Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.7%

Time: 11.2s

Alternatives: 11

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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{6 \cdot x + -6}{\left(x + 1\right) - \frac{-4}{{x}^{-0.5}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (* 6.0 x) -6.0) (- (+ x 1.0) (/ -4.0 (pow x -0.5)))))

C

double code(double x) {
	return ((6.0 * x) + -6.0) / ((x + 1.0) - (-4.0 / pow(x, -0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((6.0d0 * x) + (-6.0d0)) / ((x + 1.0d0) - ((-4.0d0) / (x ** (-0.5d0))))
end function

Java

public static double code(double x) {
	return ((6.0 * x) + -6.0) / ((x + 1.0) - (-4.0 / Math.pow(x, -0.5)));
}

Python

def code(x):
	return ((6.0 * x) + -6.0) / ((x + 1.0) - (-4.0 / math.pow(x, -0.5)))

Julia

function code(x)
	return Float64(Float64(Float64(6.0 * x) + -6.0) / Float64(Float64(x + 1.0) - Float64(-4.0 / (x ^ -0.5))))
end

MATLAB

function tmp = code(x)
	tmp = ((6.0 * x) + -6.0) / ((x + 1.0) - (-4.0 / (x ^ -0.5)));
end

Wolfram

code[x_] := N[(N[(N[(6.0 * x), $MachinePrecision] + -6.0), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] - N[(-4.0 / N[Power[x, -0.5], $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. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

   10. remove-double-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

   17. metadata-eval 99.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

3. Simplified 99.8%

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

   1. pow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{\frac{1}{2}}\right), -4\right)\right)\right)\right) \]

   2. sqr-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), -4\right)\right)\right)\right) \]

   3. pow-prod-down N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left({\left(x \cdot x\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), -4\right)\right)\right)\right) \]

   4. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x \cdot x\right), \left(\frac{\frac{1}{2}}{2}\right)\right), -4\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{\frac{1}{2}}{2}\right)\right), -4\right)\right)\right)\right) \]

   6. metadata-eval 76.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{4}\right), -4\right)\right)\right)\right) \]

6. Applied egg-rr 76.5%

   \[\leadsto \frac{6 \cdot x + -6}{x + \left(1 - \color{blue}{{\left(x \cdot x\right)}^{0.25}} \cdot -4\right)} \]
7. Step-by-step derivation

   1. unpow-prod-down N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{\frac{1}{4}} \cdot {x}^{\frac{1}{4}}\right), -4\right)\right)\right)\right) \]

   2. pow-prod-up N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{\left(\frac{1}{4} + \frac{1}{4}\right)}\right), -4\right)\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{\frac{1}{2}}\right), -4\right)\right)\right)\right) \]

   4. pow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), -4\right)\right)\right)\right) \]

   5. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)}\right), -4\right)\right)\right)\right) \]

   6. sub0-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\mathsf{neg}\left(\left(0 - x\right)\right)}\right), -4\right)\right)\right)\right) \]

   7. flip-- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{0 \cdot 0 - x \cdot x}{0 + x}\right)}\right), -4\right)\right)\right)\right) \]

   8. +-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{0 \cdot 0 - x \cdot x}{x}\right)}\right), -4\right)\right)\right)\right) \]

   9. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left(0 \cdot 0 - x \cdot x\right)\right)}{x}}\right), -4\right)\right)\right)\right) \]

   10. sqrt-div N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\sqrt{\mathsf{neg}\left(\left(0 \cdot 0 - x \cdot x\right)\right)}}{\sqrt{x}}\right), -4\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\sqrt{\mathsf{neg}\left(\left(0 - x \cdot x\right)\right)}}{\sqrt{x}}\right), -4\right)\right)\right)\right) \]

   12. sub0-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\sqrt{\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot x\right)\right)\right)}}{\sqrt{x}}\right), -4\right)\right)\right)\right) \]

   13. remove-double-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\sqrt{x \cdot x}}{\sqrt{x}}\right), -4\right)\right)\right)\right) \]

   14. sqrt-unprod N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\sqrt{x} \cdot \sqrt{x}}{\sqrt{x}}\right), -4\right)\right)\right)\right) \]

   15. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{x}{\sqrt{x}}\right), -4\right)\right)\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\sqrt{x}\right)\right), -4\right)\right)\right)\right) \]

   17. sqrt-lowering-sqrt.f64 99.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(x\right)\right), -4\right)\right)\right)\right) \]

8. Applied egg-rr 99.9%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(1 - \frac{x}{\sqrt{x}} \cdot -4\right) + \color{blue}{x}\right)\right) \]

   2. associate-+l- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 - \color{blue}{\left(\frac{x}{\sqrt{x}} \cdot -4 - x\right)}\right)\right) \]

   3. fmm-def N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 - \mathsf{fma}\left(\frac{x}{\sqrt{x}}, \color{blue}{-4}, \mathsf{neg}\left(x\right)\right)\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 - \mathsf{fma}\left(\frac{1}{\frac{\sqrt{x}}{x}}, -4, \mathsf{neg}\left(x\right)\right)\right)\right) \]

   5. associate-/r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 - \mathsf{fma}\left(\frac{1}{\sqrt{x}} \cdot x, -4, \mathsf{neg}\left(x\right)\right)\right)\right) \]

   6. inv-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 - \mathsf{fma}\left({\left(\sqrt{x}\right)}^{-1} \cdot x, -4, \mathsf{neg}\left(x\right)\right)\right)\right) \]

   7. sqrt-pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 - \mathsf{fma}\left({x}^{\left(\frac{-1}{2}\right)} \cdot x, -4, \mathsf{neg}\left(x\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 - \mathsf{fma}\left({x}^{\frac{-1}{2}} \cdot x, -4, \mathsf{neg}\left(x\right)\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 - \mathsf{fma}\left({x}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot x, -4, \mathsf{neg}\left(x\right)\right)\right)\right) \]

   10. pow-plus N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 - \mathsf{fma}\left({x}^{\left(-1 \cdot \frac{1}{2} + 1\right)}, -4, \mathsf{neg}\left(x\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 - \mathsf{fma}\left({x}^{\left(\frac{-1}{2} + 1\right)}, -4, \mathsf{neg}\left(x\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 - \mathsf{fma}\left({x}^{\frac{1}{2}}, -4, \mathsf{neg}\left(x\right)\right)\right)\right) \]

   13. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 - \mathsf{fma}\left(\sqrt{x}, -4, \mathsf{neg}\left(x\right)\right)\right)\right) \]

   14. fmm-def N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 - \left(\sqrt{x} \cdot -4 - \color{blue}{x}\right)\right)\right) \]

   15. associate-+l- N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(1 - \sqrt{x} \cdot -4\right) + \color{blue}{x}\right)\right) \]

   16. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 - \sqrt{x} \cdot -4\right)}\right)\right) \]

   17. associate-+r- N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + 1\right) - \color{blue}{\sqrt{x} \cdot -4}\right)\right) \]

   18. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{\_.f64}\left(\left(x + 1\right), \color{blue}{\left(\sqrt{x} \cdot -4\right)}\right)\right) \]

   19. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\color{blue}{\sqrt{x}} \cdot -4\right)\right)\right) \]

10. Applied egg-rr 99.9%

    \[\leadsto \frac{6 \cdot x + -6}{\color{blue}{\left(x + 1\right) - \frac{-4}{{x}^{-0.5}}}} \]
11. Add Preprocessing

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \sqrt{x} \cdot 4\\ \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;\frac{6 \cdot x + -6}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{x + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (sqrt x) 4.0))))
   (if (<= x 3.5) (/ (+ (* 6.0 x) -6.0) t_0) (* 6.0 (/ x (+ x t_0))))))

C

double code(double x) {
	double t_0 = 1.0 + (sqrt(x) * 4.0);
	double tmp;
	if (x <= 3.5) {
		tmp = ((6.0 * x) + -6.0) / t_0;
	} else {
		tmp = 6.0 * (x / (x + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (sqrt(x) * 4.0d0)
    if (x <= 3.5d0) then
        tmp = ((6.0d0 * x) + (-6.0d0)) / t_0
    else
        tmp = 6.0d0 * (x / (x + t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (Math.sqrt(x) * 4.0);
	double tmp;
	if (x <= 3.5) {
		tmp = ((6.0 * x) + -6.0) / t_0;
	} else {
		tmp = 6.0 * (x / (x + t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 + (math.sqrt(x) * 4.0)
	tmp = 0
	if x <= 3.5:
		tmp = ((6.0 * x) + -6.0) / t_0
	else:
		tmp = 6.0 * (x / (x + t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(sqrt(x) * 4.0))
	tmp = 0.0
	if (x <= 3.5)
		tmp = Float64(Float64(Float64(6.0 * x) + -6.0) / t_0);
	else
		tmp = Float64(6.0 * Float64(x / Float64(x + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 + (sqrt(x) * 4.0);
	tmp = 0.0;
	if (x <= 3.5)
		tmp = ((6.0 * x) + -6.0) / t_0;
	else
		tmp = 6.0 * (x / (x + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Sqrt[x], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.5], N[(N[(N[(6.0 * x), $MachinePrecision] + -6.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(6.0 * N[(x / N[(x + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \color{blue}{\left(1 - -4 \cdot \sqrt{x}\right)}\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot -4\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \left(\sqrt{x} \cdot 4\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{4}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 97.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), 4\right)\right)\right) \]

   7. Simplified 97.5%

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

   if 3.5 < 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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(6 \cdot x\right)}, \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot 6\right), \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

      2. *-lowering-*.f64 97.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 6\right), \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   7. Simplified 97.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(6, \color{blue}{\left(\frac{x}{x + \left(1 - \sqrt{x} \cdot -4\right)}\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{\left(1 - \sqrt{x} \cdot -4\right) + \color{blue}{x}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \color{blue}{\left(\sqrt{x} \cdot -4 - x\right)}}\right)\right) \]

      6. fmm-def N/A

         \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left(\sqrt{x}, \color{blue}{-4}, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left({x}^{\frac{1}{2}}, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left({x}^{\left(\frac{-1}{2} + 1\right)}, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left({x}^{\left(-1 \cdot \frac{1}{2} + 1\right)}, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      10. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left({x}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot x, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left({x}^{\frac{-1}{2}} \cdot x, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left({x}^{\left(\frac{-1}{2}\right)} \cdot x, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      13. sqrt-pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left({\left(\sqrt{x}\right)}^{-1} \cdot x, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      14. inv-pow N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left(\frac{1}{\sqrt{x}} \cdot x, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left(\frac{1}{\frac{\sqrt{x}}{x}}, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left(\frac{x}{\sqrt{x}}, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      17. fmm-def N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \left(\frac{x}{\sqrt{x}} \cdot -4 - \color{blue}{x}\right)}\right)\right) \]

      18. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{\left(1 - \frac{x}{\sqrt{x}} \cdot -4\right) + \color{blue}{x}}\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{x + \color{blue}{\left(1 - \frac{x}{\sqrt{x}} \cdot -4\right)}}\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(6, \mathsf{/.f64}\left(x, \color{blue}{\left(x + \left(1 - \frac{x}{\sqrt{x}} \cdot -4\right)\right)}\right)\right) \]

   9. Applied egg-rr 97.9%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / (x + (1.0d0 - ((-4.0d0) * sqrt(x))))
    else
        tmp = 6.0d0 * (x / (x + (1.0d0 + (sqrt(x) * 4.0d0))))
    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 - (-4.0 * Math.sqrt(x))));
	} else {
		tmp = 6.0 * (x / (x + (1.0 + (Math.sqrt(x) * 4.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / (x + (1.0 - (-4.0 * math.sqrt(x))))
	else:
		tmp = 6.0 * (x / (x + (1.0 + (math.sqrt(x) * 4.0))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / (x + (1.0 - (-4.0 * sqrt(x))));
	else
		tmp = 6.0 * (x / (x + (1.0 + (sqrt(x) * 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(x + N[(1.0 - N[(-4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(x / N[(x + N[(1.0 + N[(N[Sqrt[x], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{-6}, \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 98.1%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(6 \cdot x\right)}, \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot 6\right), \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

      2. *-lowering-*.f64 97.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 6\right), \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   7. Simplified 97.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(6, \color{blue}{\left(\frac{x}{x + \left(1 - \sqrt{x} \cdot -4\right)}\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{\left(1 - \sqrt{x} \cdot -4\right) + \color{blue}{x}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \color{blue}{\left(\sqrt{x} \cdot -4 - x\right)}}\right)\right) \]

      6. fmm-def N/A

         \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left(\sqrt{x}, \color{blue}{-4}, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left({x}^{\frac{1}{2}}, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left({x}^{\left(\frac{-1}{2} + 1\right)}, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left({x}^{\left(-1 \cdot \frac{1}{2} + 1\right)}, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      10. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left({x}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot x, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left({x}^{\frac{-1}{2}} \cdot x, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left({x}^{\left(\frac{-1}{2}\right)} \cdot x, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      13. sqrt-pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left({\left(\sqrt{x}\right)}^{-1} \cdot x, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      14. inv-pow N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left(\frac{1}{\sqrt{x}} \cdot x, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left(\frac{1}{\frac{\sqrt{x}}{x}}, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      16. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \mathsf{fma}\left(\frac{x}{\sqrt{x}}, -4, \mathsf{neg}\left(x\right)\right)}\right)\right) \]

      17. fmm-def N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{1 - \left(\frac{x}{\sqrt{x}} \cdot -4 - \color{blue}{x}\right)}\right)\right) \]

      18. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{\left(1 - \frac{x}{\sqrt{x}} \cdot -4\right) + \color{blue}{x}}\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(6, \left(\frac{x}{x + \color{blue}{\left(1 - \frac{x}{\sqrt{x}} \cdot -4\right)}}\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(6, \mathsf{/.f64}\left(x, \color{blue}{\left(x + \left(1 - \frac{x}{\sqrt{x}} \cdot -4\right)\right)}\right)\right) \]

   9. Applied egg-rr 97.3%

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

4. Final simplification 97.7%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (x + (1.0 - (-4.0 * sqrt(x))));
	} else {
		tmp = 6.0 / (1.0 + (4.0 / sqrt(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 - ((-4.0d0) * sqrt(x))))
    else
        tmp = 6.0d0 / (1.0d0 + (4.0d0 / sqrt(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 - (-4.0 * Math.sqrt(x))));
	} else {
		tmp = 6.0 / (1.0 + (4.0 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / (x + (1.0 - (-4.0 * math.sqrt(x))))
	else:
		tmp = 6.0 / (1.0 + (4.0 / math.sqrt(x)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / (x + (1.0 - (-4.0 * sqrt(x))));
	else
		tmp = 6.0 / (1.0 + (4.0 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(x + N[(1.0 - N[(-4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 + N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{-6}, \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 98.1%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(1 - -4 \cdot \sqrt{\frac{1}{x}}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(1 + \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{\frac{1}{x}}\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{\frac{1}{x}}\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot -4\right)\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{x}} \cdot \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{x}} \cdot 4\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \color{blue}{4}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), 4\right)\right)\right) \]

      9. /-lowering-/.f64 97.2%

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), 4\right)\right)\right) \]

   7. Simplified 97.2%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(\sqrt{\frac{1}{x}} \cdot 4 + \color{blue}{1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot 4\right), \color{blue}{1}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \sqrt{\frac{1}{x}}\right), 1\right)\right) \]

      4. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \frac{\sqrt{1}}{\sqrt{x}}\right), 1\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \frac{1}{\sqrt{x}}\right), 1\right)\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(\frac{4}{\sqrt{x}}\right), 1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \left(\sqrt{x}\right)\right), 1\right)\right) \]

      8. sqrt-lowering-sqrt.f64 97.2%

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right), 1\right)\right) \]

   9. Applied egg-rr 97.2%

      \[\leadsto \frac{6}{\color{blue}{\frac{4}{\sqrt{x}} + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

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

Alternative 5: 99.7% accurate, 1.0× speedup

Math

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

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(x + Float64(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[(x + N[(1.0 - N[(-4.0 * N[Sqrt[x], $MachinePrecision]), $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. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

   10. remove-double-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

   17. metadata-eval 99.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(6.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 4.0), $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. Final simplification 99.8%

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

Alternative 7: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + \sqrt{x} \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 + \frac{4}{\sqrt{x}}}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (sqrt(x) * 4.0));
	} else {
		tmp = 6.0 / (1.0 + (4.0 / sqrt(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) / (1.0d0 + (sqrt(x) * 4.0d0))
    else
        tmp = 6.0d0 / (1.0d0 + (4.0d0 / sqrt(x)))
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / (1.0 + (math.sqrt(x) * 4.0))
	else:
		tmp = 6.0 / (1.0 + (4.0 / math.sqrt(x)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / (1.0 + (sqrt(x) * 4.0));
	else
		tmp = 6.0 / (1.0 + (4.0 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(1.0 + N[(N[Sqrt[x], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 + N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \color{blue}{\left(1 - -4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \left(1 + \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot -4\right)\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \left(\sqrt{x} \cdot 4\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{4}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 98.1%

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), 4\right)\right)\right) \]

   7. Simplified 98.1%

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

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(1 - -4 \cdot \sqrt{\frac{1}{x}}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(1 + \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{\frac{1}{x}}\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{\frac{1}{x}}\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot -4\right)\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{x}} \cdot \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{x}} \cdot 4\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \color{blue}{4}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), 4\right)\right)\right) \]

      9. /-lowering-/.f64 97.2%

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), 4\right)\right)\right) \]

   7. Simplified 97.2%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(\sqrt{\frac{1}{x}} \cdot 4 + \color{blue}{1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot 4\right), \color{blue}{1}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \sqrt{\frac{1}{x}}\right), 1\right)\right) \]

      4. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \frac{\sqrt{1}}{\sqrt{x}}\right), 1\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \frac{1}{\sqrt{x}}\right), 1\right)\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(\frac{4}{\sqrt{x}}\right), 1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \left(\sqrt{x}\right)\right), 1\right)\right) \]

      8. sqrt-lowering-sqrt.f64 97.2%

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right), 1\right)\right) \]

   9. Applied egg-rr 97.2%

      \[\leadsto \frac{6}{\color{blue}{\frac{4}{\sqrt{x}} + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

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

Alternative 8: 52.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + \sqrt{x} \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot 1.5\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (sqrt(x) * 4.0));
	} else {
		tmp = sqrt(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 = (-6.0d0) / (1.0d0 + (sqrt(x) * 4.0d0))
    else
        tmp = sqrt(x) * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (Math.sqrt(x) * 4.0));
	} else {
		tmp = Math.sqrt(x) * 1.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / (1.0 + (math.sqrt(x) * 4.0))
	else:
		tmp = math.sqrt(x) * 1.5
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / (1.0 + (sqrt(x) * 4.0));
	else
		tmp = sqrt(x) * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \color{blue}{\left(1 - -4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \left(1 + \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot -4\right)\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \left(\sqrt{x} \cdot 4\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{4}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 98.1%

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), 4\right)\right)\right) \]

   7. Simplified 98.1%

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

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(1 - -4 \cdot \sqrt{\frac{1}{x}}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(1 + \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{\frac{1}{x}}\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{\frac{1}{x}}\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot -4\right)\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{x}} \cdot \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{x}} \cdot 4\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \color{blue}{4}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), 4\right)\right)\right) \]

      9. /-lowering-/.f64 97.2%

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), 4\right)\right)\right) \]

   7. Simplified 97.2%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{3}{2}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\frac{3}{2}}\right) \]

      3. sqrt-lowering-sqrt.f64 7.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{3}{2}\right) \]

   10. Simplified 7.1%

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

Alternative 9: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -1.5 / sqrt(x);
	} else {
		tmp = sqrt(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(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(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(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(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(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[x], $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. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \color{blue}{\left(1 - -4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \left(1 + \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot -4\right)\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \left(\sqrt{x} \cdot 4\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{4}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 98.1%

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), 4\right)\right)\right) \]

   7. Simplified 98.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(-6, \color{blue}{\left(4 \cdot \sqrt{x}\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \left(\sqrt{x} \cdot \color{blue}{4}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{4}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 6.8%

         \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), 4\right)\right) \]

   10. Simplified 6.8%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. metadata-eval N/A

          \[\leadsto \frac{\frac{-3}{2}}{\sqrt{\color{blue}{x}}} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{3}{2}\right)}{\sqrt{\color{blue}{x}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right), \color{blue}{\left(\sqrt{x}\right)}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-3}{2}, \left(\sqrt{\color{blue}{x}}\right)\right) \]

       7. sqrt-lowering-sqrt.f64 6.8%

          \[\leadsto \mathsf{/.f64}\left(\frac{-3}{2}, \mathsf{sqrt.f64}\left(x\right)\right) \]

   12. Applied egg-rr 6.8%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(1 - -4 \cdot \sqrt{\frac{1}{x}}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(1 + \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{\frac{1}{x}}\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{\frac{1}{x}}\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot -4\right)\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{x}} \cdot \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{x}} \cdot 4\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \color{blue}{4}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), 4\right)\right)\right) \]

      9. /-lowering-/.f64 97.2%

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), 4\right)\right)\right) \]

   7. Simplified 97.2%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{3}{2}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\frac{3}{2}}\right) \]

      3. sqrt-lowering-sqrt.f64 7.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{3}{2}\right) \]

   10. Simplified 7.1%

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

Alternative 10: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[Sqrt[x], $MachinePrecision] * -1.5), $MachinePrecision], N[(N[Sqrt[x], $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. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \color{blue}{\left(1 - -4 \cdot \sqrt{x}\right)}\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot -4\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \left(\sqrt{x} \cdot 4\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{4}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 98.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), 4\right)\right)\right) \]

   7. Simplified 98.1%

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

   8. Taylor expanded in x around -inf

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

      1. *-commutative N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{-3}{2}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\frac{-3}{2}}\right) \]

      3. sqrt-lowering-sqrt.f64 6.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{-3}{2}\right) \]

   10. Simplified 6.7%

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

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(1 - -4 \cdot \sqrt{\frac{1}{x}}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(1 + \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{\frac{1}{x}}\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{\frac{1}{x}}\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot -4\right)\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{x}} \cdot \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{x}} \cdot 4\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \color{blue}{4}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), 4\right)\right)\right) \]

      9. /-lowering-/.f64 97.2%

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), 4\right)\right)\right) \]

   7. Simplified 97.2%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{3}{2}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\frac{3}{2}}\right) \]

      3. sqrt-lowering-sqrt.f64 7.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{3}{2}\right) \]

   10. Simplified 7.1%

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

Alternative 11: 4.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[x], $MachinePrecision] * -1.5), $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. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

   10. remove-double-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

   17. metadata-eval 99.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \color{blue}{\left(1 - -4 \cdot \sqrt{x}\right)}\right) \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot -4\right)\right)\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)}\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \left(\sqrt{x} \cdot 4\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{4}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 54.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), 4\right)\right)\right) \]

7. Simplified 54.1%

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

8. Taylor expanded in x around -inf

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

   1. *-commutative N/A

      \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{-3}{2}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\frac{-3}{2}}\right) \]

   3. sqrt-lowering-sqrt.f64 4.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{-3}{2}\right) \]

10. Simplified 4.1%

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

Developer Target 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]

Reproduce

herbie shell --seed 2024161 
(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)))))