrt(x), 4.0, 1.0);
	} else {
		tmp = (x * 6.0) / fma(sqrt(x), 4.0, (x - -1.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x - 1.0) * 6.0)
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(4.0 * sqrt(x)) + Float64(1.0 + x))) <= 2.0)
		tmp = Float64(t_0 / fma(sqrt(x), 4.0, 1.0));
	else
		tmp = Float64(Float64(x * 6.0) / fma(sqrt(x), 4.0, Float64(x - -1.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x - 1.0), $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * 6.0), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 96.3

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

   5. Applied rewrites 96.3%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.7

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 98.7

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

   7. Applied rewrites 98.7%

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

4. Final simplification 97.4%

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

Alternative 4: 97.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 1.0) * 6.0) / Float64(Float64(4.0 * sqrt(x)) + Float64(1.0 + x))) <= 2.0)
		tmp = Float64(fma(x, 6.0, -6.0) / fma(sqrt(x), 4.0, 1.0));
	else
		tmp = Float64(Float64(x * 6.0) / fma(sqrt(x), 4.0, 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[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(x * 6.0 + -6.0), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * 6.0), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 96.3

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

   5. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 96.3

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

   7. Applied rewrites 96.3%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.7

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 98.7

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

   7. Applied rewrites 98.7%

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

4. Final simplification 97.4%

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

Alternative 5: 52.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \sqrt{x}\\ \mathbf{if}\;\frac{\left(x - 1\right) \cdot 6}{t\_0 + \left(1 + x\right)} \leq -5:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 6, -6\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= (/ (* (- x 1.0) 6.0) (+ t_0 (+ 1.0 x))) -5.0)
     (/ -6.0 (+ (fma 4.0 (sqrt x) x) 1.0))
     (/ (fma x 6.0 -6.0) t_0))))

C

double code(double x) {
	double t_0 = 4.0 * sqrt(x);
	double tmp;
	if ((((x - 1.0) * 6.0) / (t_0 + (1.0 + x))) <= -5.0) {
		tmp = -6.0 / (fma(4.0, sqrt(x), x) + 1.0);
	} else {
		tmp = fma(x, 6.0, -6.0) / t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(4.0 * sqrt(x))
	tmp = 0.0
	if (Float64(Float64(Float64(x - 1.0) * 6.0) / Float64(t_0 + Float64(1.0 + x))) <= -5.0)
		tmp = Float64(-6.0 / Float64(fma(4.0, sqrt(x), x) + 1.0));
	else
		tmp = Float64(fma(x, 6.0, -6.0) / t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x - 1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(t$95$0 + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5.0], N[(-6.0 / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * 6.0 + -6.0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 97.9

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

   5. Applied rewrites 97.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 97.9

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 100.0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 97.9%

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

   if -5 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 7.2

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

   5. Applied rewrites 7.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 7.2

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

   7. Applied rewrites 7.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 7.2%

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

4. Final simplification 56.1%

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

Alternative 6: 52.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 1\right) \cdot 6}{4 \cdot \sqrt{x} + \left(1 + x\right)} \leq -0.1:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* (- x 1.0) 6.0) (+ (* 4.0 (sqrt x)) (+ 1.0 x))) -0.1)
   (/ -6.0 (+ (fma 4.0 (sqrt x) x) 1.0))
   (/ (* x 6.0) (fma (sqrt x) 4.0 1.0))))

C

double code(double x) {
	double tmp;
	if ((((x - 1.0) * 6.0) / ((4.0 * sqrt(x)) + (1.0 + x))) <= -0.1) {
		tmp = -6.0 / (fma(4.0, sqrt(x), x) + 1.0);
	} else {
		tmp = (x * 6.0) / fma(sqrt(x), 4.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 1.0) * 6.0) / Float64(Float64(4.0 * sqrt(x)) + Float64(1.0 + x))) <= -0.1)
		tmp = Float64(-6.0 / Float64(fma(4.0, sqrt(x), x) + 1.0));
	else
		tmp = Float64(Float64(x * 6.0) / fma(sqrt(x), 4.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(x - 1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], N[(-6.0 / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * 6.0), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -0.10000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 96.9

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

   5. Applied rewrites 96.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 96.9

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

   7. Applied rewrites 96.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 100.0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 96.8%

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

   if -0.10000000000000001 < (/.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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 6.9

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

   5. Applied rewrites 6.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 6.9

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

   8. Applied rewrites 6.9%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 1\right) \cdot 6}{4 \cdot \sqrt{x} + \left(1 + x\right)} \leq -0.1:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 1\right) \cdot 6}{4 \cdot \sqrt{x} + \left(1 + x\right)} \leq -0.1:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \sqrt{x}, 0.375\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* (- x 1.0) 6.0) (+ (* 4.0 (sqrt x)) (+ 1.0 x))) -0.1)
   (/ -6.0 (+ (fma 4.0 (sqrt x) x) 1.0))
   (/ (fma 1.5 (sqrt x) 0.375) x)))

C

double code(double x) {
	double tmp;
	if ((((x - 1.0) * 6.0) / ((4.0 * sqrt(x)) + (1.0 + x))) <= -0.1) {
		tmp = -6.0 / (fma(4.0, sqrt(x), x) + 1.0);
	} else {
		tmp = fma(1.5, sqrt(x), 0.375) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 1.0) * 6.0) / Float64(Float64(4.0 * sqrt(x)) + Float64(1.0 + x))) <= -0.1)
		tmp = Float64(-6.0 / Float64(fma(4.0, sqrt(x), x) + 1.0));
	else
		tmp = Float64(fma(1.5, sqrt(x), 0.375) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(x - 1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], N[(-6.0 / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.5 * N[Sqrt[x], $MachinePrecision] + 0.375), $MachinePrecision] / x), $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)))) < -0.10000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 96.9

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

   5. Applied rewrites 96.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 96.9

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

   7. Applied rewrites 96.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 100.0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 96.8%

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

   if -0.10000000000000001 < (/.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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 1.9

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

   5. Applied rewrites 1.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 1.9%

      \[\leadsto -1.5 \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

   9. Taylor expanded in x around -inf

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

   11. Applied rewrites 6.9%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 1\right) \cdot 6}{4 \cdot \sqrt{x} + \left(1 + x\right)} \leq -0.1:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \sqrt{x}, 0.375\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 1\right) \cdot 6}{4 \cdot \sqrt{x} + \left(1 + x\right)} \leq -0.1:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \sqrt{x}, 0.375\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* (- x 1.0) 6.0) (+ (* 4.0 (sqrt x)) (+ 1.0 x))) -0.1)
   (/ -6.0 (fma (sqrt x) 4.0 1.0))
   (/ (fma 1.5 (sqrt x) 0.375) x)))

C

double code(double x) {
	double tmp;
	if ((((x - 1.0) * 6.0) / ((4.0 * sqrt(x)) + (1.0 + x))) <= -0.1) {
		tmp = -6.0 / fma(sqrt(x), 4.0, 1.0);
	} else {
		tmp = fma(1.5, sqrt(x), 0.375) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 1.0) * 6.0) / Float64(Float64(4.0 * sqrt(x)) + Float64(1.0 + x))) <= -0.1)
		tmp = Float64(-6.0 / fma(sqrt(x), 4.0, 1.0));
	else
		tmp = Float64(fma(1.5, sqrt(x), 0.375) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(x - 1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], N[(-6.0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.5 * N[Sqrt[x], $MachinePrecision] + 0.375), $MachinePrecision] / x), $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)))) < -0.10000000000000001

   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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if -0.10000000000000001 < (/.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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 1.9

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

   5. Applied rewrites 1.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 1.9%

      \[\leadsto -1.5 \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

   9. Taylor expanded in x around -inf

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

   11. Applied rewrites 6.9%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 1\right) \cdot 6}{4 \cdot \sqrt{x} + \left(1 + x\right)} \leq -0.1:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \sqrt{x}, 0.375\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lower-+.f64 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 10: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

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

Alternative 11: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-sqrt.f64 56.1

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

5. Applied rewrites 56.1%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 56.1

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

7. Applied rewrites 56.1%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sqrt.f64 99.8

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

10. Applied rewrites 99.8%

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

Alternative 12: 6.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 7.1%

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

   10. Applied rewrites 7.1%

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

   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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 1.9

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

   5. Applied rewrites 1.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 1.9%

      \[\leadsto -1.5 \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

   9. Taylor expanded in x around -inf

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

   11. Applied rewrites 6.9%

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

Alternative 13: 52.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-sqrt.f64 56.1

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

5. Applied rewrites 56.1%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 56.1

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

7. Applied rewrites 56.1%

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

Alternative 14: 6.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -1.5 / sqrt(x);
	} else {
		tmp = sqrt((1.0 / 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 = (-1.5d0) / sqrt(x)
    else
        tmp = sqrt((1.0d0 / x)) * 1.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 7.1%

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

   10. Applied rewrites 7.1%

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

   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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 1.9

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

   5. Applied rewrites 1.9%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 6.9%

      \[\leadsto 1.5 \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.0%

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

Alternative 15: 4.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(-1.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sqrt.f64 53.7

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

5. Applied rewrites 53.7%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 4.7%

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

10. Applied rewrites 4.7%

    \[\leadsto \color{blue}{\frac{-1.5}{\sqrt{x}}} \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024243 
(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)))))