Numeric.Log:$clog1p from log-domain-0.10.2.1, B

Percentage Accurate: 99.7% → 99.7%

Time: 10.6s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (/ x (+ 1.0 (sqrt (+ x 1.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ x (+ 1.0 (sqrt (+ x 1.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ x (+ 1.0 (sqrt (+ x 1.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.125 + x \cdot 0.0625\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq 3:\\ \;\;\;\;\frac{x}{\frac{x}{2} + \frac{1}{\frac{t\_0 \cdot \left(x \cdot x\right) + -2}{\left(x \cdot x\right) \cdot \left(t\_1 \cdot t\_1\right) + -4}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ -0.125 (* x 0.0625))) (t_1 (* x t_0)))
   (if (<= x 3.0)
     (/
      x
      (+
       (/ x 2.0)
       (/ 1.0 (/ (+ (* t_0 (* x x)) -2.0) (+ (* (* x x) (* t_1 t_1)) -4.0)))))
     (+ (sqrt x) -1.0))))

C

double code(double x) {
	double t_0 = -0.125 + (x * 0.0625);
	double t_1 = x * t_0;
	double tmp;
	if (x <= 3.0) {
		tmp = x / ((x / 2.0) + (1.0 / (((t_0 * (x * x)) + -2.0) / (((x * x) * (t_1 * t_1)) + -4.0))));
	} else {
		tmp = sqrt(x) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.125d0) + (x * 0.0625d0)
    t_1 = x * t_0
    if (x <= 3.0d0) then
        tmp = x / ((x / 2.0d0) + (1.0d0 / (((t_0 * (x * x)) + (-2.0d0)) / (((x * x) * (t_1 * t_1)) + (-4.0d0)))))
    else
        tmp = sqrt(x) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -0.125 + (x * 0.0625);
	double t_1 = x * t_0;
	double tmp;
	if (x <= 3.0) {
		tmp = x / ((x / 2.0) + (1.0 / (((t_0 * (x * x)) + -2.0) / (((x * x) * (t_1 * t_1)) + -4.0))));
	} else {
		tmp = Math.sqrt(x) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = -0.125 + (x * 0.0625)
	t_1 = x * t_0
	tmp = 0
	if x <= 3.0:
		tmp = x / ((x / 2.0) + (1.0 / (((t_0 * (x * x)) + -2.0) / (((x * x) * (t_1 * t_1)) + -4.0))))
	else:
		tmp = math.sqrt(x) + -1.0
	return tmp

Julia

function code(x)
	t_0 = Float64(-0.125 + Float64(x * 0.0625))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (x <= 3.0)
		tmp = Float64(x / Float64(Float64(x / 2.0) + Float64(1.0 / Float64(Float64(Float64(t_0 * Float64(x * x)) + -2.0) / Float64(Float64(Float64(x * x) * Float64(t_1 * t_1)) + -4.0)))));
	else
		tmp = Float64(sqrt(x) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -0.125 + (x * 0.0625);
	t_1 = x * t_0;
	tmp = 0.0;
	if (x <= 3.0)
		tmp = x / ((x / 2.0) + (1.0 / (((t_0 * (x * x)) + -2.0) / (((x * x) * (t_1 * t_1)) + -4.0))));
	else
		tmp = sqrt(x) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.125 + N[(x * 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[x, 3.0], N[(x / N[(N[(x / 2.0), $MachinePrecision] + N[(1.0 / N[(N[(N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3

   1. Initial program 100.0%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(2 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right) + \color{blue}{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right), \color{blue}{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), 2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \frac{-1}{8}\right)\right)\right)\right), 2\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{-1}{8} + \frac{1}{16} \cdot x\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(\frac{1}{16} \cdot x\right)\right)\right)\right)\right), 2\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(x \cdot \frac{1}{16}\right)\right)\right)\right)\right), 2\right)\right) \]

      11. *-lowering-*.f64 99.2%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right)\right)\right)\right), 2\right)\right) \]

   5. Simplified 99.2%

      \[\leadsto \frac{x}{\color{blue}{x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right) + 2}} \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\left(\frac{1}{2} \cdot x + \left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot x\right) + 2\right)\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{2} \cdot x + \color{blue}{\left(\left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot x + 2\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{\left(\left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot x + 2\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \left(\color{blue}{\left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot x} + 2\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \left(\left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot \color{blue}{x} + 2\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{x}{2}\right), \left(\color{blue}{\left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot x} + 2\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \left(\color{blue}{\left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot x} + 2\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{+.f64}\left(\left(\left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot x\right), \color{blue}{2}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{+.f64}\left(\left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot x\right) \cdot x\right), 2\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{+.f64}\left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right), 2\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right), \left(x \cdot x\right)\right), 2\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \left(x \cdot \frac{1}{16}\right)\right), \left(x \cdot x\right)\right), 2\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right), \left(x \cdot x\right)\right), 2\right)\right)\right) \]

      14. *-lowering-*.f64 99.2%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right), \mathsf{*.f64}\left(x, x\right)\right), 2\right)\right)\right) \]

   7. Applied egg-rr 99.2%

      \[\leadsto \frac{x}{\color{blue}{\frac{x}{2} + \left(\left(-0.125 + x \cdot 0.0625\right) \cdot \left(x \cdot x\right) + 2\right)}} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \left(\frac{\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2}{\color{blue}{\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right) - 2}}\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \left(\frac{1}{\color{blue}{\frac{\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right) - 2}{\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2}}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right) - 2}{\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right) - 2\right), \color{blue}{\left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2\right)}\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right)} - 2 \cdot 2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right)} - 2 \cdot 2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right), \left(x \cdot x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right)} \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \left(x \cdot \frac{1}{16}\right)\right), \left(x \cdot x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right), \left(\left(\color{blue}{\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right), \left(x \cdot x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right), \left(\left(\left(\frac{-1}{8} + \color{blue}{x \cdot \frac{1}{16}}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right), \left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right), \mathsf{*.f64}\left(x, x\right)\right), -2\right), \left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right)} - 2 \cdot 2\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right), \mathsf{*.f64}\left(x, x\right)\right), -2\right), \left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 4\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 99.2%

      \[\leadsto \frac{x}{\frac{x}{2} + \color{blue}{\frac{1}{\frac{\left(-0.125 + x \cdot 0.0625\right) \cdot \left(x \cdot x\right) + -2}{\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(-0.125 + x \cdot 0.0625\right)\right) \cdot \left(x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)\right) + -4}}}} \]

   if 3 < x

   1. Initial program 99.3%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

         \[\leadsto \sqrt{x} + -1 \]

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

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

      4. sqrt-lowering-sqrt.f64 97.8%

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

   5. Simplified 97.8%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.125 + x \cdot 0.0625\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq 3.7:\\ \;\;\;\;\frac{x}{\frac{x}{2} + \frac{1}{\frac{t\_0 \cdot \left(x \cdot x\right) + -2}{\left(x \cdot x\right) \cdot \left(t\_1 \cdot t\_1\right) + -4}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ -0.125 (* x 0.0625))) (t_1 (* x t_0)))
   (if (<= x 3.7)
     (/
      x
      (+
       (/ x 2.0)
       (/ 1.0 (/ (+ (* t_0 (* x x)) -2.0) (+ (* (* x x) (* t_1 t_1)) -4.0)))))
     (sqrt x))))

C

double code(double x) {
	double t_0 = -0.125 + (x * 0.0625);
	double t_1 = x * t_0;
	double tmp;
	if (x <= 3.7) {
		tmp = x / ((x / 2.0) + (1.0 / (((t_0 * (x * x)) + -2.0) / (((x * x) * (t_1 * t_1)) + -4.0))));
	} else {
		tmp = sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.125d0) + (x * 0.0625d0)
    t_1 = x * t_0
    if (x <= 3.7d0) then
        tmp = x / ((x / 2.0d0) + (1.0d0 / (((t_0 * (x * x)) + (-2.0d0)) / (((x * x) * (t_1 * t_1)) + (-4.0d0)))))
    else
        tmp = sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -0.125 + (x * 0.0625);
	double t_1 = x * t_0;
	double tmp;
	if (x <= 3.7) {
		tmp = x / ((x / 2.0) + (1.0 / (((t_0 * (x * x)) + -2.0) / (((x * x) * (t_1 * t_1)) + -4.0))));
	} else {
		tmp = Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = -0.125 + (x * 0.0625)
	t_1 = x * t_0
	tmp = 0
	if x <= 3.7:
		tmp = x / ((x / 2.0) + (1.0 / (((t_0 * (x * x)) + -2.0) / (((x * x) * (t_1 * t_1)) + -4.0))))
	else:
		tmp = math.sqrt(x)
	return tmp

Julia

function code(x)
	t_0 = Float64(-0.125 + Float64(x * 0.0625))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (x <= 3.7)
		tmp = Float64(x / Float64(Float64(x / 2.0) + Float64(1.0 / Float64(Float64(Float64(t_0 * Float64(x * x)) + -2.0) / Float64(Float64(Float64(x * x) * Float64(t_1 * t_1)) + -4.0)))));
	else
		tmp = sqrt(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -0.125 + (x * 0.0625);
	t_1 = x * t_0;
	tmp = 0.0;
	if (x <= 3.7)
		tmp = x / ((x / 2.0) + (1.0 / (((t_0 * (x * x)) + -2.0) / (((x * x) * (t_1 * t_1)) + -4.0))));
	else
		tmp = sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.125 + N[(x * 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[x, 3.7], N[(x / N[(N[(x / 2.0), $MachinePrecision] + N[(1.0 / N[(N[(N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[x], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.7000000000000002

   1. Initial program 100.0%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(2 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right) + \color{blue}{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right), \color{blue}{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), 2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \frac{-1}{8}\right)\right)\right)\right), 2\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{-1}{8} + \frac{1}{16} \cdot x\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(\frac{1}{16} \cdot x\right)\right)\right)\right)\right), 2\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(x \cdot \frac{1}{16}\right)\right)\right)\right)\right), 2\right)\right) \]

      11. *-lowering-*.f64 99.2%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right)\right)\right)\right), 2\right)\right) \]

   5. Simplified 99.2%

      \[\leadsto \frac{x}{\color{blue}{x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right) + 2}} \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\left(\frac{1}{2} \cdot x + \left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot x\right) + 2\right)\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{2} \cdot x + \color{blue}{\left(\left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot x + 2\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{\left(\left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot x + 2\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \left(\color{blue}{\left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot x} + 2\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \left(\left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot \color{blue}{x} + 2\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{x}{2}\right), \left(\color{blue}{\left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot x} + 2\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \left(\color{blue}{\left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot x} + 2\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{+.f64}\left(\left(\left(x \cdot \left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)\right) \cdot x\right), \color{blue}{2}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{+.f64}\left(\left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot x\right) \cdot x\right), 2\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{+.f64}\left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right), 2\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right), \left(x \cdot x\right)\right), 2\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \left(x \cdot \frac{1}{16}\right)\right), \left(x \cdot x\right)\right), 2\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right), \left(x \cdot x\right)\right), 2\right)\right)\right) \]

      14. *-lowering-*.f64 99.2%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right), \mathsf{*.f64}\left(x, x\right)\right), 2\right)\right)\right) \]

   7. Applied egg-rr 99.2%

      \[\leadsto \frac{x}{\color{blue}{\frac{x}{2} + \left(\left(-0.125 + x \cdot 0.0625\right) \cdot \left(x \cdot x\right) + 2\right)}} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \left(\frac{\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2}{\color{blue}{\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right) - 2}}\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \left(\frac{1}{\color{blue}{\frac{\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right) - 2}{\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2}}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right) - 2}{\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right) - 2\right), \color{blue}{\left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2\right)}\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right)} - 2 \cdot 2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right)} - 2 \cdot 2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right), \left(x \cdot x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right)} \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \left(x \cdot \frac{1}{16}\right)\right), \left(x \cdot x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right), \left(\left(\color{blue}{\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right), \left(x \cdot x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right), \left(\left(\left(\frac{-1}{8} + \color{blue}{x \cdot \frac{1}{16}}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right), \left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 2 \cdot 2\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right), \mathsf{*.f64}\left(x, x\right)\right), -2\right), \left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right)} - 2 \cdot 2\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, 2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right), \mathsf{*.f64}\left(x, x\right)\right), -2\right), \left(\left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{-1}{8} + x \cdot \frac{1}{16}\right) \cdot \left(x \cdot x\right)\right) - 4\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 99.2%

      \[\leadsto \frac{x}{\frac{x}{2} + \color{blue}{\frac{1}{\frac{\left(-0.125 + x \cdot 0.0625\right) \cdot \left(x \cdot x\right) + -2}{\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(-0.125 + x \cdot 0.0625\right)\right) \cdot \left(x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)\right) + -4}}}} \]

   if 3.7000000000000002 < x

   1. Initial program 99.3%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. sqrt-lowering-sqrt.f64 96.0%

         \[\leadsto \mathsf{sqrt.f64}\left(x\right) \]

   5. Simplified 96.0%

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

Alternative 4: 68.3% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{1}{2 + \frac{x}{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (/ 1.0 (+ 2.0 (/ x 2.0)))))

C

double code(double x) {
	return x * (1.0 / (2.0 + (x / 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 / (2.0d0 + (x / 2.0d0)))
end function

Java

public static double code(double x) {
	return x * (1.0 / (2.0 + (x / 2.0)));
}

Python

def code(x):
	return x * (1.0 / (2.0 + (x / 2.0)))

Julia

function code(x)
	return Float64(x * Float64(1.0 / Float64(2.0 + Float64(x / 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 / (2.0 + (x / 2.0)));
end

Wolfram

code[x_] := N[(x * N[(1.0 / N[(2.0 + N[(x / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(2 + \frac{1}{2} \cdot x\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{2} \cdot x + \color{blue}{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{2}\right)\right) \]

   3. *-lowering-*.f64 71.5%

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

5. Simplified 71.5%

   \[\leadsto \frac{x}{\color{blue}{0.5 \cdot x + 2}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{2} \cdot x + 2}{x}}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{1}{\frac{1}{2} \cdot x + 2} \cdot \color{blue}{x} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{1}{2} \cdot x + 2}\right), \color{blue}{x}\right) \]

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

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

   5. +-commutative N/A

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

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

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. div-inv N/A

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

   10. /-lowering-/.f64 71.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(x, 2\right)\right)\right), x\right) \]

7. Applied egg-rr 71.5%

   \[\leadsto \color{blue}{\frac{1}{2 + \frac{x}{2}} \cdot x} \]

8. Final simplification 71.5%

   \[\leadsto x \cdot \frac{1}{2 + \frac{x}{2}} \]
9. Add Preprocessing

Alternative 5: 68.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{2 + x \cdot 0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ 2.0 (* x 0.5))))

C

double code(double x) {
	return x / (2.0 + (x * 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (2.0d0 + (x * 0.5d0))
end function

Java

public static double code(double x) {
	return x / (2.0 + (x * 0.5));
}

Python

def code(x):
	return x / (2.0 + (x * 0.5))

Julia

function code(x)
	return Float64(x / Float64(2.0 + Float64(x * 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = x / (2.0 + (x * 0.5));
end

Wolfram

code[x_] := N[(x / N[(2.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(2 + \frac{1}{2} \cdot x\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{2} \cdot x + \color{blue}{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{2}\right)\right) \]

   3. *-lowering-*.f64 71.5%

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

5. Simplified 71.5%

   \[\leadsto \frac{x}{\color{blue}{0.5 \cdot x + 2}} \]

6. Final simplification 71.5%

   \[\leadsto \frac{x}{2 + x \cdot 0.5} \]
7. Add Preprocessing

Alternative 6: 68.1% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{0.5 + \frac{2}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ 0.5 (/ 2.0 x))))

C

double code(double x) {
	return 1.0 / (0.5 + (2.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (0.5d0 + (2.0d0 / x))
end function

Java

public static double code(double x) {
	return 1.0 / (0.5 + (2.0 / x));
}

Python

def code(x):
	return 1.0 / (0.5 + (2.0 / x))

Julia

function code(x)
	return Float64(1.0 / Float64(0.5 + Float64(2.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (0.5 + (2.0 / x));
end

Wolfram

code[x_] := N[(1.0 / N[(0.5 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(2 + \frac{1}{2} \cdot x\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{2} \cdot x + \color{blue}{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{2}\right)\right) \]

   3. *-lowering-*.f64 71.5%

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

5. Simplified 71.5%

   \[\leadsto \frac{x}{\color{blue}{0.5 \cdot x + 2}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{2} \cdot x + 2}{x}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{2} \cdot x + 2}{x}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x + 2\right), \color{blue}{x}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(2 + \frac{1}{2} \cdot x\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{1}{2} \cdot x\right)\right), x\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(x \cdot \frac{1}{2}\right)\right), x\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(x \cdot \frac{1}{2}\right)\right), x\right)\right) \]

   8. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{x}{2}\right)\right), x\right)\right) \]

   9. /-lowering-/.f64 71.4%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(x, 2\right)\right), x\right)\right) \]

7. Applied egg-rr 71.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{2 + \frac{x}{2}}{x}}} \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{2} + 2 \cdot \frac{1}{x}\right)}\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(2 \cdot \frac{1}{x}\right)}\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{2 \cdot 1}{\color{blue}{x}}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{2}{x}\right)\right)\right) \]

   4. /-lowering-/.f64 71.4%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(2, \color{blue}{x}\right)\right)\right) \]

10. Simplified 71.4%

    \[\leadsto \frac{1}{\color{blue}{0.5 + \frac{2}{x}}} \]
11. Add Preprocessing

Alternative 7: 67.5% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x 2.0))

C

double code(double x) {
	return x / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / 2.0d0
end function

Java

public static double code(double x) {
	return x / 2.0;
}

Python

def code(x):
	return x / 2.0

Julia

function code(x)
	return Float64(x / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = x / 2.0;
end

Wolfram

code[x_] := N[(x / 2.0), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{2}\right) \]
4. Step-by-step derivation

5. Simplified 70.4%

   \[\leadsto \frac{x}{\color{blue}{2}} \]
6. Add Preprocessing

Alternative 8: 4.9% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 99.8%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(2 + \frac{1}{2} \cdot x\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{2} \cdot x + \color{blue}{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{2}\right)\right) \]

   3. *-lowering-*.f64 71.5%

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

5. Simplified 71.5%

   \[\leadsto \frac{x}{\color{blue}{0.5 \cdot x + 2}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{2} \]
7. Step-by-step derivation

8. Simplified 4.8%

   \[\leadsto \color{blue}{2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))