Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 7.4s

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, 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]

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%

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

   2. sub-neg 99.9%

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

   3. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-/l* 99.9%

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

   2. sub-neg 99.9%

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

   3. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. frac-2neg 99.9%

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

   2. div-inv 99.9%

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

   3. metadata-eval 99.9%

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

   4. distribute-neg-frac 99.9%

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

   5. +-commutative 99.9%

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

   6. fma-def 99.9%

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

5. Applied egg-rr 99.9%

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

   1. associate-/r/ 99.8%

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

   2. associate-*l/ 99.9%

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

   3. *-lft-identity 99.9%

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

   4. +-commutative 99.9%

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

   5. fma-udef 99.9%

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

   6. distribute-neg-in 99.9%

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

   7. metadata-eval 99.9%

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

   8. distribute-lft-neg-in 99.9%

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

   9. remove-double-neg 99.9%

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

   10. *-commutative 99.9%

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

   11. +-commutative 99.9%

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

   12. distribute-neg-in 99.9%

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

   13. metadata-eval 99.9%

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

   14. unsub-neg 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 3: 94.8% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-/l* 99.9%

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

   2. sub-neg 99.9%

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

   3. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. associate-/l* 99.8%

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

   2. metadata-eval 99.8%

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

   3. sub-neg 99.8%

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

   4. frac-2neg 99.8%

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

   5. div-inv 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

   8. distribute-rgt-neg-in 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. *-rgt-identity 99.6%

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

   13. sub-neg 99.6%

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

   14. *-rgt-identity 99.6%

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

   15. +-commutative 99.6%

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

   16. fma-def 99.6%

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

5. Applied egg-rr 99.6%

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

   1. fma-udef 99.6%

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

   2. associate-+r+ 99.6%

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

   3. add-sqr-sqrt 99.6%

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

   4. sqrt-unprod 99.6%

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

   5. *-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. swap-sqr 99.6%

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

   8. add-sqr-sqrt 99.6%

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

   9. metadata-eval 99.6%

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in x around inf 95.1%

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

9. Final simplification 95.1%

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

Alternative 4: 95.1% accurate, 16.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = 6.0d0 * (x + (-1.0d0))
    else
        tmp = 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 6.0 * (x + -1.0);
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = 6.0 * (x + -1.0)
	else:
		tmp = 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(6.0 * Float64(x + -1.0));
	else
		tmp = 6.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = 6.0 * (x + -1.0);
	else
		tmp = 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[(6.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], 6.0]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg 99.9%

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

      2. div-inv 99.9%

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

      3. metadata-eval 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-/r/ 99.9%

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

      2. associate-*l/ 99.9%

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

      3. *-lft-identity 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-udef 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. distribute-lft-neg-in 99.9%

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

      9. remove-double-neg 99.9%

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

      10. *-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

      14. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

      3. associate-+r- 99.9%

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

      4. fma-def 99.9%

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

      5. distribute-lft-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. distribute-lft-neg-in 99.9%

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

      10. distribute-rgt-neg-in 99.9%

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

      11. distribute-lft-in 99.9%

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

      12. sub-neg 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around 0 94.6%

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

       1. associate-/r/ 94.6%

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

       2. metadata-eval 94.6%

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

       3. neg-mul-1 94.6%

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

       4. *-commutative 94.6%

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

       5. distribute-lft-neg-in 94.6%

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

       6. sub-neg 94.6%

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

       7. distribute-neg-in 94.6%

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

       8. metadata-eval 94.6%

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

       9. remove-double-neg 94.6%

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

       10. +-commutative 94.6%

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

   12. Applied egg-rr 94.6%

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

   if 2 < x

   1. Initial program 99.6%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 96.1%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;6 \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

Alternative 5: 95.2% accurate, 16.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 6.0 * (x + -1.0);
	} else {
		tmp = 6.0 - (6.0 / 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 * (x + (-1.0d0))
    else
        tmp = 6.0d0 - (6.0d0 / x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg 99.9%

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

      2. div-inv 99.9%

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

      3. metadata-eval 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-/r/ 99.9%

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

      2. associate-*l/ 99.9%

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

      3. *-lft-identity 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-udef 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. distribute-lft-neg-in 99.9%

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

      9. remove-double-neg 99.9%

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

      10. *-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

      14. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

      3. associate-+r- 99.9%

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

      4. fma-def 99.9%

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

      5. distribute-lft-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. distribute-lft-neg-in 99.9%

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

      10. distribute-rgt-neg-in 99.9%

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

      11. distribute-lft-in 99.9%

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

      12. sub-neg 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around 0 95.2%

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

       1. associate-/r/ 95.2%

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

       2. metadata-eval 95.2%

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

       3. neg-mul-1 95.2%

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

       4. *-commutative 95.2%

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

       5. distribute-lft-neg-in 95.2%

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

       6. sub-neg 95.2%

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

       7. distribute-neg-in 95.2%

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

       8. metadata-eval 95.2%

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

       9. remove-double-neg 95.2%

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

       10. +-commutative 95.2%

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

   12. Applied egg-rr 95.2%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 95.5%

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

   5. Taylor expanded in x around 0 95.5%

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

      1. associate-*r/ 95.5%

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

      2. metadata-eval 95.5%

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

   7. Simplified 95.5%

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

4. Final simplification 95.4%

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

Alternative 6: 95.1% 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 99.9%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 95.2%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 95.5%

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

4. Final simplification 95.4%

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

Alternative 7: 49.3% 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%

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

   2. sub-neg 99.9%

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

   3. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 45.1%

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

5. Final simplification 45.1%

   \[\leadsto -6 \]

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 2023308 
(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)))))