Given's Rotation SVD example, simplified

Percentage Accurate: 76.6% → 99.5%

Time: 10.5s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\_m\right) \leq 2:\\ \;\;\;\;{x\_m}^{2} \cdot \left(0.125 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot \left(0.0673828125 + {x\_m}^{2} \cdot -0.056243896484375\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x\_m}}{1 + \sqrt[3]{{\left(0.5 + \frac{0.5}{x\_m}\right)}^{1.5}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (hypot 1.0 x_m) 2.0)
   (*
    (pow x_m 2.0)
    (+
     0.125
     (*
      (pow x_m 2.0)
      (-
       (* (pow x_m 2.0) (+ 0.0673828125 (* (pow x_m 2.0) -0.056243896484375)))
       0.0859375))))
   (/ (- 0.5 (/ 0.5 x_m)) (+ 1.0 (cbrt (pow (+ 0.5 (/ 0.5 x_m)) 1.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (hypot(1.0, x_m) <= 2.0) {
		tmp = pow(x_m, 2.0) * (0.125 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * (0.0673828125 + (pow(x_m, 2.0) * -0.056243896484375))) - 0.0859375)));
	} else {
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + cbrt(pow((0.5 + (0.5 / x_m)), 1.5)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.hypot(1.0, x_m) <= 2.0) {
		tmp = Math.pow(x_m, 2.0) * (0.125 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * (0.0673828125 + (Math.pow(x_m, 2.0) * -0.056243896484375))) - 0.0859375)));
	} else {
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + Math.cbrt(Math.pow((0.5 + (0.5 / x_m)), 1.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (hypot(1.0, x_m) <= 2.0)
		tmp = Float64((x_m ^ 2.0) * Float64(0.125 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * Float64(0.0673828125 + Float64((x_m ^ 2.0) * -0.056243896484375))) - 0.0859375))));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / x_m)) / Float64(1.0 + cbrt((Float64(0.5 + Float64(0.5 / x_m)) ^ 1.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision], 2.0], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.125 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.0673828125 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.056243896484375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Power[N[Power[N[(0.5 + N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.7%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 49.7%

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

      2. metadata-eval 49.7%

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

      3. associate-*r/ 49.7%

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

      4. metadata-eval 49.7%

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

   3. Simplified 49.7%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + -0.056243896484375 \cdot {x}^{2}\right) - 0.0859375\right)\right)} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.2%

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

      1. flip-- 98.2%

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

      2. metadata-eval 98.2%

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

      3. add-sqr-sqrt 99.7%

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

      4. associate--r+ 99.7%

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

      5. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. add-cbrt-cube 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{\sqrt[3]{\left(\sqrt{0.5 + \frac{0.5}{x}} \cdot \sqrt{0.5 + \frac{0.5}{x}}\right) \cdot \sqrt{0.5 + \frac{0.5}{x}}}}} \]

      2. pow1/3 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{{\left(\left(\sqrt{0.5 + \frac{0.5}{x}} \cdot \sqrt{0.5 + \frac{0.5}{x}}\right) \cdot \sqrt{0.5 + \frac{0.5}{x}}\right)}^{0.3333333333333333}}} \]

      3. pow3 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + {\color{blue}{\left({\left(\sqrt{0.5 + \frac{0.5}{x}}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. sqrt-pow2 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + {\color{blue}{\left({\left(0.5 + \frac{0.5}{x}\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333}} \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + {\left({\left(0.5 + \frac{0.5}{x}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   9. Applied egg-rr 99.7%

      \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{{\left({\left(0.5 + \frac{0.5}{x}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   10. Step-by-step derivation

       1. unpow1/3 99.7%

          \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{x}\right)}^{1.5}}}} \]

   11. Simplified 99.7%

       \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{x}\right)}^{1.5}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + {x}^{2} \cdot -0.056243896484375\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + \sqrt[3]{{\left(0.5 + \frac{0.5}{x}\right)}^{1.5}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\_m\right) \leq 1.00005:\\ \;\;\;\;{x\_m}^{2} \cdot \left(0.125 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot 0.0673828125 - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \sqrt{\frac{0.25}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\_m\right)}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (hypot 1.0 x_m) 1.00005)
   (*
    (pow x_m 2.0)
    (+ 0.125 (* (pow x_m 2.0) (- (* (pow x_m 2.0) 0.0673828125) 0.0859375))))
   (/
    (- 0.5 (sqrt (/ 0.25 (fma x_m x_m 1.0))))
    (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x_m))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (hypot(1.0, x_m) <= 1.00005) {
		tmp = pow(x_m, 2.0) * (0.125 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * 0.0673828125) - 0.0859375)));
	} else {
		tmp = (0.5 - sqrt((0.25 / fma(x_m, x_m, 1.0)))) / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x_m)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (hypot(1.0, x_m) <= 1.00005)
		tmp = Float64((x_m ^ 2.0) * Float64(0.125 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * 0.0673828125) - 0.0859375))));
	else
		tmp = Float64(Float64(0.5 - sqrt(Float64(0.25 / fma(x_m, x_m, 1.0)))) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x_m))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision], 1.00005], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.125 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.0673828125), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[Sqrt[N[(0.25 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.00005000000000011

   1. Initial program 49.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 49.5%

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

      2. metadata-eval 49.5%

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

      3. associate-*r/ 49.5%

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

      4. metadata-eval 49.5%

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

   3. Simplified 49.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)} \]

   if 1.00005000000000011 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.3%

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

      2. metadata-eval 98.3%

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

      3. associate-*r/ 98.3%

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

      4. metadata-eval 98.3%

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

   3. Simplified 98.3%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.3%

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

      2. metadata-eval 98.3%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate--r+ 99.8%

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

      5. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. frac-times 99.8%

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

      4. metadata-eval 99.8%

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

      5. hypot-undefine 99.8%

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

      6. hypot-undefine 99.8%

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

      7. rem-square-sqrt 99.8%

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

      8. metadata-eval 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

   8. Applied egg-rr 99.8%

      \[\leadsto \frac{0.5 - \color{blue}{\sqrt{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.00005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot 0.0673828125 - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \sqrt{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\_m\right) \leq 2:\\ \;\;\;\;{x\_m}^{2} \cdot \left(0.125 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot 0.0673828125 - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x\_m}}{1 + \sqrt[3]{{\left(0.5 + \frac{0.5}{x\_m}\right)}^{1.5}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (hypot 1.0 x_m) 2.0)
   (*
    (pow x_m 2.0)
    (+ 0.125 (* (pow x_m 2.0) (- (* (pow x_m 2.0) 0.0673828125) 0.0859375))))
   (/ (- 0.5 (/ 0.5 x_m)) (+ 1.0 (cbrt (pow (+ 0.5 (/ 0.5 x_m)) 1.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (hypot(1.0, x_m) <= 2.0) {
		tmp = pow(x_m, 2.0) * (0.125 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * 0.0673828125) - 0.0859375)));
	} else {
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + cbrt(pow((0.5 + (0.5 / x_m)), 1.5)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.hypot(1.0, x_m) <= 2.0) {
		tmp = Math.pow(x_m, 2.0) * (0.125 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * 0.0673828125) - 0.0859375)));
	} else {
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + Math.cbrt(Math.pow((0.5 + (0.5 / x_m)), 1.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (hypot(1.0, x_m) <= 2.0)
		tmp = Float64((x_m ^ 2.0) * Float64(0.125 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * 0.0673828125) - 0.0859375))));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / x_m)) / Float64(1.0 + cbrt((Float64(0.5 + Float64(0.5 / x_m)) ^ 1.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision], 2.0], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.125 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.0673828125), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Power[N[Power[N[(0.5 + N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.7%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 49.7%

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

      2. metadata-eval 49.7%

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

      3. associate-*r/ 49.7%

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

      4. metadata-eval 49.7%

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

   3. Simplified 49.7%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.2%

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

      1. flip-- 98.2%

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

      2. metadata-eval 98.2%

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

      3. add-sqr-sqrt 99.7%

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

      4. associate--r+ 99.7%

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

      5. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. add-cbrt-cube 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{\sqrt[3]{\left(\sqrt{0.5 + \frac{0.5}{x}} \cdot \sqrt{0.5 + \frac{0.5}{x}}\right) \cdot \sqrt{0.5 + \frac{0.5}{x}}}}} \]

      2. pow1/3 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{{\left(\left(\sqrt{0.5 + \frac{0.5}{x}} \cdot \sqrt{0.5 + \frac{0.5}{x}}\right) \cdot \sqrt{0.5 + \frac{0.5}{x}}\right)}^{0.3333333333333333}}} \]

      3. pow3 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + {\color{blue}{\left({\left(\sqrt{0.5 + \frac{0.5}{x}}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. sqrt-pow2 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + {\color{blue}{\left({\left(0.5 + \frac{0.5}{x}\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333}} \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + {\left({\left(0.5 + \frac{0.5}{x}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   9. Applied egg-rr 99.7%

      \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{{\left({\left(0.5 + \frac{0.5}{x}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   10. Step-by-step derivation

       1. unpow1/3 99.7%

          \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{x}\right)}^{1.5}}}} \]

   11. Simplified 99.7%

       \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{x}\right)}^{1.5}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot 0.0673828125 - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + \sqrt[3]{{\left(0.5 + \frac{0.5}{x}\right)}^{1.5}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\_m\right) \leq 2:\\ \;\;\;\;{x\_m}^{2} \cdot \left(0.125 + {x\_m}^{2} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x\_m}}{1 + \sqrt[3]{{\left(0.5 + \frac{0.5}{x\_m}\right)}^{1.5}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (hypot 1.0 x_m) 2.0)
   (* (pow x_m 2.0) (+ 0.125 (* (pow x_m 2.0) -0.0859375)))
   (/ (- 0.5 (/ 0.5 x_m)) (+ 1.0 (cbrt (pow (+ 0.5 (/ 0.5 x_m)) 1.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (hypot(1.0, x_m) <= 2.0) {
		tmp = pow(x_m, 2.0) * (0.125 + (pow(x_m, 2.0) * -0.0859375));
	} else {
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + cbrt(pow((0.5 + (0.5 / x_m)), 1.5)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.hypot(1.0, x_m) <= 2.0) {
		tmp = Math.pow(x_m, 2.0) * (0.125 + (Math.pow(x_m, 2.0) * -0.0859375));
	} else {
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + Math.cbrt(Math.pow((0.5 + (0.5 / x_m)), 1.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (hypot(1.0, x_m) <= 2.0)
		tmp = Float64((x_m ^ 2.0) * Float64(0.125 + Float64((x_m ^ 2.0) * -0.0859375)));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / x_m)) / Float64(1.0 + cbrt((Float64(0.5 + Float64(0.5 / x_m)) ^ 1.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision], 2.0], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.125 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Power[N[Power[N[(0.5 + N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.7%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 49.7%

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

      2. metadata-eval 49.7%

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

      3. associate-*r/ 49.7%

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

      4. metadata-eval 49.7%

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

   3. Simplified 49.7%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto {x}^{2} \cdot \left(0.125 + \color{blue}{{x}^{2} \cdot -0.0859375}\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot -0.0859375\right)} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.2%

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

      1. flip-- 98.2%

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

      2. metadata-eval 98.2%

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

      3. add-sqr-sqrt 99.7%

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

      4. associate--r+ 99.7%

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

      5. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. add-cbrt-cube 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{\sqrt[3]{\left(\sqrt{0.5 + \frac{0.5}{x}} \cdot \sqrt{0.5 + \frac{0.5}{x}}\right) \cdot \sqrt{0.5 + \frac{0.5}{x}}}}} \]

      2. pow1/3 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{{\left(\left(\sqrt{0.5 + \frac{0.5}{x}} \cdot \sqrt{0.5 + \frac{0.5}{x}}\right) \cdot \sqrt{0.5 + \frac{0.5}{x}}\right)}^{0.3333333333333333}}} \]

      3. pow3 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + {\color{blue}{\left({\left(\sqrt{0.5 + \frac{0.5}{x}}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. sqrt-pow2 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + {\color{blue}{\left({\left(0.5 + \frac{0.5}{x}\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333}} \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + {\left({\left(0.5 + \frac{0.5}{x}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   9. Applied egg-rr 99.7%

      \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{{\left({\left(0.5 + \frac{0.5}{x}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   10. Step-by-step derivation

       1. unpow1/3 99.7%

          \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{x}\right)}^{1.5}}}} \]

   11. Simplified 99.7%

       \[\leadsto \frac{0.5 - \frac{0.5}{x}}{1 + \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{x}\right)}^{1.5}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\_m\right) \leq 2:\\ \;\;\;\;{x\_m}^{2} \cdot \left(0.125 + {x\_m}^{2} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x\_m}}{1 + \sqrt{0.5 + \frac{0.5}{x\_m}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (hypot 1.0 x_m) 2.0)
   (* (pow x_m 2.0) (+ 0.125 (* (pow x_m 2.0) -0.0859375)))
   (/ (- 0.5 (/ 0.5 x_m)) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x_m)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (hypot(1.0, x_m) <= 2.0) {
		tmp = pow(x_m, 2.0) * (0.125 + (pow(x_m, 2.0) * -0.0859375));
	} else {
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + sqrt((0.5 + (0.5 / x_m))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.hypot(1.0, x_m) <= 2.0) {
		tmp = Math.pow(x_m, 2.0) * (0.125 + (Math.pow(x_m, 2.0) * -0.0859375));
	} else {
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + Math.sqrt((0.5 + (0.5 / x_m))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.hypot(1.0, x_m) <= 2.0:
		tmp = math.pow(x_m, 2.0) * (0.125 + (math.pow(x_m, 2.0) * -0.0859375))
	else:
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + math.sqrt((0.5 + (0.5 / x_m))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (hypot(1.0, x_m) <= 2.0)
		tmp = Float64((x_m ^ 2.0) * Float64(0.125 + Float64((x_m ^ 2.0) * -0.0859375)));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / x_m)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / x_m)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (hypot(1.0, x_m) <= 2.0)
		tmp = (x_m ^ 2.0) * (0.125 + ((x_m ^ 2.0) * -0.0859375));
	else
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + sqrt((0.5 + (0.5 / x_m))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision], 2.0], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.125 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.7%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 49.7%

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

      2. metadata-eval 49.7%

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

      3. associate-*r/ 49.7%

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

      4. metadata-eval 49.7%

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

   3. Simplified 49.7%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto {x}^{2} \cdot \left(0.125 + \color{blue}{{x}^{2} \cdot -0.0859375}\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot -0.0859375\right)} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.2%

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

      1. flip-- 98.2%

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

      2. metadata-eval 98.2%

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

      3. add-sqr-sqrt 99.7%

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

      4. associate--r+ 99.7%

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

      5. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

Alternative 6: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\_m\right) \leq 2:\\ \;\;\;\;{x\_m}^{2} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x\_m}}{1 + \sqrt{0.5 + \frac{0.5}{x\_m}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (hypot 1.0 x_m) 2.0)
   (* (pow x_m 2.0) 0.125)
   (/ (- 0.5 (/ 0.5 x_m)) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x_m)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (hypot(1.0, x_m) <= 2.0) {
		tmp = pow(x_m, 2.0) * 0.125;
	} else {
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + sqrt((0.5 + (0.5 / x_m))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.hypot(1.0, x_m) <= 2.0) {
		tmp = Math.pow(x_m, 2.0) * 0.125;
	} else {
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + Math.sqrt((0.5 + (0.5 / x_m))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.hypot(1.0, x_m) <= 2.0:
		tmp = math.pow(x_m, 2.0) * 0.125
	else:
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + math.sqrt((0.5 + (0.5 / x_m))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (hypot(1.0, x_m) <= 2.0)
		tmp = Float64((x_m ^ 2.0) * 0.125);
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / x_m)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / x_m)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (hypot(1.0, x_m) <= 2.0)
		tmp = (x_m ^ 2.0) * 0.125;
	else
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + sqrt((0.5 + (0.5 / x_m))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision], 2.0], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.125), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.7%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 49.7%

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

      2. metadata-eval 49.7%

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

      3. associate-*r/ 49.7%

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

      4. metadata-eval 49.7%

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

   3. Simplified 49.7%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.2%

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

      1. flip-- 98.2%

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

      2. metadata-eval 98.2%

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

      3. add-sqr-sqrt 99.7%

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

      4. associate--r+ 99.7%

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

      5. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;{x}^{2} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\_m\right) \leq 2:\\ \;\;\;\;{x\_m}^{2} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (hypot 1.0 x_m) 2.0)
   (* (pow x_m 2.0) 0.125)
   (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (hypot(1.0, x_m) <= 2.0) {
		tmp = pow(x_m, 2.0) * 0.125;
	} else {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.hypot(1.0, x_m) <= 2.0) {
		tmp = Math.pow(x_m, 2.0) * 0.125;
	} else {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.hypot(1.0, x_m) <= 2.0:
		tmp = math.pow(x_m, 2.0) * 0.125
	else:
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (hypot(1.0, x_m) <= 2.0)
		tmp = Float64((x_m ^ 2.0) * 0.125);
	else
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (hypot(1.0, x_m) <= 2.0)
		tmp = (x_m ^ 2.0) * 0.125;
	else
		tmp = 0.5 / (1.0 + sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision], 2.0], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.125), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.7%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 49.7%

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

      2. metadata-eval 49.7%

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

      3. associate-*r/ 49.7%

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

      4. metadata-eval 49.7%

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

   3. Simplified 49.7%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate--r+ 100.0%

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

      5. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 98.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;{x}^{2} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55:\\ \;\;\;\;{x\_m}^{2} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.55) (* (pow x_m 2.0) 0.125) (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = pow(x_m, 2.0) * 0.125;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.55d0) then
        tmp = (x_m ** 2.0d0) * 0.125d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = Math.pow(x_m, 2.0) * 0.125;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.55:
		tmp = math.pow(x_m, 2.0) * 0.125
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.55)
		tmp = Float64((x_m ^ 2.0) * 0.125);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.55)
		tmp = (x_m ^ 2.0) * 0.125;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.55], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.125), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 64.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 64.2%

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

      2. metadata-eval 64.2%

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

      3. associate-*r/ 64.2%

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

      4. metadata-eval 64.2%

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

   3. Simplified 64.2%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 70.8%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]

   if 1.55000000000000004 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 96.3%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;{x}^{2} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.15 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.15e-77) 0.0 (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.15e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.15d-77) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.15e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.15e-77:
		tmp = 0.0
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.15e-77)
		tmp = 0.0;
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.15e-77)
		tmp = 0.0;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.15e-77], 0.0, N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1500000000000001e-77

   1. Initial program 69.9%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 69.9%

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

      2. metadata-eval 69.9%

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

      3. associate-*r/ 69.9%

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

      4. metadata-eval 69.9%

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

   3. Simplified 69.9%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 38.5%

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

   if 2.1500000000000001e-77 < x

   1. Initial program 77.9%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 77.9%

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

      2. metadata-eval 77.9%

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

      3. associate-*r/ 77.9%

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

      4. metadata-eval 77.9%

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

   3. Simplified 77.9%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 75.1%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.3% accurate, 34.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.15 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.15e-77) 0.0 0.25))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.15e-77) {
		tmp = 0.0;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.15d-77) then
        tmp = 0.0d0
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.15e-77) {
		tmp = 0.0;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.15e-77:
		tmp = 0.0
	else:
		tmp = 0.25
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.15e-77)
		tmp = 0.0;
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.15e-77)
		tmp = 0.0;
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.15e-77], 0.0, 0.25]

Derivation

1. Split input into 2 regimes

2. if x < 2.1500000000000001e-77

   1. Initial program 69.9%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 69.9%

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

      2. metadata-eval 69.9%

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

      3. associate-*r/ 69.9%

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

      4. metadata-eval 69.9%

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

   3. Simplified 69.9%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 38.5%

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

   if 2.1500000000000001e-77 < x

   1. Initial program 77.9%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 77.9%

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

      2. metadata-eval 77.9%

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

      3. associate-*r/ 77.9%

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

      4. metadata-eval 77.9%

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

   3. Simplified 77.9%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 77.9%

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

      2. metadata-eval 77.9%

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

      3. add-sqr-sqrt 79.2%

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

      4. associate--r+ 79.2%

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

      5. metadata-eval 79.2%

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

   6. Applied egg-rr 79.2%

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

   7. Taylor expanded in x around 0 19.8%

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

   8. Taylor expanded in x around inf 19.0%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.9% accurate, 210.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 72.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
2. Step-by-step derivation

   1. distribute-lft-in 72.4%

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

   2. metadata-eval 72.4%

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

   3. associate-*r/ 72.4%

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

   4. metadata-eval 72.4%

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

3. Simplified 72.4%

   \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 27.5%

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

6. Final simplification 27.5%

   \[\leadsto 0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024102 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))