Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.8%

Time: 5.6s

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: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{6}{\frac{x + \frac{x \cdot 16 + -1}{\sqrt{x \cdot 16} + -1}}{x + -1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  6.0
  (/ (+ x (/ (+ (* x 16.0) -1.0) (+ (sqrt (* x 16.0)) -1.0))) (+ x -1.0))))

C

double code(double x) {
	return 6.0 / ((x + (((x * 16.0) + -1.0) / (sqrt((x * 16.0)) + -1.0))) / (x + -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0 / ((x + (((x * 16.0d0) + (-1.0d0)) / (sqrt((x * 16.0d0)) + (-1.0d0)))) / (x + (-1.0d0)))
end function

Java

public static double code(double x) {
	return 6.0 / ((x + (((x * 16.0) + -1.0) / (Math.sqrt((x * 16.0)) + -1.0))) / (x + -1.0));
}

Python

def code(x):
	return 6.0 / ((x + (((x * 16.0) + -1.0) / (math.sqrt((x * 16.0)) + -1.0))) / (x + -1.0))

Julia

function code(x)
	return Float64(6.0 / Float64(Float64(x + Float64(Float64(Float64(x * 16.0) + -1.0) / Float64(sqrt(Float64(x * 16.0)) + -1.0))) / Float64(x + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 6.0 / ((x + (((x * 16.0) + -1.0) / (sqrt((x * 16.0)) + -1.0))) / (x + -1.0));
end

Wolfram

code[x_] := N[(6.0 / N[(N[(x + N[(N[(N[(x * 16.0), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[Sqrt[N[(x * 16.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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. Step-by-step derivation

   1. *-un-lft-identity 99.8%

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

   2. times-frac 99.9%

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

   3. metadata-eval 99.9%

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

   4. sub-neg 99.9%

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

   5. metadata-eval 99.9%

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

   6. +-commutative 99.9%

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

   7. fma-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. clear-num 99.9%

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

   2. un-div-inv 99.9%

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

5. Applied egg-rr 99.9%

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

   1. fma-udef 99.9%

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

   2. flip-+ 99.9%

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

   3. *-commutative 99.9%

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

   4. *-commutative 99.9%

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

   5. swap-sqr 99.9%

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

   6. add-sqr-sqrt 99.9%

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

   7. metadata-eval 99.9%

      \[\leadsto \frac{6}{\frac{x + \frac{x \cdot \color{blue}{16} - 1 \cdot 1}{4 \cdot \sqrt{x} - 1}}{x + -1}} \]

   8. metadata-eval 99.9%

      \[\leadsto \frac{6}{\frac{x + \frac{x \cdot 16 - \color{blue}{1}}{4 \cdot \sqrt{x} - 1}}{x + -1}} \]

   9. add-sqr-sqrt 99.9%

      \[\leadsto \frac{6}{\frac{x + \frac{x \cdot 16 - 1}{\color{blue}{\sqrt{4 \cdot \sqrt{x}} \cdot \sqrt{4 \cdot \sqrt{x}}} - 1}}{x + -1}} \]

   10. sqrt-unprod 99.9%

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

   11. *-commutative 99.9%

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

   12. *-commutative 99.9%

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

   13. swap-sqr 99.9%

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

   14. add-sqr-sqrt 99.9%

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

   15. metadata-eval 99.9%

       \[\leadsto \frac{6}{\frac{x + \frac{x \cdot 16 - 1}{\sqrt{x \cdot \color{blue}{16}} - 1}}{x + -1}} \]

7. Applied egg-rr 99.9%

   \[\leadsto \frac{6}{\frac{x + \color{blue}{\frac{x \cdot 16 - 1}{\sqrt{x \cdot 16} - 1}}}{x + -1}} \]

8. Final simplification 99.9%

   \[\leadsto \frac{6}{\frac{x + \frac{x \cdot 16 + -1}{\sqrt{x \cdot 16} + -1}}{x + -1}} \]

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{6}{\frac{x + \left(1 + \sqrt{x \cdot 16}\right)}{x + -1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 6.0 (/ (+ x (+ 1.0 (sqrt (* x 16.0)))) (+ x -1.0))))

C

double code(double x) {
	return 6.0 / ((x + (1.0 + sqrt((x * 16.0)))) / (x + -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0 / ((x + (1.0d0 + sqrt((x * 16.0d0)))) / (x + (-1.0d0)))
end function

Java

public static double code(double x) {
	return 6.0 / ((x + (1.0 + Math.sqrt((x * 16.0)))) / (x + -1.0));
}

Python

def code(x):
	return 6.0 / ((x + (1.0 + math.sqrt((x * 16.0)))) / (x + -1.0))

Julia

function code(x)
	return Float64(6.0 / Float64(Float64(x + Float64(1.0 + sqrt(Float64(x * 16.0)))) / Float64(x + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 6.0 / ((x + (1.0 + sqrt((x * 16.0)))) / (x + -1.0));
end

Wolfram

code[x_] := N[(6.0 / N[(N[(x + N[(1.0 + N[Sqrt[N[(x * 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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. Step-by-step derivation

   1. *-un-lft-identity 99.8%

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

   2. times-frac 99.9%

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

   3. metadata-eval 99.9%

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

   4. sub-neg 99.9%

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

   5. metadata-eval 99.9%

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

   6. +-commutative 99.9%

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

   7. fma-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. clear-num 99.9%

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

   2. un-div-inv 99.9%

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

5. Applied egg-rr 99.9%

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

   1. fma-udef 99.9%

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

   2. add-sqr-sqrt 99.9%

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

   3. sqrt-unprod 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. swap-sqr 99.9%

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

   7. add-sqr-sqrt 99.9%

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

   8. metadata-eval 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

   \[\leadsto \frac{6}{\frac{x + \left(1 + \sqrt{x \cdot 16}\right)}{x + -1}} \]

Alternative 3: 95.4% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \frac{6}{\frac{x + 1}{x + -1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return 6.0 / ((x + 1.0) / (x + -1.0));
}

Python

def code(x):
	return 6.0 / ((x + 1.0) / (x + -1.0))

Julia

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

MATLAB

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

Wolfram

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

   1. *-un-lft-identity 99.8%

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

   2. times-frac 99.9%

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

   3. metadata-eval 99.9%

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

   4. sub-neg 99.9%

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

   5. metadata-eval 99.9%

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

   6. +-commutative 99.9%

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

   7. fma-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. clear-num 99.9%

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

   2. un-div-inv 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in x around 0 95.3%

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

7. Final simplification 95.3%

   \[\leadsto \frac{6}{\frac{x + 1}{x + -1}} \]

Alternative 4: 95.4% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) -6.0 (- 6.0 (/ 12.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0;
	} else {
		tmp = 6.0 - (12.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = -6.0d0
    else
        tmp = 6.0d0 - (12.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0;
	} else {
		tmp = 6.0 - (12.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0
	else:
		tmp = 6.0 - (12.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = -6.0;
	else
		tmp = Float64(6.0 - Float64(12.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0;
	else
		tmp = 6.0 - (12.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], -6.0, N[(6.0 - N[(12.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 95.0%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. metadata-eval 99.7%

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

      5. metadata-eval 99.7%

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

      6. fma-def 99.7%

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

      7. metadata-eval 99.7%

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

      8. associate-+l+ 99.7%

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

      9. +-commutative 99.7%

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

      10. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 95.3%

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

   5. Taylor expanded in x around inf 95.5%

      \[\leadsto \color{blue}{6 - 12 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 95.5%

         \[\leadsto 6 - \color{blue}{\frac{12 \cdot 1}{x}} \]

      2. metadata-eval 95.5%

         \[\leadsto 6 - \frac{\color{blue}{12}}{x} \]

   7. Simplified 95.5%

      \[\leadsto \color{blue}{6 - \frac{12}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]

Alternative 5: 95.4% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;x \cdot 12 - 6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.0) (- (* x 12.0) 6.0) (- 6.0 (/ 12.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = (x * 12.0) - 6.0;
	} else {
		tmp = 6.0 - (12.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = (x * 12.0d0) - 6.0d0
    else
        tmp = 6.0d0 - (12.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = (x * 12.0) - 6.0;
	} else {
		tmp = 6.0 - (12.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = (x * 12.0) - 6.0
	else:
		tmp = 6.0 - (12.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(x * 12.0) - 6.0);
	else
		tmp = Float64(6.0 - Float64(12.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (x * 12.0) - 6.0;
	else
		tmp = 6.0 - (12.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[(N[(x * 12.0), $MachinePrecision] - 6.0), $MachinePrecision], N[(6.0 - N[(12.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. fma-def 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-+l+ 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 95.0%

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

   5. Taylor expanded in x around 0 95.0%

      \[\leadsto \color{blue}{12 \cdot x - 6} \]

   if 2 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. metadata-eval 99.7%

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

      5. metadata-eval 99.7%

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

      6. fma-def 99.7%

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

      7. metadata-eval 99.7%

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

      8. associate-+l+ 99.7%

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

      9. +-commutative 99.7%

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

      10. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 95.3%

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

   5. Taylor expanded in x around inf 95.5%

      \[\leadsto \color{blue}{6 - 12 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 95.5%

         \[\leadsto 6 - \color{blue}{\frac{12 \cdot 1}{x}} \]

      2. metadata-eval 95.5%

         \[\leadsto 6 - \frac{\color{blue}{12}}{x} \]

   7. Simplified 95.5%

      \[\leadsto \color{blue}{6 - \frac{12}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;x \cdot 12 - 6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]

Alternative 6: 95.4% accurate, 36.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) -6.0 6.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0;
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = -6.0d0
    else
        tmp = 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0;
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0
	else:
		tmp = 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = -6.0;
	else
		tmp = 6.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0;
	else
		tmp = 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], -6.0, 6.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 95.0%

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

   if 1 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 95.5%

      \[\leadsto \color{blue}{6} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

Alternative 7: 48.1% 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. Taylor expanded in x around 0 47.6%

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

3. Final simplification 47.6%

   \[\leadsto -6 \]

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 2023338 
(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)))))