Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 7.5s

Alternatives: 7

Speedup: 1.0×

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. fma-udef 99.8%

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

   3. *-un-lft-identity 99.8%

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

   4. times-frac 99.9%

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

   5. metadata-eval 99.9%

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

   6. sub-neg 99.9%

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

   7. metadata-eval 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 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]

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. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 99.8%

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

4. Final simplification 99.8%

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

Alternative 4: 49.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf 46.8%

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

4. Final simplification 46.8%

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

Alternative 5: 50.2% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 6 - \frac{6}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 6.0 (/ 6.0 x)))

C

double code(double x) {
	return 6.0 - (6.0 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0 - (6.0d0 / x)
end function

Java

public static double code(double x) {
	return 6.0 - (6.0 / x);
}

Python

def code(x):
	return 6.0 - (6.0 / x)

Julia

function code(x)
	return Float64(6.0 - Float64(6.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = 6.0 - (6.0 / x);
end

Wolfram

code[x_] := N[(6.0 - N[(6.0 / 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. Step-by-step derivation

   1. associate-*l/ 99.9%

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

   2. sub-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. fma-def 99.9%

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

   5. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 48.0%

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

6. Taylor expanded in x around 0 48.0%

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

   1. associate-*r/ 48.0%

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

   2. metadata-eval 48.0%

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

8. Simplified 48.0%

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

9. Final simplification 48.0%

   \[\leadsto 6 - \frac{6}{x} \]
10. Add Preprocessing

Alternative 6: 48.6% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ -6 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -6.0)

C

double code(double x) {
	return -6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -6.0d0
end function

Java

public static double code(double x) {
	return -6.0;
}

Python

def code(x):
	return -6.0

Julia

function code(x)
	return -6.0
end

MATLAB

function tmp = code(x)
	tmp = -6.0;
end

Wolfram

code[x_] := -6.0

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.9%

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

   2. sub-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. fma-def 99.9%

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

   5. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 52.9%

   \[\leadsto \color{blue}{-6} \]

6. Final simplification 52.9%

   \[\leadsto -6 \]
7. Add Preprocessing

Alternative 7: 48.3% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 6 \end{array} \]

FPCore

(FPCore (x) :precision binary64 6.0)

C

double code(double x) {
	return 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0
end function

Java

public static double code(double x) {
	return 6.0;
}

Python

def code(x):
	return 6.0

Julia

function code(x)
	return 6.0
end

MATLAB

function tmp = code(x)
	tmp = 6.0;
end

Wolfram

code[x_] := 6.0

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.9%

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

   2. sub-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. fma-def 99.9%

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

   5. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 46.1%

   \[\leadsto \color{blue}{6} \]

6. Final simplification 46.1%

   \[\leadsto 6 \]
7. Add Preprocessing

Developer target: 99.9% accurate, 1.0× 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 2024033 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))