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

Percentage Accurate: 99.7% → 99.5%

Time: 6.9s

Alternatives: 6

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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\left(--0.5\right) - x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-6) (* x (- (- -0.5) (* x 0.125))) (+ (sqrt (+ x 1.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 5.6e-6) {
		tmp = x * (-(-0.5) - (x * 0.125));
	} else {
		tmp = sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.6d-6) then
        tmp = x * (-(-0.5d0) - (x * 0.125d0))
    else
        tmp = sqrt((x + 1.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.6e-6) {
		tmp = x * (-(-0.5) - (x * 0.125));
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.6e-6:
		tmp = x * (-(-0.5) - (x * 0.125))
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.6e-6)
		tmp = Float64(x * Float64(Float64(-(-0.5)) - Float64(x * 0.125)));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.6e-6)
		tmp = x * (-(-0.5) - (x * 0.125));
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.6e-6], N[(x * N[((--0.5) - N[(x * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.59999999999999975e-6

   1. Initial program 100.0%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Step-by-step derivation

      1. frac-2neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \color{blue}{0 - \frac{x}{-\left(1 + \sqrt{x + 1}\right)}} \]

      4. div-inv 100.0%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{1}{-\left(1 + \sqrt{x + 1}\right)}} \]

      5. cancel-sign-sub-inv 100.0%

         \[\leadsto \color{blue}{0 + \left(-x\right) \cdot \frac{1}{-\left(1 + \sqrt{x + 1}\right)}} \]

      6. distribute-lft-neg-out 100.0%

         \[\leadsto 0 + \color{blue}{\left(-x \cdot \frac{1}{-\left(1 + \sqrt{x + 1}\right)}\right)} \]

      7. div-inv 100.0%

         \[\leadsto 0 + \left(-\color{blue}{\frac{x}{-\left(1 + \sqrt{x + 1}\right)}}\right) \]

      8. remove-double-neg 100.0%

         \[\leadsto 0 + \left(-\frac{\color{blue}{-\left(-x\right)}}{-\left(1 + \sqrt{x + 1}\right)}\right) \]

      9. neg-sub0 100.0%

         \[\leadsto 0 + \left(-\frac{-\color{blue}{\left(0 - x\right)}}{-\left(1 + \sqrt{x + 1}\right)}\right) \]

      10. metadata-eval 100.0%

          \[\leadsto 0 + \left(-\frac{-\left(\color{blue}{\left(1 - 1\right)} - x\right)}{-\left(1 + \sqrt{x + 1}\right)}\right) \]

      11. associate--r+ 7.7%

          \[\leadsto 0 + \left(-\frac{-\color{blue}{\left(1 - \left(1 + x\right)\right)}}{-\left(1 + \sqrt{x + 1}\right)}\right) \]

      12. metadata-eval 7.7%

          \[\leadsto 0 + \left(-\frac{-\left(\color{blue}{1 \cdot 1} - \left(1 + x\right)\right)}{-\left(1 + \sqrt{x + 1}\right)}\right) \]

      13. +-commutative 7.7%

          \[\leadsto 0 + \left(-\frac{-\left(1 \cdot 1 - \color{blue}{\left(x + 1\right)}\right)}{-\left(1 + \sqrt{x + 1}\right)}\right) \]

      14. rem-square-sqrt 7.6%

          \[\leadsto 0 + \left(-\frac{-\left(1 \cdot 1 - \color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}}\right)}{-\left(1 + \sqrt{x + 1}\right)}\right) \]

      15. frac-2neg 7.6%

          \[\leadsto 0 + \left(-\color{blue}{\frac{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}{1 + \sqrt{x + 1}}}\right) \]

      16. flip-- 7.6%

          \[\leadsto 0 + \left(-\color{blue}{\left(1 - \sqrt{x + 1}\right)}\right) \]

   3. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{0 + \left(-\left(1 - \sqrt{x + 1}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg 7.6%

         \[\leadsto \color{blue}{0 - \left(1 - \sqrt{x + 1}\right)} \]

      2. associate--r- 7.6%

         \[\leadsto \color{blue}{\left(0 - 1\right) + \sqrt{x + 1}} \]

      3. metadata-eval 7.6%

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

      4. +-commutative 7.6%

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

   5. Simplified 7.6%

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

      1. +-commutative 7.6%

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

      2. flip-+ 7.6%

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

      3. clear-num 7.6%

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

      4. metadata-eval 7.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(0 - 1\right)} - \sqrt{x + 1}}{-1 \cdot -1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}} \]

      5. associate--r+ 7.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{0 - \left(1 + \sqrt{x + 1}\right)}}{-1 \cdot -1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}} \]

      6. +-commutative 7.6%

         \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(\sqrt{x + 1} + 1\right)}}{-1 \cdot -1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}} \]

      7. pow1/2 7.6%

         \[\leadsto \frac{1}{\frac{0 - \left(\color{blue}{{\left(x + 1\right)}^{0.5}} + 1\right)}{-1 \cdot -1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}} \]

      8. metadata-eval 7.6%

         \[\leadsto \frac{1}{\frac{0 - \left({\left(x + 1\right)}^{\color{blue}{\left(0.25 + 0.25\right)}} + 1\right)}{-1 \cdot -1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}} \]

      9. pow-prod-up 7.6%

         \[\leadsto \frac{1}{\frac{0 - \left(\color{blue}{{\left(x + 1\right)}^{0.25} \cdot {\left(x + 1\right)}^{0.25}} + 1\right)}{-1 \cdot -1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}} \]

      10. fma-udef 7.6%

          \[\leadsto \frac{1}{\frac{0 - \color{blue}{\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, 1\right)}}{-1 \cdot -1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}} \]

      11. neg-sub0 7.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{-\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, 1\right)}}{-1 \cdot -1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}} \]

      12. metadata-eval 7.6%

          \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, 1\right)}{\color{blue}{1} - \sqrt{x + 1} \cdot \sqrt{x + 1}}} \]

      13. rem-square-sqrt 7.7%

          \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, 1\right)}{1 - \color{blue}{\left(x + 1\right)}}} \]

      14. +-commutative 7.7%

          \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, 1\right)}{1 - \color{blue}{\left(1 + x\right)}}} \]

      15. associate--r+ 99.7%

          \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, 1\right)}{\color{blue}{\left(1 - 1\right) - x}}} \]

      16. metadata-eval 99.7%

          \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, 1\right)}{\color{blue}{0} - x}} \]

      17. neg-sub0 99.7%

          \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, 1\right)}{\color{blue}{-x}}} \]

      18. frac-2neg 99.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, 1\right)}{x}}} \]

      19. div-inv 99.7%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, 1\right) \cdot \frac{1}{x}}} \]

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around 0 99.7%

      \[\leadsto \frac{\color{blue}{0.5 + -0.125 \cdot x}}{\frac{1}{x}} \]
   9. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \frac{0.5 + \color{blue}{x \cdot -0.125}}{\frac{1}{x}} \]

   10. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{0.5 + x \cdot -0.125}}{\frac{1}{x}} \]
   11. Step-by-step derivation

       1. frac-2neg 99.7%

          \[\leadsto \color{blue}{\frac{-\left(0.5 + x \cdot -0.125\right)}{-\frac{1}{x}}} \]

       2. div-inv 99.7%

          \[\leadsto \color{blue}{\left(-\left(0.5 + x \cdot -0.125\right)\right) \cdot \frac{1}{-\frac{1}{x}}} \]

       3. distribute-neg-in 99.7%

          \[\leadsto \color{blue}{\left(\left(-0.5\right) + \left(-x \cdot -0.125\right)\right)} \cdot \frac{1}{-\frac{1}{x}} \]

       4. metadata-eval 99.7%

          \[\leadsto \left(\color{blue}{-0.5} + \left(-x \cdot -0.125\right)\right) \cdot \frac{1}{-\frac{1}{x}} \]

       5. distribute-rgt-neg-in 99.7%

          \[\leadsto \left(-0.5 + \color{blue}{x \cdot \left(--0.125\right)}\right) \cdot \frac{1}{-\frac{1}{x}} \]

       6. metadata-eval 99.7%

          \[\leadsto \left(-0.5 + x \cdot \color{blue}{0.125}\right) \cdot \frac{1}{-\frac{1}{x}} \]

       7. frac-2neg 99.7%

          \[\leadsto \left(-0.5 + x \cdot 0.125\right) \cdot \color{blue}{\frac{-1}{-\left(-\frac{1}{x}\right)}} \]

       8. remove-double-neg 99.7%

          \[\leadsto \left(-0.5 + x \cdot 0.125\right) \cdot \frac{-1}{\color{blue}{\frac{1}{x}}} \]

       9. distribute-neg-frac 99.7%

          \[\leadsto \left(-0.5 + x \cdot 0.125\right) \cdot \color{blue}{\left(-\frac{1}{\frac{1}{x}}\right)} \]

       10. remove-double-div 100.0%

           \[\leadsto \left(-0.5 + x \cdot 0.125\right) \cdot \left(-\color{blue}{x}\right) \]

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(-0.5 + x \cdot 0.125\right) \cdot \left(-x\right)} \]

   if 5.59999999999999975e-6 < x

   1. Initial program 99.2%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Step-by-step derivation

      1. frac-2neg 99.2%

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

      2. distribute-frac-neg 99.2%

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

      3. neg-sub0 99.2%

         \[\leadsto \color{blue}{0 - \frac{x}{-\left(1 + \sqrt{x + 1}\right)}} \]

      4. div-inv 99.2%

         \[\leadsto 0 - \color{blue}{x \cdot \frac{1}{-\left(1 + \sqrt{x + 1}\right)}} \]

      5. cancel-sign-sub-inv 99.2%

         \[\leadsto \color{blue}{0 + \left(-x\right) \cdot \frac{1}{-\left(1 + \sqrt{x + 1}\right)}} \]

      6. distribute-lft-neg-out 99.2%

         \[\leadsto 0 + \color{blue}{\left(-x \cdot \frac{1}{-\left(1 + \sqrt{x + 1}\right)}\right)} \]

      7. div-inv 99.2%

         \[\leadsto 0 + \left(-\color{blue}{\frac{x}{-\left(1 + \sqrt{x + 1}\right)}}\right) \]

      8. remove-double-neg 99.2%

         \[\leadsto 0 + \left(-\frac{\color{blue}{-\left(-x\right)}}{-\left(1 + \sqrt{x + 1}\right)}\right) \]

      9. neg-sub0 99.2%

         \[\leadsto 0 + \left(-\frac{-\color{blue}{\left(0 - x\right)}}{-\left(1 + \sqrt{x + 1}\right)}\right) \]

      10. metadata-eval 99.2%

          \[\leadsto 0 + \left(-\frac{-\left(\color{blue}{\left(1 - 1\right)} - x\right)}{-\left(1 + \sqrt{x + 1}\right)}\right) \]

      11. associate--r+ 98.8%

          \[\leadsto 0 + \left(-\frac{-\color{blue}{\left(1 - \left(1 + x\right)\right)}}{-\left(1 + \sqrt{x + 1}\right)}\right) \]

      12. metadata-eval 98.8%

          \[\leadsto 0 + \left(-\frac{-\left(\color{blue}{1 \cdot 1} - \left(1 + x\right)\right)}{-\left(1 + \sqrt{x + 1}\right)}\right) \]

      13. +-commutative 98.8%

          \[\leadsto 0 + \left(-\frac{-\left(1 \cdot 1 - \color{blue}{\left(x + 1\right)}\right)}{-\left(1 + \sqrt{x + 1}\right)}\right) \]

      14. rem-square-sqrt 99.5%

          \[\leadsto 0 + \left(-\frac{-\left(1 \cdot 1 - \color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}}\right)}{-\left(1 + \sqrt{x + 1}\right)}\right) \]

      15. frac-2neg 99.5%

          \[\leadsto 0 + \left(-\color{blue}{\frac{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}{1 + \sqrt{x + 1}}}\right) \]

      16. flip-- 99.6%

          \[\leadsto 0 + \left(-\color{blue}{\left(1 - \sqrt{x + 1}\right)}\right) \]

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{0 + \left(-\left(1 - \sqrt{x + 1}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg 99.6%

         \[\leadsto \color{blue}{0 - \left(1 - \sqrt{x + 1}\right)} \]

      2. associate--r- 99.6%

         \[\leadsto \color{blue}{\left(0 - 1\right) + \sqrt{x + 1}} \]

      3. metadata-eval 99.6%

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

      4. +-commutative 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\left(--0.5\right) - x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{0.25}\\ \frac{x}{\mathsf{fma}\left(t_0, t_0, 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (+ x 1.0) 0.25))) (/ x (fma t_0 t_0 1.0))))

C

double code(double x) {
	double t_0 = pow((x + 1.0), 0.25);
	return x / fma(t_0, t_0, 1.0);
}

Julia

function code(x)
	t_0 = Float64(x + 1.0) ^ 0.25
	return Float64(x / fma(t_0, t_0, 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x + 1.0), $MachinePrecision], 0.25], $MachinePrecision]}, N[(x / N[(t$95$0 * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.7%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Step-by-step derivation

   1. +-commutative 99.7%

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

   2. pow1/2 99.7%

      \[\leadsto \frac{x}{\color{blue}{{\left(x + 1\right)}^{0.5}} + 1} \]

   3. rem-square-sqrt 99.7%

      \[\leadsto \frac{x}{{\color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}}^{0.5} + 1} \]

   4. sqr-pow 99.6%

      \[\leadsto \frac{x}{\color{blue}{{\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}^{\left(\frac{0.5}{2}\right)}} + 1} \]

   5. fma-def 99.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}^{\left(\frac{0.5}{2}\right)}, {\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}^{\left(\frac{0.5}{2}\right)}, 1\right)}} \]

   6. rem-square-sqrt 99.7%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(x + 1\right)}}^{\left(\frac{0.5}{2}\right)}, {\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}^{\left(\frac{0.5}{2}\right)}, 1\right)} \]

   7. metadata-eval 99.7%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(x + 1\right)}^{\color{blue}{0.25}}, {\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}^{\left(\frac{0.5}{2}\right)}, 1\right)} \]

   8. rem-square-sqrt 99.7%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\color{blue}{\left(x + 1\right)}}^{\left(\frac{0.5}{2}\right)}, 1\right)} \]

   9. metadata-eval 99.7%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{\color{blue}{0.25}}, 1\right)} \]

3. Applied egg-rr 99.7%

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, 1\right)}} \]

4. Final simplification 99.7%

   \[\leadsto \frac{x}{\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, 1\right)} \]

Alternative 3: 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 99.7%

   \[\frac{x}{1 + \sqrt{x + 1}} \]

2. Final simplification 99.7%

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

Alternative 4: 68.4% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\frac{x}{1 + \sqrt{x + 1}} \]

2. Taylor expanded in x around 0 69.5%

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

   1. +-commutative 69.5%

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

4. Simplified 69.5%

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

5. Final simplification 69.5%

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

Alternative 5: 67.7% 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 99.7%

   \[\frac{x}{1 + \sqrt{x + 1}} \]

2. Taylor expanded in x around 0 68.6%

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

3. Final simplification 68.6%

   \[\leadsto \frac{x}{2} \]

Alternative 6: 4.8% 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 99.7%

   \[\frac{x}{1 + \sqrt{x + 1}} \]

2. Taylor expanded in x around 0 69.5%

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

   1. +-commutative 69.5%

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

4. Simplified 69.5%

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

   1. +-commutative 69.5%

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

   2. flip-+ 68.9%

      \[\leadsto \frac{x}{\color{blue}{\frac{2 \cdot 2 - \left(0.5 \cdot x\right) \cdot \left(0.5 \cdot x\right)}{2 - 0.5 \cdot x}}} \]

   3. associate-/r/ 68.9%

      \[\leadsto \color{blue}{\frac{x}{2 \cdot 2 - \left(0.5 \cdot x\right) \cdot \left(0.5 \cdot x\right)} \cdot \left(2 - 0.5 \cdot x\right)} \]

   4. swap-sqr 68.9%

      \[\leadsto \frac{x}{2 \cdot 2 - \color{blue}{\left(0.5 \cdot 0.5\right) \cdot \left(x \cdot x\right)}} \cdot \left(2 - 0.5 \cdot x\right) \]

   5. metadata-eval 68.9%

      \[\leadsto \frac{x}{2 \cdot 2 - \color{blue}{0.25} \cdot \left(x \cdot x\right)} \cdot \left(2 - 0.5 \cdot x\right) \]

   6. cancel-sign-sub-inv 68.9%

      \[\leadsto \frac{x}{\color{blue}{2 \cdot 2 + \left(-0.25\right) \cdot \left(x \cdot x\right)}} \cdot \left(2 - 0.5 \cdot x\right) \]

   7. metadata-eval 68.9%

      \[\leadsto \frac{x}{\color{blue}{4} + \left(-0.25\right) \cdot \left(x \cdot x\right)} \cdot \left(2 - 0.5 \cdot x\right) \]

   8. metadata-eval 68.9%

      \[\leadsto \frac{x}{4 + \color{blue}{-0.25} \cdot \left(x \cdot x\right)} \cdot \left(2 - 0.5 \cdot x\right) \]

   9. sub-neg 68.9%

      \[\leadsto \frac{x}{4 + -0.25 \cdot \left(x \cdot x\right)} \cdot \color{blue}{\left(2 + \left(-0.5 \cdot x\right)\right)} \]

   10. *-commutative 68.9%

       \[\leadsto \frac{x}{4 + -0.25 \cdot \left(x \cdot x\right)} \cdot \left(2 + \left(-\color{blue}{x \cdot 0.5}\right)\right) \]

   11. distribute-rgt-neg-in 68.9%

       \[\leadsto \frac{x}{4 + -0.25 \cdot \left(x \cdot x\right)} \cdot \left(2 + \color{blue}{x \cdot \left(-0.5\right)}\right) \]

   12. metadata-eval 68.9%

       \[\leadsto \frac{x}{4 + -0.25 \cdot \left(x \cdot x\right)} \cdot \left(2 + x \cdot \color{blue}{-0.5}\right) \]

6. Applied egg-rr 68.9%

   \[\leadsto \color{blue}{\frac{x}{4 + -0.25 \cdot \left(x \cdot x\right)} \cdot \left(2 + x \cdot -0.5\right)} \]

7. Taylor expanded in x around inf 4.9%

   \[\leadsto \color{blue}{2} \]

8. Final simplification 4.9%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))