2isqrt (example 3.6)

Percentage Accurate: 69.3% → 99.7%

Time: 11.6s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 90000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{x + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 90000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (/ (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))) (+ x 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 90000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = (1.0 / (sqrt((1.0 + x)) + sqrt(x))) / (x + 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 90000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = (1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) / (x + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 90000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = (1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) / (x + 0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 90000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) / (x + 0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 90000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) / Float64(x + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 90000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = (1.0 / (sqrt((1.0 + x)) + sqrt(x))) / (x + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 90000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9e4

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. pow1/2 99.5%

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

      9. pow-flip 99.9%

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

      10. metadata-eval 99.9%

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

      11. pow1/2 99.9%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. expm1-log1p 99.9%

         \[\leadsto {x}^{-0.5} - \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} - \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)\right) \]

      3. expm1-def 99.9%

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

      4. associate--l- 99.9%

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

      5. fma-udef 99.9%

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

      6. distribute-lft1-in 99.9%

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

      7. metadata-eval 99.9%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 99.9%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 99.9%

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

      10. expm1-def 99.9%

          \[\leadsto {x}^{-0.5} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} \]

      11. expm1-log1p 100.0%

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

   5. Simplified 100.0%

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

   if 9e4 < x

   1. Initial program 37.2%

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

      1. frac-sub 37.4%

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

      2. *-un-lft-identity 37.4%

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

      3. *-rgt-identity 37.4%

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

      4. +-commutative 37.4%

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

      5. sqrt-unprod 37.4%

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

      6. +-commutative 37.4%

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

   3. Applied egg-rr 37.4%

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

   4. Taylor expanded in x around inf 37.4%

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

      1. +-commutative 37.4%

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

   6. Simplified 37.4%

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

      1. flip-- 38.1%

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

      2. add-sqr-sqrt 25.3%

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

      3. add-sqr-sqrt 38.1%

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

   8. Applied egg-rr 38.1%

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

      1. associate--l+ 99.7%

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

      2. +-inverses 99.7%

         \[\leadsto \frac{\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}}}{x + 0.5} \]

      3. metadata-eval 99.7%

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

      4. +-commutative 99.7%

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

   10. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}}{x + 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 90000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{x + 0.5}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\sqrt{1 + x}} \cdot 0 + \frac{\frac{1}{\mathsf{fma}\left(x, {\left(1 + x\right)}^{-0.5}, \sqrt{x}\right)}}{1 + x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (* (/ -1.0 (sqrt (+ 1.0 x))) 0.0)
  (/ (/ 1.0 (fma x (pow (+ 1.0 x) -0.5) (sqrt x))) (+ 1.0 x))))

C

double code(double x) {
	return ((-1.0 / sqrt((1.0 + x))) * 0.0) + ((1.0 / fma(x, pow((1.0 + x), -0.5), sqrt(x))) / (1.0 + x));
}

Julia

function code(x)
	return Float64(Float64(Float64(-1.0 / sqrt(Float64(1.0 + x))) * 0.0) + Float64(Float64(1.0 / fma(x, (Float64(1.0 + x) ^ -0.5), sqrt(x))) / Float64(1.0 + x)))
end

Wolfram

code[x_] := N[(N[(N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision] + N[(N[(1.0 / N[(x * N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.8%

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

   1. inv-pow 69.8%

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

   2. add-cube-cbrt 57.9%

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

   3. unpow-prod-down 57.1%

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

   4. frac-2neg 57.1%

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

   5. metadata-eval 57.1%

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

   6. div-inv 57.1%

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

   7. metadata-eval 57.1%

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

   8. distribute-neg-frac 57.1%

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

   9. metadata-eval 57.1%

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

   10. frac-2neg 57.1%

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

   11. prod-diff 53.5%

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

3. Applied egg-rr 53.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{-1}, {\left(\sqrt[3]{\sqrt{x}}\right)}^{-1}, -\frac{-1}{\sqrt{1 + x}} \cdot -1\right) + \mathsf{fma}\left(-\frac{-1}{\sqrt{1 + x}}, -1, \frac{-1}{\sqrt{1 + x}} \cdot -1\right)} \]
4. Step-by-step derivation

   1. +-commutative 53.9%

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

   2. fma-udef 53.9%

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

   3. distribute-lft-neg-in 53.9%

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

   4. distribute-rgt-neg-in 53.9%

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

   5. metadata-eval 53.9%

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

   6. distribute-lft-out 53.9%

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

   7. metadata-eval 53.9%

      \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot \color{blue}{0} + \mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{-1}, {\left(\sqrt[3]{\sqrt{x}}\right)}^{-1}, -\frac{-1}{\sqrt{1 + x}} \cdot -1\right) \]

   8. distribute-rgt-neg-in 53.9%

      \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{-1}, {\left(\sqrt[3]{\sqrt{x}}\right)}^{-1}, \color{blue}{\frac{-1}{\sqrt{1 + x}} \cdot \left(--1\right)}\right) \]

   9. metadata-eval 53.9%

      \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{-1}, {\left(\sqrt[3]{\sqrt{x}}\right)}^{-1}, \frac{-1}{\sqrt{1 + x}} \cdot \color{blue}{1}\right) \]

   10. unpow-1 53.9%

       \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt[3]{x}}}, {\left(\sqrt[3]{\sqrt{x}}\right)}^{-1}, \frac{-1}{\sqrt{1 + x}} \cdot 1\right) \]

   11. unpow-1 53.9%

       \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \mathsf{fma}\left(\frac{1}{\sqrt[3]{x}}, \color{blue}{\frac{1}{\sqrt[3]{\sqrt{x}}}}, \frac{-1}{\sqrt{1 + x}} \cdot 1\right) \]

   12. *-rgt-identity 53.9%

       \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \mathsf{fma}\left(\frac{1}{\sqrt[3]{x}}, \frac{1}{\sqrt[3]{\sqrt{x}}}, \color{blue}{\frac{-1}{\sqrt{1 + x}}}\right) \]

5. Simplified 53.9%

   \[\leadsto \color{blue}{\frac{-1}{\sqrt{1 + x}} \cdot 0 + \mathsf{fma}\left(\frac{1}{\sqrt[3]{x}}, \frac{1}{\sqrt[3]{\sqrt{x}}}, \frac{-1}{\sqrt{1 + x}}\right)} \]

7. Applied egg-rr 69.6%

   \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \color{blue}{\left(\frac{\frac{1}{x}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} - \frac{\frac{1}{x + 1}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\right)} \]
8. Step-by-step derivation

   1. associate-/l/ 65.8%

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

9. Simplified 65.8%

   \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \color{blue}{\left(\frac{1}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot x} - \frac{\frac{1}{x + 1}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\right)} \]
10. Step-by-step derivation

    1. associate-/r* 63.7%

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

    2. clear-num 64.7%

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

    3. frac-sub 69.6%

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

    4. +-commutative 69.6%

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

    5. +-commutative 69.6%

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

    6. +-commutative 69.6%

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

11. Applied egg-rr 69.6%

    \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \color{blue}{\frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \cdot \frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{1}{1 + x}} - x \cdot 1}{x \cdot \frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{1}{1 + x}}}} \]
12. Step-by-step derivation

    1. associate-*r/ 70.0%

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

    2. lft-mult-inverse 70.1%

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

    3. remove-double-div 70.1%

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

    4. *-rgt-identity 70.1%

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

    5. associate-/r/ 70.1%

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

    6. /-rgt-identity 70.1%

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

    7. associate--l+ 98.5%

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

    8. +-inverses 98.5%

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

    9. metadata-eval 98.5%

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

    10. associate-*r* 98.5%

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

13. Simplified 98.7%

    \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \color{blue}{\frac{1}{\left({x}^{0.5} + {\left(1 + x\right)}^{-0.5} \cdot x\right) \cdot \left(1 + x\right)}} \]
14. Step-by-step derivation

    1. expm1-log1p-u 95.0%

       \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left({x}^{0.5} + {\left(1 + x\right)}^{-0.5} \cdot x\right) \cdot \left(1 + x\right)}\right)\right)} \]

    2. expm1-udef 65.9%

       \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\left({x}^{0.5} + {\left(1 + x\right)}^{-0.5} \cdot x\right) \cdot \left(1 + x\right)}\right)} - 1\right)} \]

    3. *-commutative 65.9%

       \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left(1 + x\right) \cdot \left({x}^{0.5} + {\left(1 + x\right)}^{-0.5} \cdot x\right)}}\right)} - 1\right) \]

    4. pow1/2 65.9%

       \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \left(e^{\mathsf{log1p}\left(\frac{1}{\left(1 + x\right) \cdot \left(\color{blue}{\sqrt{x}} + {\left(1 + x\right)}^{-0.5} \cdot x\right)}\right)} - 1\right) \]

    5. +-commutative 65.9%

       \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \left(e^{\mathsf{log1p}\left(\frac{1}{\left(1 + x\right) \cdot \color{blue}{\left({\left(1 + x\right)}^{-0.5} \cdot x + \sqrt{x}\right)}}\right)} - 1\right) \]

    6. *-commutative 65.9%

       \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \left(e^{\mathsf{log1p}\left(\frac{1}{\left(1 + x\right) \cdot \left(\color{blue}{x \cdot {\left(1 + x\right)}^{-0.5}} + \sqrt{x}\right)}\right)} - 1\right) \]

    7. fma-def 65.9%

       \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \left(e^{\mathsf{log1p}\left(\frac{1}{\left(1 + x\right) \cdot \color{blue}{\mathsf{fma}\left(x, {\left(1 + x\right)}^{-0.5}, \sqrt{x}\right)}}\right)} - 1\right) \]

15. Applied egg-rr 65.9%

    \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, {\left(1 + x\right)}^{-0.5}, \sqrt{x}\right)}\right)} - 1\right)} \]
16. Step-by-step derivation

    1. expm1-def 95.0%

       \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, {\left(1 + x\right)}^{-0.5}, \sqrt{x}\right)}\right)\right)} \]

    2. expm1-log1p 98.7%

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

    3. *-commutative 98.7%

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

    4. associate-/r* 99.6%

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

    5. +-commutative 99.6%

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

    6. +-commutative 99.6%

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

17. Simplified 99.6%

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

18. Final simplification 99.6%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

   1. inv-pow 69.8%

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

   2. add-cube-cbrt 57.9%

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

   3. unpow-prod-down 57.1%

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

   4. frac-2neg 57.1%

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

   5. metadata-eval 57.1%

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

   6. div-inv 57.1%

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

   7. metadata-eval 57.1%

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

   8. distribute-neg-frac 57.1%

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

   9. metadata-eval 57.1%

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

   10. frac-2neg 57.1%

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

   11. prod-diff 53.5%

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

3. Applied egg-rr 53.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{-1}, {\left(\sqrt[3]{\sqrt{x}}\right)}^{-1}, -\frac{-1}{\sqrt{1 + x}} \cdot -1\right) + \mathsf{fma}\left(-\frac{-1}{\sqrt{1 + x}}, -1, \frac{-1}{\sqrt{1 + x}} \cdot -1\right)} \]
4. Step-by-step derivation

   1. +-commutative 53.9%

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

   2. fma-udef 53.9%

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

   3. distribute-lft-neg-in 53.9%

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

   4. distribute-rgt-neg-in 53.9%

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

   5. metadata-eval 53.9%

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

   6. distribute-lft-out 53.9%

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

   7. metadata-eval 53.9%

      \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot \color{blue}{0} + \mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{-1}, {\left(\sqrt[3]{\sqrt{x}}\right)}^{-1}, -\frac{-1}{\sqrt{1 + x}} \cdot -1\right) \]

   8. distribute-rgt-neg-in 53.9%

      \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{-1}, {\left(\sqrt[3]{\sqrt{x}}\right)}^{-1}, \color{blue}{\frac{-1}{\sqrt{1 + x}} \cdot \left(--1\right)}\right) \]

   9. metadata-eval 53.9%

      \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{-1}, {\left(\sqrt[3]{\sqrt{x}}\right)}^{-1}, \frac{-1}{\sqrt{1 + x}} \cdot \color{blue}{1}\right) \]

   10. unpow-1 53.9%

       \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt[3]{x}}}, {\left(\sqrt[3]{\sqrt{x}}\right)}^{-1}, \frac{-1}{\sqrt{1 + x}} \cdot 1\right) \]

   11. unpow-1 53.9%

       \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \mathsf{fma}\left(\frac{1}{\sqrt[3]{x}}, \color{blue}{\frac{1}{\sqrt[3]{\sqrt{x}}}}, \frac{-1}{\sqrt{1 + x}} \cdot 1\right) \]

   12. *-rgt-identity 53.9%

       \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \mathsf{fma}\left(\frac{1}{\sqrt[3]{x}}, \frac{1}{\sqrt[3]{\sqrt{x}}}, \color{blue}{\frac{-1}{\sqrt{1 + x}}}\right) \]

5. Simplified 53.9%

   \[\leadsto \color{blue}{\frac{-1}{\sqrt{1 + x}} \cdot 0 + \mathsf{fma}\left(\frac{1}{\sqrt[3]{x}}, \frac{1}{\sqrt[3]{\sqrt{x}}}, \frac{-1}{\sqrt{1 + x}}\right)} \]

7. Applied egg-rr 69.6%

   \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \color{blue}{\left(\frac{\frac{1}{x}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} - \frac{\frac{1}{x + 1}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\right)} \]
8. Step-by-step derivation

   1. associate-/l/ 65.8%

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

9. Simplified 65.8%

   \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \color{blue}{\left(\frac{1}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot x} - \frac{\frac{1}{x + 1}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\right)} \]
10. Step-by-step derivation

    1. associate-/r* 63.7%

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

    2. clear-num 64.7%

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

    3. frac-sub 69.6%

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

    4. +-commutative 69.6%

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

    5. +-commutative 69.6%

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

    6. +-commutative 69.6%

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

11. Applied egg-rr 69.6%

    \[\leadsto \frac{-1}{\sqrt{1 + x}} \cdot 0 + \color{blue}{\frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \cdot \frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{1}{1 + x}} - x \cdot 1}{x \cdot \frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{1}{1 + x}}}} \]
12. Step-by-step derivation

    1. associate-*r/ 70.0%

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

    2. lft-mult-inverse 70.1%

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

    3. remove-double-div 70.1%

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

    4. *-rgt-identity 70.1%

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

    5. associate-/r/ 70.1%

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

    6. /-rgt-identity 70.1%

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

    7. associate--l+ 98.5%

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

    8. +-inverses 98.5%

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

    9. metadata-eval 98.5%

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

    10. associate-*r* 98.5%

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

13. Simplified 98.7%

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

14. Taylor expanded in x around 0 98.7%

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

15. Final simplification 98.7%

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

Alternative 4: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 112000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 112000000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 112000000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 112000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 112000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 112000000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 112000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 112000000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 112000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.12e8

   1. Initial program 99.3%

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

      1. *-un-lft-identity 99.3%

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

      2. clear-num 99.3%

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

      3. associate-/r/ 99.3%

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

      4. prod-diff 99.3%

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

      5. *-un-lft-identity 99.3%

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

      6. fma-neg 99.3%

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

      7. *-un-lft-identity 99.3%

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

      8. pow1/2 99.3%

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

      9. pow-flip 99.6%

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

      10. metadata-eval 99.6%

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

      11. pow1/2 99.6%

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

      12. pow-flip 99.7%

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

      13. +-commutative 99.7%

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

      14. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. associate-+l- 99.7%

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

      2. expm1-log1p 99.6%

         \[\leadsto {x}^{-0.5} - \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} - \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)\right) \]

      3. expm1-def 99.5%

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

      4. associate--l- 99.5%

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

      5. fma-udef 99.5%

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

      6. distribute-lft1-in 99.5%

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

      7. metadata-eval 99.5%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 99.5%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 99.5%

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

      10. expm1-def 99.6%

          \[\leadsto {x}^{-0.5} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} \]

      11. expm1-log1p 99.7%

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

   5. Simplified 99.7%

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

   if 1.12e8 < x

   1. Initial program 37.0%

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

      1. *-un-lft-identity 37.0%

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

      2. clear-num 37.0%

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

      3. associate-/r/ 37.0%

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

      4. prod-diff 37.0%

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

      5. *-un-lft-identity 37.0%

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

      6. fma-neg 37.0%

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

      7. *-un-lft-identity 37.0%

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

      8. pow1/2 37.0%

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

      9. pow-flip 30.7%

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

      10. metadata-eval 30.7%

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

      11. pow1/2 30.7%

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

      12. pow-flip 37.1%

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

      13. +-commutative 37.1%

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

      14. metadata-eval 37.1%

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

   3. Applied egg-rr 37.1%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. associate-+l- 37.1%

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

      2. expm1-log1p 37.1%

         \[\leadsto {x}^{-0.5} - \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} - \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)\right) \]

      3. expm1-def 5.2%

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

      4. associate--l- 5.2%

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

      5. fma-udef 5.2%

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

      6. distribute-lft1-in 5.2%

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

      7. metadata-eval 5.2%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 5.2%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 5.2%

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

      10. expm1-def 37.1%

          \[\leadsto {x}^{-0.5} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} \]

      11. expm1-log1p 37.1%

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

   5. Simplified 37.1%

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

   6. Taylor expanded in x around inf 61.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 112000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 5: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

         \[\leadsto {x}^{-0.5} - \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} - \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)\right) \]

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

          \[\leadsto {x}^{-0.5} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} \]

      11. expm1-log1p 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.7%

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

   if 1 < x

   1. Initial program 38.2%

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

      1. *-un-lft-identity 38.2%

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

      2. clear-num 38.2%

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

      3. associate-/r/ 38.2%

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

      4. prod-diff 38.2%

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

      5. *-un-lft-identity 38.2%

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

      6. fma-neg 38.2%

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

      7. *-un-lft-identity 38.2%

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

      8. pow1/2 38.2%

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

      9. pow-flip 32.0%

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

      10. metadata-eval 32.0%

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

      11. pow1/2 32.0%

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

      12. pow-flip 38.2%

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

      13. +-commutative 38.2%

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

      14. metadata-eval 38.2%

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

   3. Applied egg-rr 38.2%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. associate-+l- 38.2%

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

      2. expm1-log1p 38.2%

         \[\leadsto {x}^{-0.5} - \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} - \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)\right) \]

      3. expm1-def 6.9%

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

      4. associate--l- 6.9%

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

      5. fma-udef 6.9%

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

      6. distribute-lft1-in 6.9%

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

      7. metadata-eval 6.9%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 6.9%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 6.9%

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

      10. expm1-def 38.2%

          \[\leadsto {x}^{-0.5} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} \]

      11. expm1-log1p 38.2%

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

   5. Simplified 38.2%

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

   6. Taylor expanded in x around inf 60.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 6: 68.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot \left(1 + x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/ 1.0 (sqrt (* x (+ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = 1.0 / sqrt((x * (1.0 + x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = 1.0d0 / sqrt((x * (1.0d0 + x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = 1.0 / Math.sqrt((x * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = 1.0 / math.sqrt((x * (1.0 + x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(1.0 / sqrt(Float64(x * Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = 1.0 / sqrt((x * (1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 / N[Sqrt[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

         \[\leadsto {x}^{-0.5} - \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} - \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)\right) \]

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

          \[\leadsto {x}^{-0.5} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} \]

      11. expm1-log1p 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.7%

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

   if 1.3999999999999999 < x

   1. Initial program 38.2%

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

      1. frac-sub 38.4%

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

      2. *-un-lft-identity 38.4%

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

      3. *-rgt-identity 38.4%

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

      4. +-commutative 38.4%

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

      5. sqrt-unprod 38.4%

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

      6. +-commutative 38.4%

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

   3. Applied egg-rr 38.4%

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

   4. Taylor expanded in x around 0 37.9%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot \left(1 + x\right)}}\\ \end{array} \]

Alternative 7: 68.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot \left(1 + x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.7) (+ (pow x -0.5) -1.0) (/ 1.0 (sqrt (* x (+ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.7) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / sqrt((x * (1.0 + x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.7d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = 1.0d0 / sqrt((x * (1.0d0 + x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.7) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / Math.sqrt((x * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.7:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = 1.0 / math.sqrt((x * (1.0 + x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(1.0 / sqrt(Float64(x * Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = 1.0 / sqrt((x * (1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.7], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 / N[Sqrt[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

         \[\leadsto {x}^{-0.5} - \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} - \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)\right) \]

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

          \[\leadsto {x}^{-0.5} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} \]

      11. expm1-log1p 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{{x}^{-0.5} - 1} \]

   if 0.69999999999999996 < x

   1. Initial program 38.2%

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

      1. frac-sub 38.4%

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

      2. *-un-lft-identity 38.4%

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

      3. *-rgt-identity 38.4%

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

      4. +-commutative 38.4%

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

      5. sqrt-unprod 38.4%

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

      6. +-commutative 38.4%

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

   3. Applied egg-rr 38.4%

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

   4. Taylor expanded in x around 0 37.9%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot \left(1 + x\right)}}\\ \end{array} \]

Alternative 8: 53.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.6) (+ (pow x -0.5) -1.0) (/ 1.0 (+ x 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.6) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / (x + 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.6d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = 1.0d0 / (x + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.6) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / (x + 0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.6:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = 1.0 / (x + 0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.6)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(1.0 / Float64(x + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.6)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = 1.0 / (x + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.6], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 / N[(x + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.599999999999999978

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

         \[\leadsto {x}^{-0.5} - \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} - \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)\right) \]

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

          \[\leadsto {x}^{-0.5} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} \]

      11. expm1-log1p 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{{x}^{-0.5} - 1} \]

   if 0.599999999999999978 < x

   1. Initial program 38.2%

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

      1. frac-sub 38.4%

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

      2. *-un-lft-identity 38.4%

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

      3. *-rgt-identity 38.4%

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

      4. +-commutative 38.4%

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

      5. sqrt-unprod 38.4%

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

      6. +-commutative 38.4%

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

   3. Applied egg-rr 38.4%

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

   4. Taylor expanded in x around inf 37.3%

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

      1. +-commutative 37.3%

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

   6. Simplified 37.3%

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

   7. Taylor expanded in x around 0 7.6%

      \[\leadsto \frac{\color{blue}{1}}{x + 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 0.5}\\ \end{array} \]

Alternative 9: 66.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

   1. *-un-lft-identity 69.8%

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

   2. clear-num 69.8%

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

   3. associate-/r/ 69.8%

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

   4. prod-diff 69.8%

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

   5. *-un-lft-identity 69.8%

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

   6. fma-neg 69.8%

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

   7. *-un-lft-identity 69.8%

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

   8. pow1/2 69.8%

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

   9. pow-flip 67.0%

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

   10. metadata-eval 67.0%

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

   11. pow1/2 67.0%

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

   12. pow-flip 70.1%

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

   13. +-commutative 70.1%

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

   14. metadata-eval 70.1%

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

3. Applied egg-rr 70.1%

   \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
4. Step-by-step derivation

   1. associate-+l- 70.1%

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

   2. expm1-log1p 70.1%

      \[\leadsto {x}^{-0.5} - \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} - \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)\right) \]

   3. expm1-def 54.9%

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

   4. associate--l- 54.9%

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

   5. fma-udef 54.9%

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

   6. distribute-lft1-in 54.9%

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

   7. metadata-eval 54.9%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

   8. mul0-lft 54.9%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

   9. metadata-eval 54.9%

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

   10. expm1-def 70.1%

       \[\leadsto {x}^{-0.5} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} \]

   11. expm1-log1p 70.1%

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

5. Simplified 70.1%

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

7. Applied egg-rr 52.8%

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

   1. associate-*r/ 52.8%

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

   2. *-rgt-identity 52.8%

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

9. Simplified 52.8%

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

10. Taylor expanded in x around inf 52.2%

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

    1. inv-pow 52.2%

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

    2. sqrt-pow1 52.3%

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

    3. metadata-eval 52.3%

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

    4. expm1-log1p-u 48.5%

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

    5. expm1-udef 63.6%

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

    6. log1p-udef 63.6%

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

    7. add-exp-log 67.4%

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

12. Applied egg-rr 67.4%

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

13. Final simplification 67.4%

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

Alternative 10: 52.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

   1. *-un-lft-identity 69.8%

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

   2. clear-num 69.8%

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

   3. associate-/r/ 69.8%

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

   4. prod-diff 69.8%

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

   5. *-un-lft-identity 69.8%

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

   6. fma-neg 69.8%

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

   7. *-un-lft-identity 69.8%

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

   8. pow1/2 69.8%

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

   9. pow-flip 67.0%

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

   10. metadata-eval 67.0%

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

   11. pow1/2 67.0%

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

   12. pow-flip 70.1%

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

   13. +-commutative 70.1%

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

   14. metadata-eval 70.1%

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

3. Applied egg-rr 70.1%

   \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
4. Step-by-step derivation

   1. associate-+l- 70.1%

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

   2. expm1-log1p 70.1%

      \[\leadsto {x}^{-0.5} - \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} - \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)\right) \]

   3. expm1-def 54.9%

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

   4. associate--l- 54.9%

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

   5. fma-udef 54.9%

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

   6. distribute-lft1-in 54.9%

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

   7. metadata-eval 54.9%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

   8. mul0-lft 54.9%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

   9. metadata-eval 54.9%

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

   10. expm1-def 70.1%

       \[\leadsto {x}^{-0.5} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} \]

   11. expm1-log1p 70.1%

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

5. Simplified 70.1%

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

7. Applied egg-rr 52.8%

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

   1. associate-*r/ 52.8%

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

   2. *-rgt-identity 52.8%

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

9. Simplified 52.8%

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

10. Taylor expanded in x around 0 53.9%

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

    1. associate-/r* 53.9%

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

12. Simplified 53.9%

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

13. Taylor expanded in x around 0 53.9%

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

    1. distribute-rgt-in 53.9%

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

    2. *-lft-identity 53.9%

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

    3. pow-plus 54.1%

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

    4. metadata-eval 54.1%

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

    5. unpow1/2 54.1%

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

15. Simplified 54.1%

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

16. Final simplification 54.1%

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

Alternative 11: 50.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow x -0.5))

C

double code(double x) {
	return pow(x, -0.5);
}

Fortran

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

Java

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

Python

def code(x):
	return math.pow(x, -0.5)

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

function tmp = code(x)
	tmp = x ^ -0.5;
end

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 69.8%

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

   1. *-un-lft-identity 69.8%

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

   2. clear-num 69.8%

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

   3. associate-/r/ 69.8%

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

   4. prod-diff 69.8%

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

   5. *-un-lft-identity 69.8%

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

   6. fma-neg 69.8%

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

   7. *-un-lft-identity 69.8%

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

   8. pow1/2 69.8%

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

   9. pow-flip 67.0%

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

   10. metadata-eval 67.0%

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

   11. pow1/2 67.0%

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

   12. pow-flip 70.1%

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

   13. +-commutative 70.1%

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

   14. metadata-eval 70.1%

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

3. Applied egg-rr 70.1%

   \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
4. Step-by-step derivation

   1. associate-+l- 70.1%

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

   2. expm1-log1p 70.1%

      \[\leadsto {x}^{-0.5} - \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} - \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)\right) \]

   3. expm1-def 54.9%

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

   4. associate--l- 54.9%

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

   5. fma-udef 54.9%

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

   6. distribute-lft1-in 54.9%

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

   7. metadata-eval 54.9%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

   8. mul0-lft 54.9%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

   9. metadata-eval 54.9%

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

   10. expm1-def 70.1%

       \[\leadsto {x}^{-0.5} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} \]

   11. expm1-log1p 70.1%

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

5. Simplified 70.1%

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

7. Applied egg-rr 52.8%

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

   1. associate-*r/ 52.8%

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

   2. *-rgt-identity 52.8%

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

9. Simplified 52.8%

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

10. Taylor expanded in x around inf 52.2%

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

    1. inv-pow 52.2%

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

    2. sqrt-pow1 52.3%

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

    3. metadata-eval 52.3%

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

    4. *-un-lft-identity 52.3%

       \[\leadsto \color{blue}{1 \cdot {x}^{-0.5}} \]

12. Applied egg-rr 52.3%

    \[\leadsto \color{blue}{1 \cdot {x}^{-0.5}} \]
13. Step-by-step derivation

    1. *-lft-identity 52.3%

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

14. Simplified 52.3%

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

15. Final simplification 52.3%

    \[\leadsto {x}^{-0.5} \]

Alternative 12: 7.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

   1. frac-sub 69.9%

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

   2. *-un-lft-identity 69.9%

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

   3. *-rgt-identity 69.9%

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

   4. +-commutative 69.9%

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

   5. sqrt-unprod 69.9%

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

   6. +-commutative 69.9%

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

3. Applied egg-rr 69.9%

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

4. Taylor expanded in x around inf 21.6%

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

   1. +-commutative 21.6%

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

6. Simplified 21.6%

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

7. Taylor expanded in x around 0 7.3%

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

8. Final simplification 7.3%

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

Alternative 13: 5.8% accurate, 209.0× 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 69.8%

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

   1. frac-sub 69.9%

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

   2. *-un-lft-identity 69.9%

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

   3. *-rgt-identity 69.9%

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

   4. +-commutative 69.9%

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

   5. sqrt-unprod 69.9%

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

   6. +-commutative 69.9%

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

3. Applied egg-rr 69.9%

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

4. Taylor expanded in x around inf 21.6%

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

   1. +-commutative 21.6%

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

6. Simplified 21.6%

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

7. Taylor expanded in x around 0 5.8%

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

8. Final simplification 5.8%

   \[\leadsto 2 \]

Developer target: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023318 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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