Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.6% → 99.9%

Time: 6.7s

Alternatives: 10

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
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. Final simplification 99.9%

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

Alternative 2: 52.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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

   5. Applied rewrites 97.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 97.6

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

   7. Applied rewrites 97.6%

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

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

   1. Initial program 99.7%

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

   3. 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. 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. lower-*.f64 7.1

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

   8. Applied rewrites 7.1%

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

4. Final simplification 57.3%

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

Alternative 3: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.8

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 99.8

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

   12. lift-+.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. sub-neg N/A

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

   16. lower--.f64 N/A

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

   17. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 52.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   5. Applied rewrites 97.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 97.6

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

   7. Applied rewrites 97.6%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
   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 7.1%

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

4. Final simplification 57.3%

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

Alternative 5: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. distribute-rgt-in N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. metadata-eval 99.8

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. sub-neg N/A

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

   19. lower--.f64 N/A

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

   20. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 6: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.8

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 99.8

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

   12. lift-+.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. sub-neg N/A

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

   16. lower--.f64 N/A

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

   17. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. div-add N/A

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

   10. div-inv N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.7

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

8. Applied rewrites 99.7%

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

Alternative 7: 52.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[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 97.5

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

   5. Applied rewrites 97.5%

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

   if 1 < x

   1. Initial program 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 7.1%

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

4. Final simplification 57.2%

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

Alternative 8: 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 57.3

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

5. Applied rewrites 57.3%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 57.3

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

7. Applied rewrites 57.3%

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

Alternative 9: 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 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 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 7.1%

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

4. Final simplification 7.1%

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

Alternative 10: 4.4% 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 54.9

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

5. Applied rewrites 54.9%

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

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 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 2024298 
(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)))))