Main:bigenough3 from C

Percentage Accurate: 52.8% → 99.7%

Time: 8.1s

Alternatives: 8

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{2}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (pow (+ (sqrt (+ 1.0 x)) (sqrt x)) 2.0) -0.5))

C

double code(double x) {
	return pow(pow((sqrt((1.0 + x)) + sqrt(x)), 2.0), -0.5);
}

Fortran

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

Java

public static double code(double x) {
	return Math.pow(Math.pow((Math.sqrt((1.0 + x)) + Math.sqrt(x)), 2.0), -0.5);
}

Python

def code(x):
	return math.pow(math.pow((math.sqrt((1.0 + x)) + math.sqrt(x)), 2.0), -0.5)

Julia

function code(x)
	return (Float64(sqrt(Float64(1.0 + x)) + sqrt(x)) ^ 2.0) ^ -0.5
end

MATLAB

function tmp = code(x)
	tmp = ((sqrt((1.0 + x)) + sqrt(x)) ^ 2.0) ^ -0.5;
end

Wolfram

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

Derivation

1. Initial program 51.2%

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

   1. flip-- 51.6%

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

   2. div-inv 51.6%

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

   3. add-sqr-sqrt 51.5%

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

   4. add-sqr-sqrt 51.8%

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

   5. associate--l+ 51.8%

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

3. Applied egg-rr 51.8%

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

   1. associate-*r/ 51.8%

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

   2. *-rgt-identity 51.8%

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

   3. +-commutative 51.8%

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

   4. associate-+l- 99.7%

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

   5. +-inverses 99.7%

      \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

5. Simplified 99.7%

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

   1. +-commutative 99.7%

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

   2. add-sqr-sqrt 99.6%

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

   3. fma-def 99.7%

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

   4. pow1/2 99.7%

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

   5. sqrt-pow1 99.7%

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

   6. metadata-eval 99.7%

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

   7. pow1/2 99.7%

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

   8. sqrt-pow1 99.7%

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

   9. metadata-eval 99.7%

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

   10. +-commutative 99.7%

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

7. Applied egg-rr 99.7%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({x}^{0.25}, {x}^{0.25}, \sqrt{x + 1}\right)}} \]
8. Step-by-step derivation

   1. clear-num 99.7%

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

   2. inv-pow 99.7%

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

   3. /-rgt-identity 99.7%

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

   4. fma-udef 99.6%

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

   5. pow-prod-up 99.7%

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

   6. metadata-eval 99.7%

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

   7. pow1/2 99.7%

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

   8. metadata-eval 99.7%

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

   9. pow-sqr 99.5%

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

   10. pow-prod-down 99.8%

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

   11. pow2 99.8%

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

   12. +-commutative 99.8%

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

   13. +-commutative 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

    \[\leadsto {\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{2}\right)}^{-0.5} \]

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 4e-8) (* 0.5 (pow x -0.5)) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 4e-8) {
		tmp = 0.5 * pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 4d-8) then
        tmp = 0.5d0 * (x ** (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 4e-8) {
		tmp = 0.5 * Math.pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 4e-8:
		tmp = 0.5 * math.pow(x, -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 4e-8)
		tmp = Float64(0.5 * (x ^ -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 4e-8)
		tmp = 0.5 * (x ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-8], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 4.0000000000000001e-8

   1. Initial program 4.0%

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

      1. flip-- 4.6%

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

      2. div-inv 4.6%

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

      3. add-sqr-sqrt 4.7%

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

      4. add-sqr-sqrt 4.6%

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

      5. associate--l+ 4.6%

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

   3. Applied egg-rr 4.6%

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

      1. associate-*r/ 4.6%

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

      2. *-rgt-identity 4.6%

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

      3. +-commutative 4.6%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. flip3-+ 65.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{{\left(\sqrt{1 + x}\right)}^{3} + {\left(\sqrt{x}\right)}^{3}}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}}} \]

      2. associate-/r/ 65.4%

         \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt{1 + x}\right)}^{3} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right)} \]

      3. sqrt-pow2 65.4%

         \[\leadsto \frac{1}{\color{blue}{{\left(1 + x\right)}^{\left(\frac{3}{2}\right)}} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      4. +-commutative 65.4%

         \[\leadsto \frac{1}{{\color{blue}{\left(x + 1\right)}}^{\left(\frac{3}{2}\right)} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      5. metadata-eval 65.4%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{\color{blue}{1.5}} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      6. sqrt-pow2 65.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval 65.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{\color{blue}{1.5}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      8. add-sqr-sqrt 65.5%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\color{blue}{\left(1 + x\right)} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      9. add-sqr-sqrt 65.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(1 + x\right) + \left(\color{blue}{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      10. associate-+r- 65.3%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(\left(1 + x\right) + x\right) - \sqrt{1 + x} \cdot \sqrt{x}\right)} \]

      11. +-commutative 65.3%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\color{blue}{\left(x + 1\right)} + x\right) - \sqrt{1 + x} \cdot \sqrt{x}\right) \]

      12. sqrt-unprod 47.9%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\left(x + 1\right) + x\right) - \color{blue}{\sqrt{\left(1 + x\right) \cdot x}}\right) \]

   7. Applied egg-rr 47.9%

      \[\leadsto \color{blue}{\frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\left(x + 1\right) + x\right) - \sqrt{\left(x + 1\right) \cdot x}\right)} \]

   8. Taylor expanded in x around inf 99.8%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)\right)} \]

      2. expm1-udef 5.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)} - 1\right)} \]

      3. inv-pow 5.9%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{x}^{-1}}}\right)} - 1\right) \]

      4. sqrt-pow1 5.9%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 5.9%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   10. Applied egg-rr 5.9%

       \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

       2. expm1-log1p 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]

   12. Simplified 100.0%

       \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]

   if 4.0000000000000001e-8 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.2%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

   1. flip-- 51.6%

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

   2. div-inv 51.6%

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

   3. add-sqr-sqrt 51.5%

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

   4. add-sqr-sqrt 51.8%

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

   5. associate--l+ 51.8%

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

3. Applied egg-rr 51.8%

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

   1. associate-*r/ 51.8%

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

   2. *-rgt-identity 51.8%

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

   3. +-commutative 51.8%

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

   4. associate-+l- 99.7%

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

   5. +-inverses 99.7%

      \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\frac{1}{1 + \left(\sqrt{x} + x \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4) (/ 1.0 (+ 1.0 (+ (sqrt x) (* x 0.5)))) (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0 / (1.0 + (sqrt(x) + (x * 0.5)));
	} else {
		tmp = 0.5 * pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.4d0) then
        tmp = 1.0d0 / (1.0d0 + (sqrt(x) + (x * 0.5d0)))
    else
        tmp = 0.5d0 * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0 / (1.0 + (Math.sqrt(x) + (x * 0.5)));
	} else {
		tmp = 0.5 * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = 1.0 / (1.0 + (math.sqrt(x) + (x * 0.5)))
	else:
		tmp = 0.5 * math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = Float64(1.0 / Float64(1.0 + Float64(sqrt(x) + Float64(x * 0.5))));
	else
		tmp = Float64(0.5 * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = 1.0 / (1.0 + (sqrt(x) + (x * 0.5)));
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], N[(1.0 / N[(1.0 + N[(N[Sqrt[x], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 99.9%

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

      1. flip-- 99.8%

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

      2. div-inv 99.8%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.8%

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

      5. associate--l+ 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l- 99.9%

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

      5. +-inverses 99.9%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. fma-def 99.9%

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

      4. pow1/2 99.9%

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

      5. sqrt-pow1 99.9%

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

      6. metadata-eval 99.9%

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

      7. pow1/2 99.9%

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

      8. sqrt-pow1 99.9%

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

      9. metadata-eval 99.9%

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

      10. +-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around 0 97.9%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(\sqrt{x} + 0.5 \cdot x\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 97.9%

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

   10. Simplified 97.9%

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

   if 2.39999999999999991 < x

   1. Initial program 6.9%

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

      1. flip-- 7.7%

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

      2. div-inv 7.7%

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

      3. add-sqr-sqrt 7.5%

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

      4. add-sqr-sqrt 8.1%

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

      5. associate--l+ 8.1%

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

   3. Applied egg-rr 8.1%

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

      1. associate-*r/ 8.1%

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

      2. *-rgt-identity 8.1%

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

      3. +-commutative 8.1%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. flip3-+ 66.7%

         \[\leadsto \frac{1}{\color{blue}{\frac{{\left(\sqrt{1 + x}\right)}^{3} + {\left(\sqrt{x}\right)}^{3}}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}}} \]

      2. associate-/r/ 66.7%

         \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt{1 + x}\right)}^{3} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right)} \]

      3. sqrt-pow2 66.6%

         \[\leadsto \frac{1}{\color{blue}{{\left(1 + x\right)}^{\left(\frac{3}{2}\right)}} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      4. +-commutative 66.6%

         \[\leadsto \frac{1}{{\color{blue}{\left(x + 1\right)}}^{\left(\frac{3}{2}\right)} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      5. metadata-eval 66.6%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{\color{blue}{1.5}} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      6. sqrt-pow2 66.5%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval 66.5%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{\color{blue}{1.5}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      8. add-sqr-sqrt 66.7%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\color{blue}{\left(1 + x\right)} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      9. add-sqr-sqrt 66.5%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(1 + x\right) + \left(\color{blue}{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      10. associate-+r- 66.5%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(\left(1 + x\right) + x\right) - \sqrt{1 + x} \cdot \sqrt{x}\right)} \]

      11. +-commutative 66.5%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\color{blue}{\left(x + 1\right)} + x\right) - \sqrt{1 + x} \cdot \sqrt{x}\right) \]

      12. sqrt-unprod 49.8%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\left(x + 1\right) + x\right) - \color{blue}{\sqrt{\left(1 + x\right) \cdot x}}\right) \]

   7. Applied egg-rr 49.8%

      \[\leadsto \color{blue}{\frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\left(x + 1\right) + x\right) - \sqrt{\left(x + 1\right) \cdot x}\right)} \]

   8. Taylor expanded in x around inf 97.7%

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

      1. expm1-log1p-u 97.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)\right)} \]

      2. expm1-udef 7.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)} - 1\right)} \]

      3. inv-pow 7.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{x}^{-1}}}\right)} - 1\right) \]

      4. sqrt-pow1 7.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 7.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   10. Applied egg-rr 7.3%

       \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 97.9%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

       2. expm1-log1p 97.9%

          \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]

   12. Simplified 97.9%

       \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\frac{1}{1 + \left(\sqrt{x} + x \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 5: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1}{1 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (/ 1.0 (+ 1.0 (sqrt x))) (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 / (1.0 + sqrt(x));
	} else {
		tmp = 0.5 * pow(x, -0.5);
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 / (1.0 + Math.sqrt(x));
	} else {
		tmp = 0.5 * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 / (1.0 + math.sqrt(x))
	else:
		tmp = 0.5 * math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 / Float64(1.0 + sqrt(x)));
	else
		tmp = Float64(0.5 * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 / (1.0 + sqrt(x));
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(1.0 / N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. flip-- 99.8%

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

      2. div-inv 99.8%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.8%

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

      5. associate--l+ 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l- 99.9%

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

      5. +-inverses 99.9%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. fma-def 99.9%

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

      4. pow1/2 99.9%

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

      5. sqrt-pow1 99.9%

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

      6. metadata-eval 99.9%

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

      7. pow1/2 99.9%

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

      8. sqrt-pow1 99.9%

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

      9. metadata-eval 99.9%

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

      10. +-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around 0 95.7%

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

   if 1 < x

   1. Initial program 6.9%

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

      1. flip-- 7.7%

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

      2. div-inv 7.7%

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

      3. add-sqr-sqrt 7.5%

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

      4. add-sqr-sqrt 8.1%

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

      5. associate--l+ 8.1%

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

   3. Applied egg-rr 8.1%

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

      1. associate-*r/ 8.1%

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

      2. *-rgt-identity 8.1%

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

      3. +-commutative 8.1%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. flip3-+ 66.7%

         \[\leadsto \frac{1}{\color{blue}{\frac{{\left(\sqrt{1 + x}\right)}^{3} + {\left(\sqrt{x}\right)}^{3}}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}}} \]

      2. associate-/r/ 66.7%

         \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt{1 + x}\right)}^{3} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right)} \]

      3. sqrt-pow2 66.6%

         \[\leadsto \frac{1}{\color{blue}{{\left(1 + x\right)}^{\left(\frac{3}{2}\right)}} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      4. +-commutative 66.6%

         \[\leadsto \frac{1}{{\color{blue}{\left(x + 1\right)}}^{\left(\frac{3}{2}\right)} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      5. metadata-eval 66.6%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{\color{blue}{1.5}} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      6. sqrt-pow2 66.5%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval 66.5%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{\color{blue}{1.5}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      8. add-sqr-sqrt 66.7%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\color{blue}{\left(1 + x\right)} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      9. add-sqr-sqrt 66.5%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(1 + x\right) + \left(\color{blue}{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      10. associate-+r- 66.5%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(\left(1 + x\right) + x\right) - \sqrt{1 + x} \cdot \sqrt{x}\right)} \]

      11. +-commutative 66.5%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\color{blue}{\left(x + 1\right)} + x\right) - \sqrt{1 + x} \cdot \sqrt{x}\right) \]

      12. sqrt-unprod 49.8%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\left(x + 1\right) + x\right) - \color{blue}{\sqrt{\left(1 + x\right) \cdot x}}\right) \]

   7. Applied egg-rr 49.8%

      \[\leadsto \color{blue}{\frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\left(x + 1\right) + x\right) - \sqrt{\left(x + 1\right) \cdot x}\right)} \]

   8. Taylor expanded in x around inf 97.7%

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

      1. expm1-log1p-u 97.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)\right)} \]

      2. expm1-udef 7.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)} - 1\right)} \]

      3. inv-pow 7.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{x}^{-1}}}\right)} - 1\right) \]

      4. sqrt-pow1 7.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 7.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   10. Applied egg-rr 7.3%

       \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 97.9%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

       2. expm1-log1p 97.9%

          \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]

   12. Simplified 97.9%

       \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1}{1 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 6: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.34:\\ \;\;\;\;\frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.34) (/ 1.0 (+ 1.0 (* x 0.5))) (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.34) {
		tmp = 1.0 / (1.0 + (x * 0.5));
	} else {
		tmp = 0.5 * pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.34d0) then
        tmp = 1.0d0 / (1.0d0 + (x * 0.5d0))
    else
        tmp = 0.5d0 * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.34) {
		tmp = 1.0 / (1.0 + (x * 0.5));
	} else {
		tmp = 0.5 * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.34:
		tmp = 1.0 / (1.0 + (x * 0.5))
	else:
		tmp = 0.5 * math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.34)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * 0.5)));
	else
		tmp = Float64(0.5 * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.34)
		tmp = 1.0 / (1.0 + (x * 0.5));
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.34], N[(1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.340000000000000024

   1. Initial program 99.9%

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

      1. flip-- 99.8%

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

      2. div-inv 99.8%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--l+ 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l- 99.9%

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

      5. +-inverses 99.9%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. fma-def 99.9%

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

      4. pow1/2 99.9%

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

      5. sqrt-pow1 99.9%

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

      6. metadata-eval 99.9%

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

      7. pow1/2 99.9%

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

      8. sqrt-pow1 99.9%

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

      9. metadata-eval 99.9%

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

      10. +-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around 0 99.1%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(\sqrt{x} + 0.5 \cdot x\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in x around inf 94.6%

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

       1. *-commutative 94.6%

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

   13. Simplified 94.6%

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

   if 0.340000000000000024 < x

   1. Initial program 8.3%

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

      1. flip-- 9.1%

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

      2. div-inv 9.1%

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

      3. add-sqr-sqrt 8.8%

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

      4. add-sqr-sqrt 9.4%

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

      5. associate--l+ 9.4%

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

   3. Applied egg-rr 9.4%

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

      1. associate-*r/ 9.4%

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

      2. *-rgt-identity 9.4%

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

      3. +-commutative 9.4%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. flip3-+ 67.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{{\left(\sqrt{1 + x}\right)}^{3} + {\left(\sqrt{x}\right)}^{3}}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}}} \]

      2. associate-/r/ 67.1%

         \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt{1 + x}\right)}^{3} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right)} \]

      3. sqrt-pow2 67.1%

         \[\leadsto \frac{1}{\color{blue}{{\left(1 + x\right)}^{\left(\frac{3}{2}\right)}} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      4. +-commutative 67.1%

         \[\leadsto \frac{1}{{\color{blue}{\left(x + 1\right)}}^{\left(\frac{3}{2}\right)} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      5. metadata-eval 67.1%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{\color{blue}{1.5}} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      6. sqrt-pow2 67.0%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      7. metadata-eval 67.0%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{\color{blue}{1.5}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      8. add-sqr-sqrt 67.2%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\color{blue}{\left(1 + x\right)} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      9. add-sqr-sqrt 67.0%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(1 + x\right) + \left(\color{blue}{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      10. associate-+r- 67.0%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(\left(1 + x\right) + x\right) - \sqrt{1 + x} \cdot \sqrt{x}\right)} \]

      11. +-commutative 67.0%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\color{blue}{\left(x + 1\right)} + x\right) - \sqrt{1 + x} \cdot \sqrt{x}\right) \]

      12. sqrt-unprod 50.5%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\left(x + 1\right) + x\right) - \color{blue}{\sqrt{\left(1 + x\right) \cdot x}}\right) \]

   7. Applied egg-rr 50.5%

      \[\leadsto \color{blue}{\frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\left(x + 1\right) + x\right) - \sqrt{\left(x + 1\right) \cdot x}\right)} \]

   8. Taylor expanded in x around inf 96.5%

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

      1. expm1-log1p-u 96.5%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)\right)} \]

      2. expm1-udef 7.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)} - 1\right)} \]

      3. inv-pow 7.5%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{x}^{-1}}}\right)} - 1\right) \]

      4. sqrt-pow1 7.5%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 7.5%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   10. Applied egg-rr 7.5%

       \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 96.7%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

       2. expm1-log1p 96.7%

          \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]

   12. Simplified 96.7%

       \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.34:\\ \;\;\;\;\frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 7: 50.9% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

   1. flip-- 51.6%

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

   2. div-inv 51.6%

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

   3. add-sqr-sqrt 51.5%

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

   4. add-sqr-sqrt 51.8%

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

   5. associate--l+ 51.8%

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

3. Applied egg-rr 51.8%

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

   1. associate-*r/ 51.8%

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

   2. *-rgt-identity 51.8%

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

   3. +-commutative 51.8%

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

   4. associate-+l- 99.7%

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

   5. +-inverses 99.7%

      \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

5. Simplified 99.7%

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

   1. +-commutative 99.7%

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

   2. add-sqr-sqrt 99.6%

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

   3. fma-def 99.7%

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

   4. pow1/2 99.7%

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

   5. sqrt-pow1 99.7%

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

   6. metadata-eval 99.7%

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

   7. pow1/2 99.7%

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

   8. sqrt-pow1 99.7%

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

   9. metadata-eval 99.7%

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

   10. +-commutative 99.7%

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in x around 0 50.3%

   \[\leadsto \frac{1}{\color{blue}{1 + \left(\sqrt{x} + 0.5 \cdot x\right)}} \]
9. Step-by-step derivation

   1. *-commutative 50.3%

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

10. Simplified 50.3%

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

11. Taylor expanded in x around inf 48.2%

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

    1. *-commutative 48.2%

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

13. Simplified 48.2%

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

14. Final simplification 48.2%

    \[\leadsto \frac{1}{1 + x \cdot 0.5} \]

Alternative 8: 50.8% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 51.2%

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

2. Taylor expanded in x around 0 48.1%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 48.1%

   \[\leadsto 1 \]

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023334 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))