Given's Rotation SVD example, simplified

Percentage Accurate: 76.2% → 99.6%

Time: 11.6s

Alternatives: 10

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: 76.2% 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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\mathsf{fma}\left(-3.814697265625 \cdot 10^{-6}, {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-1.5}\right)}^{6}, 3.814697265625 \cdot 10^{-6}\right)}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \left(0.000244140625 + \frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{t_0}}{1 + \sqrt{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (hypot 1.0 x)))))
   (if (<= (hypot 1.0 x) 1.0)
     (* 0.125 (pow x 2.0))
     (/
      (/
       (/
        (/
         (fma
          -3.814697265625e-6
          (pow (pow (fma x x 1.0) -1.5) 6.0)
          3.814697265625e-6)
         (+
          (/ 0.000244140625 (pow (fma x x 1.0) 6.0))
          (+ 0.000244140625 (/ 0.000244140625 (pow (fma x x 1.0) 3.0)))))
        (+
         (/ 0.0625 (pow (fma x x 1.0) 2.0))
         (- 0.0625 (/ -0.0625 (fma x x 1.0)))))
       t_0)
      (+ 1.0 (sqrt t_0))))))

C

double code(double x) {
	double t_0 = 0.5 + (0.5 / hypot(1.0, x));
	double tmp;
	if (hypot(1.0, x) <= 1.0) {
		tmp = 0.125 * pow(x, 2.0);
	} else {
		tmp = (((fma(-3.814697265625e-6, pow(pow(fma(x, x, 1.0), -1.5), 6.0), 3.814697265625e-6) / ((0.000244140625 / pow(fma(x, x, 1.0), 6.0)) + (0.000244140625 + (0.000244140625 / pow(fma(x, x, 1.0), 3.0))))) / ((0.0625 / pow(fma(x, x, 1.0), 2.0)) + (0.0625 - (-0.0625 / fma(x, x, 1.0))))) / t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(0.5 / hypot(1.0, x)))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0)
		tmp = Float64(0.125 * (x ^ 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(fma(-3.814697265625e-6, ((fma(x, x, 1.0) ^ -1.5) ^ 6.0), 3.814697265625e-6) / Float64(Float64(0.000244140625 / (fma(x, x, 1.0) ^ 6.0)) + Float64(0.000244140625 + Float64(0.000244140625 / (fma(x, x, 1.0) ^ 3.0))))) / Float64(Float64(0.0625 / (fma(x, x, 1.0) ^ 2.0)) + Float64(0.0625 - Float64(-0.0625 / fma(x, x, 1.0))))) / t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0], N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-3.814697265625e-6 * N[Power[N[Power[N[(x * x + 1.0), $MachinePrecision], -1.5], $MachinePrecision], 6.0], $MachinePrecision] + 3.814697265625e-6), $MachinePrecision] / N[(N[(0.000244140625 / N[Power[N[(x * x + 1.0), $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.000244140625 + N[(0.000244140625 / N[Power[N[(x * x + 1.0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.0625 / N[Power[N[(x * x + 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.0625 - N[(-0.0625 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1

   1. Initial program 55.3%

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

      1. distribute-lft-in 55.3%

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

      2. metadata-eval 55.3%

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

      3. associate-*r/ 55.3%

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

      4. metadata-eval 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. 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. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. metadata-eval 99.9%

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

      4. frac-times 99.9%

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

      5. metadata-eval 99.9%

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

      6. hypot-udef 99.9%

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

      7. hypot-udef 99.9%

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

      8. rem-square-sqrt 99.9%

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

      9. metadata-eval 99.9%

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

      10. pow2 99.9%

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

   9. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\left(0.25 - \frac{0.25}{1 + {x}^{2}}\right) \cdot \frac{1}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 99.9%

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

       2. *-rgt-identity 99.9%

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

       3. sub-neg 99.9%

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

       4. distribute-neg-frac 99.9%

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

       5. metadata-eval 99.9%

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

       6. +-commutative 99.9%

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

       7. unpow2 99.9%

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

       8. fma-def 99.9%

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

   11. Simplified 99.9%

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

       1. +-commutative 99.9%

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

       2. flip3-+ 99.9%

          \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(\frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)}\right)}^{3} + {0.25}^{3}}{\frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} + \left(0.25 \cdot 0.25 - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       3. cube-div 99.9%

          \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{{-0.25}^{3}}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}} + {0.25}^{3}}{\frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} + \left(0.25 \cdot 0.25 - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       4. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{-0.015625}}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + {0.25}^{3}}{\frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} + \left(0.25 \cdot 0.25 - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       5. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + \color{blue}{0.015625}}{\frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} + \left(0.25 \cdot 0.25 - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       6. frac-times 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\color{blue}{\frac{-0.25 \cdot -0.25}{\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)}} + \left(0.25 \cdot 0.25 - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       7. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{\color{blue}{0.0625}}{\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)} + \left(0.25 \cdot 0.25 - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       8. pow2 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{0.0625}{\color{blue}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}}} + \left(0.25 \cdot 0.25 - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       9. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(\color{blue}{0.0625} - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       10. *-commutative 99.9%

           \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \color{blue}{0.25 \cdot \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)}}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       11. associate-*r/ 99.9%

           \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \color{blue}{\frac{0.25 \cdot -0.25}{\mathsf{fma}\left(x, x, 1\right)}}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       12. metadata-eval 99.9%

           \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{\color{blue}{-0.0625}}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   13. Applied egg-rr 99.9%

       \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   14. Step-by-step derivation

       1. flip3-+ 100.0%

          \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{{\left(\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}^{3} + {0.015625}^{3}}{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + \left(0.015625 \cdot 0.015625 - \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot 0.015625\right)}}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       2. div-inv 99.9%

          \[\leadsto \frac{\frac{\frac{\color{blue}{\left({\left(\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}^{3} + {0.015625}^{3}\right) \cdot \frac{1}{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + \left(0.015625 \cdot 0.015625 - \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot 0.015625\right)}}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       3. div-inv 99.9%

          \[\leadsto \frac{\frac{\frac{\left({\color{blue}{\left(-0.015625 \cdot \frac{1}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}}^{3} + {0.015625}^{3}\right) \cdot \frac{1}{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + \left(0.015625 \cdot 0.015625 - \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot 0.015625\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       4. unpow-prod-down 99.9%

          \[\leadsto \frac{\frac{\frac{\left(\color{blue}{{-0.015625}^{3} \cdot {\left(\frac{1}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}^{3}} + {0.015625}^{3}\right) \cdot \frac{1}{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + \left(0.015625 \cdot 0.015625 - \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot 0.015625\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       5. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{\left(\color{blue}{-3.814697265625 \cdot 10^{-6}} \cdot {\left(\frac{1}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}^{3} + {0.015625}^{3}\right) \cdot \frac{1}{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + \left(0.015625 \cdot 0.015625 - \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot 0.015625\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       6. pow-flip 99.9%

          \[\leadsto \frac{\frac{\frac{\left(-3.814697265625 \cdot 10^{-6} \cdot {\color{blue}{\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\left(-3\right)}\right)}}^{3} + {0.015625}^{3}\right) \cdot \frac{1}{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + \left(0.015625 \cdot 0.015625 - \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot 0.015625\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       7. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{\left(-3.814697265625 \cdot 10^{-6} \cdot {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\color{blue}{-3}}\right)}^{3} + {0.015625}^{3}\right) \cdot \frac{1}{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + \left(0.015625 \cdot 0.015625 - \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot 0.015625\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       8. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{\left(-3.814697265625 \cdot 10^{-6} \cdot {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-3}\right)}^{3} + \color{blue}{3.814697265625 \cdot 10^{-6}}\right) \cdot \frac{1}{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + \left(0.015625 \cdot 0.015625 - \frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} \cdot 0.015625\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   15. Applied egg-rr 99.9%

       \[\leadsto \frac{\frac{\frac{\color{blue}{\left(-3.814697265625 \cdot 10^{-6} \cdot {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-3}\right)}^{3} + 3.814697265625 \cdot 10^{-6}\right) \cdot \frac{1}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \left(0.000244140625 - \frac{-0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   16. Step-by-step derivation

       1. associate-*r/ 99.9%

          \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{\left(-3.814697265625 \cdot 10^{-6} \cdot {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-3}\right)}^{3} + 3.814697265625 \cdot 10^{-6}\right) \cdot 1}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \left(0.000244140625 - \frac{-0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       2. *-rgt-identity 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{-3.814697265625 \cdot 10^{-6} \cdot {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-3}\right)}^{3} + 3.814697265625 \cdot 10^{-6}}}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \left(0.000244140625 - \frac{-0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       3. fma-def 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(-3.814697265625 \cdot 10^{-6}, {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-3}\right)}^{3}, 3.814697265625 \cdot 10^{-6}\right)}}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \left(0.000244140625 - \frac{-0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       4. sqr-pow 100.0%

          \[\leadsto \frac{\frac{\frac{\frac{\mathsf{fma}\left(-3.814697265625 \cdot 10^{-6}, {\color{blue}{\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\left(\frac{-3}{2}\right)} \cdot {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\left(\frac{-3}{2}\right)}\right)}}^{3}, 3.814697265625 \cdot 10^{-6}\right)}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \left(0.000244140625 - \frac{-0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       5. cube-prod 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{\mathsf{fma}\left(-3.814697265625 \cdot 10^{-6}, \color{blue}{{\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\left(\frac{-3}{2}\right)}\right)}^{3} \cdot {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\left(\frac{-3}{2}\right)}\right)}^{3}}, 3.814697265625 \cdot 10^{-6}\right)}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \left(0.000244140625 - \frac{-0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       6. pow-sqr 100.0%

          \[\leadsto \frac{\frac{\frac{\frac{\mathsf{fma}\left(-3.814697265625 \cdot 10^{-6}, \color{blue}{{\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\left(\frac{-3}{2}\right)}\right)}^{\left(2 \cdot 3\right)}}, 3.814697265625 \cdot 10^{-6}\right)}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \left(0.000244140625 - \frac{-0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       7. metadata-eval 100.0%

          \[\leadsto \frac{\frac{\frac{\frac{\mathsf{fma}\left(-3.814697265625 \cdot 10^{-6}, {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\color{blue}{-1.5}}\right)}^{\left(2 \cdot 3\right)}, 3.814697265625 \cdot 10^{-6}\right)}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \left(0.000244140625 - \frac{-0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       8. metadata-eval 100.0%

          \[\leadsto \frac{\frac{\frac{\frac{\mathsf{fma}\left(-3.814697265625 \cdot 10^{-6}, {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-1.5}\right)}^{\color{blue}{6}}, 3.814697265625 \cdot 10^{-6}\right)}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \left(0.000244140625 - \frac{-0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       9. sub-neg 100.0%

          \[\leadsto \frac{\frac{\frac{\frac{\mathsf{fma}\left(-3.814697265625 \cdot 10^{-6}, {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-1.5}\right)}^{6}, 3.814697265625 \cdot 10^{-6}\right)}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \color{blue}{\left(0.000244140625 + \left(-\frac{-0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)\right)}}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       10. distribute-neg-frac 100.0%

           \[\leadsto \frac{\frac{\frac{\frac{\mathsf{fma}\left(-3.814697265625 \cdot 10^{-6}, {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-1.5}\right)}^{6}, 3.814697265625 \cdot 10^{-6}\right)}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \left(0.000244140625 + \color{blue}{\frac{--0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}}\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       11. metadata-eval 100.0%

           \[\leadsto \frac{\frac{\frac{\frac{\mathsf{fma}\left(-3.814697265625 \cdot 10^{-6}, {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-1.5}\right)}^{6}, 3.814697265625 \cdot 10^{-6}\right)}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \left(0.000244140625 + \frac{\color{blue}{0.000244140625}}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   17. Simplified 100.0%

       \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(-3.814697265625 \cdot 10^{-6}, {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-1.5}\right)}^{6}, 3.814697265625 \cdot 10^{-6}\right)}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \left(0.000244140625 + \frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\mathsf{fma}\left(-3.814697265625 \cdot 10^{-6}, {\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-1.5}\right)}^{6}, 3.814697265625 \cdot 10^{-6}\right)}{\frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{6}} + \left(0.000244140625 + \frac{0.000244140625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}\right)}}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{t_0}}{1 + \sqrt{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (hypot 1.0 x)))))
   (if (<= (hypot 1.0 x) 1.0)
     (* 0.125 (pow x 2.0))
     (/
      (/
       (/
        (+ (/ -0.015625 (pow (fma x x 1.0) 3.0)) 0.015625)
        (+
         (/ 0.0625 (pow (fma x x 1.0) 2.0))
         (- 0.0625 (/ -0.0625 (fma x x 1.0)))))
       t_0)
      (+ 1.0 (sqrt t_0))))))

C

double code(double x) {
	double t_0 = 0.5 + (0.5 / hypot(1.0, x));
	double tmp;
	if (hypot(1.0, x) <= 1.0) {
		tmp = 0.125 * pow(x, 2.0);
	} else {
		tmp = ((((-0.015625 / pow(fma(x, x, 1.0), 3.0)) + 0.015625) / ((0.0625 / pow(fma(x, x, 1.0), 2.0)) + (0.0625 - (-0.0625 / fma(x, x, 1.0))))) / t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(0.5 / hypot(1.0, x)))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0)
		tmp = Float64(0.125 * (x ^ 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.015625 / (fma(x, x, 1.0) ^ 3.0)) + 0.015625) / Float64(Float64(0.0625 / (fma(x, x, 1.0) ^ 2.0)) + Float64(0.0625 - Float64(-0.0625 / fma(x, x, 1.0))))) / t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0], N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.015625 / N[Power[N[(x * x + 1.0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + 0.015625), $MachinePrecision] / N[(N[(0.0625 / N[Power[N[(x * x + 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.0625 - N[(-0.0625 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1

   1. Initial program 55.3%

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

      1. distribute-lft-in 55.3%

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

      2. metadata-eval 55.3%

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

      3. associate-*r/ 55.3%

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

      4. metadata-eval 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. 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. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. metadata-eval 99.9%

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

      4. frac-times 99.9%

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

      5. metadata-eval 99.9%

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

      6. hypot-udef 99.9%

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

      7. hypot-udef 99.9%

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

      8. rem-square-sqrt 99.9%

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

      9. metadata-eval 99.9%

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

      10. pow2 99.9%

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

   9. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\left(0.25 - \frac{0.25}{1 + {x}^{2}}\right) \cdot \frac{1}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 99.9%

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

       2. *-rgt-identity 99.9%

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

       3. sub-neg 99.9%

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

       4. distribute-neg-frac 99.9%

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

       5. metadata-eval 99.9%

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

       6. +-commutative 99.9%

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

       7. unpow2 99.9%

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

       8. fma-def 99.9%

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

   11. Simplified 99.9%

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

       1. +-commutative 99.9%

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

       2. flip3-+ 99.9%

          \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(\frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)}\right)}^{3} + {0.25}^{3}}{\frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} + \left(0.25 \cdot 0.25 - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       3. cube-div 99.9%

          \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{{-0.25}^{3}}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}}} + {0.25}^{3}}{\frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} + \left(0.25 \cdot 0.25 - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       4. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{-0.015625}}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + {0.25}^{3}}{\frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} + \left(0.25 \cdot 0.25 - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       5. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + \color{blue}{0.015625}}{\frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} + \left(0.25 \cdot 0.25 - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       6. frac-times 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\color{blue}{\frac{-0.25 \cdot -0.25}{\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)}} + \left(0.25 \cdot 0.25 - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       7. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{\color{blue}{0.0625}}{\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)} + \left(0.25 \cdot 0.25 - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       8. pow2 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{0.0625}{\color{blue}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}}} + \left(0.25 \cdot 0.25 - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       9. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(\color{blue}{0.0625} - \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} \cdot 0.25\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       10. *-commutative 99.9%

           \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \color{blue}{0.25 \cdot \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)}}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       11. associate-*r/ 99.9%

           \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \color{blue}{\frac{0.25 \cdot -0.25}{\mathsf{fma}\left(x, x, 1\right)}}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       12. metadata-eval 99.9%

           \[\leadsto \frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{\color{blue}{-0.0625}}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   13. Applied egg-rr 99.9%

       \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{-0.015625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{3}} + 0.015625}{\frac{0.0625}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}} + \left(0.0625 - \frac{-0.0625}{\mathsf{fma}\left(x, x, 1\right)}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 + \sqrt{\frac{0.25}{t_0} - \frac{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{t_0}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ 0.5 (hypot 1.0 x)))))
   (if (<= (hypot 1.0 x) 1.0)
     (* 0.125 (pow x 2.0))
     (/ t_0 (+ 1.0 (sqrt (- (/ 0.25 t_0) (/ (/ 0.25 (fma x x 1.0)) t_0))))))))

C

double code(double x) {
	double t_0 = 0.5 - (0.5 / hypot(1.0, x));
	double tmp;
	if (hypot(1.0, x) <= 1.0) {
		tmp = 0.125 * pow(x, 2.0);
	} else {
		tmp = t_0 / (1.0 + sqrt(((0.25 / t_0) - ((0.25 / fma(x, x, 1.0)) / t_0))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(0.5 / hypot(1.0, x)))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0)
		tmp = Float64(0.125 * (x ^ 2.0));
	else
		tmp = Float64(t_0 / Float64(1.0 + sqrt(Float64(Float64(0.25 / t_0) - Float64(Float64(0.25 / fma(x, x, 1.0)) / t_0)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 - N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0], N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.0 + N[Sqrt[N[(N[(0.25 / t$95$0), $MachinePrecision] - N[(N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1

   1. Initial program 55.3%

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

      1. distribute-lft-in 55.3%

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

      2. metadata-eval 55.3%

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

      3. associate-*r/ 55.3%

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

      4. metadata-eval 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. 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. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

      1. flip-+ 99.9%

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

      2. div-sub 99.9%

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

      3. metadata-eval 99.9%

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

      4. frac-times 99.9%

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

      5. metadata-eval 99.9%

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

      6. hypot-udef 99.9%

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

      7. hypot-udef 99.9%

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

      8. rem-square-sqrt 99.9%

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

      9. metadata-eval 99.9%

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

      10. +-commutative 99.9%

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

      11. fma-def 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

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

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t_0}{1 + \sqrt{0.5 + t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.0)
     (* 0.125 (pow x 2.0))
     (/ (- 0.5 t_0) (+ 1.0 (sqrt (+ 0.5 t_0)))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.0) {
		tmp = 0.125 * pow(x, 2.0);
	} else {
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0) {
		tmp = 0.125 * Math.pow(x, 2.0);
	} else {
		tmp = (0.5 - t_0) / (1.0 + Math.sqrt((0.5 + t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	tmp = 0
	if math.hypot(1.0, x) <= 1.0:
		tmp = 0.125 * math.pow(x, 2.0)
	else:
		tmp = (0.5 - t_0) / (1.0 + math.sqrt((0.5 + t_0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0)
		tmp = Float64(0.125 * (x ^ 2.0));
	else
		tmp = Float64(Float64(0.5 - t_0) / Float64(1.0 + sqrt(Float64(0.5 + t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0)
		tmp = 0.125 * (x ^ 2.0);
	else
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0], N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1

   1. Initial program 55.3%

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

      1. distribute-lft-in 55.3%

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

      2. metadata-eval 55.3%

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

      3. associate-*r/ 55.3%

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

      4. metadata-eval 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. 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. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 100.0%

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

Alternative 5: 98.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.5)
   (* 0.125 (pow x 2.0))
   (/ (- 0.5 (/ 0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x)))))))

C

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

Java

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

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.5:
		tmp = 0.125 * math.pow(x, 2.0)
	else:
		tmp = (0.5 - (0.5 / x)) / (1.0 + math.sqrt((0.5 + (0.5 / x))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.5)
		tmp = 0.125 * (x ^ 2.0);
	else
		tmp = (0.5 - (0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.5], N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.5

   1. Initial program 55.6%

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

      1. distribute-lft-in 55.6%

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

      2. metadata-eval 55.6%

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

      3. associate-*r/ 55.6%

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

      4. metadata-eval 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in x around 0 99.4%

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

   if 1.5 < (hypot.f64 1 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. Taylor expanded in x around inf 97.9%

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

      1. associate-*r/ 97.9%

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

      2. metadata-eval 97.9%

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

   6. Simplified 97.9%

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

      1. flip-- 97.9%

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

      2. metadata-eval 97.9%

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

      3. add-sqr-sqrt 99.4%

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

      4. associate--r+ 99.4%

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

      5. metadata-eval 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 99.4%

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

Alternative 6: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $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 1 x) < 2

   1. Initial program 55.9%

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

      1. distribute-lft-in 55.9%

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

      2. metadata-eval 55.9%

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

      3. associate-*r/ 55.9%

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

      4. metadata-eval 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in x around 0 98.8%

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

   if 2 < (hypot.f64 1 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. Taylor expanded in x around inf 97.9%

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

      1. flip-- 97.9%

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

      2. div-inv 97.9%

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

      3. metadata-eval 97.9%

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

      4. rem-square-sqrt 99.4%

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

      5. metadata-eval 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   8. Simplified 99.4%

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

4. Final simplification 99.1%

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

Alternative 7: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 1.52\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.52) (not (<= x 1.52)))
   (- 1.0 (sqrt 0.5))
   (* 0.125 (pow x 2.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.52) || !(x <= 1.52)) {
		tmp = 1.0 - sqrt(0.5);
	} else {
		tmp = 0.125 * pow(x, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.52d0)) .or. (.not. (x <= 1.52d0))) then
        tmp = 1.0d0 - sqrt(0.5d0)
    else
        tmp = 0.125d0 * (x ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.52) || !(x <= 1.52)) {
		tmp = 1.0 - Math.sqrt(0.5);
	} else {
		tmp = 0.125 * Math.pow(x, 2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.52) or not (x <= 1.52):
		tmp = 1.0 - math.sqrt(0.5)
	else:
		tmp = 0.125 * math.pow(x, 2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.52) || !(x <= 1.52))
		tmp = Float64(1.0 - sqrt(0.5));
	else
		tmp = Float64(0.125 * (x ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.52) || ~((x <= 1.52)))
		tmp = 1.0 - sqrt(0.5);
	else
		tmp = 0.125 * (x ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.52], N[Not[LessEqual[x, 1.52]], $MachinePrecision]], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.52 or 1.52 < 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. Taylor expanded in x around inf 97.9%

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

   if -1.52 < x < 1.52

   1. Initial program 55.9%

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

      1. distribute-lft-in 55.9%

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

      2. metadata-eval 55.9%

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

      3. associate-*r/ 55.9%

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

      4. metadata-eval 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 1.52\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \end{array} \]

Alternative 8: 74.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-77} \lor \neg \left(x \leq 2.2 \cdot 10^{-77}\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.2e-77) (not (<= x 2.2e-77))) (- 1.0 (sqrt 0.5)) 0.0))

C

double code(double x) {
	double tmp;
	if ((x <= -2.2e-77) || !(x <= 2.2e-77)) {
		tmp = 1.0 - sqrt(0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.2d-77)) .or. (.not. (x <= 2.2d-77))) then
        tmp = 1.0d0 - sqrt(0.5d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -2.2e-77) || !(x <= 2.2e-77)) {
		tmp = 1.0 - Math.sqrt(0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -2.2e-77) or not (x <= 2.2e-77):
		tmp = 1.0 - math.sqrt(0.5)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -2.2e-77) || !(x <= 2.2e-77))
		tmp = Float64(1.0 - sqrt(0.5));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -2.2e-77) || ~((x <= 2.2e-77)))
		tmp = 1.0 - sqrt(0.5);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -2.2e-77], N[Not[LessEqual[x, 2.2e-77]], $MachinePrecision]], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.20000000000000007e-77 or 2.20000000000000007e-77 < x

   1. Initial program 84.9%

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

      1. distribute-lft-in 84.9%

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

      2. metadata-eval 84.9%

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

      3. associate-*r/ 84.9%

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

      4. metadata-eval 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in x around inf 83.7%

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

   if -2.20000000000000007e-77 < x < 2.20000000000000007e-77

   1. Initial program 67.7%

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

      1. distribute-lft-in 67.7%

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

      2. metadata-eval 67.7%

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

      3. associate-*r/ 67.7%

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

      4. metadata-eval 67.7%

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

   3. Simplified 67.7%

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

   4. Taylor expanded in x around 0 67.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-77} \lor \neg \left(x \leq 2.2 \cdot 10^{-77}\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 37.9% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-77}:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.1e-77) 0.25 (if (<= x 2.1e-77) 0.0 0.25)))

C

double code(double x) {
	double tmp;
	if (x <= -2.1e-77) {
		tmp = 0.25;
	} else if (x <= 2.1e-77) {
		tmp = 0.0;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.1d-77)) then
        tmp = 0.25d0
    else if (x <= 2.1d-77) then
        tmp = 0.0d0
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.1e-77) {
		tmp = 0.25;
	} else if (x <= 2.1e-77) {
		tmp = 0.0;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.1e-77:
		tmp = 0.25
	elif x <= 2.1e-77:
		tmp = 0.0
	else:
		tmp = 0.25
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.1e-77)
		tmp = 0.25;
	elseif (x <= 2.1e-77)
		tmp = 0.0;
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.1e-77)
		tmp = 0.25;
	elseif (x <= 2.1e-77)
		tmp = 0.0;
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.1e-77], 0.25, If[LessEqual[x, 2.1e-77], 0.0, 0.25]]

Derivation

1. Split input into 2 regimes

2. if x < -2.10000000000000015e-77 or 2.10000000000000015e-77 < x

   1. Initial program 84.9%

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

      1. distribute-lft-in 84.9%

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

      2. metadata-eval 84.9%

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

      3. associate-*r/ 84.9%

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

      4. metadata-eval 84.9%

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

   3. Simplified 84.9%

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

      1. flip-- 84.9%

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

      2. div-inv 84.9%

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

      3. metadata-eval 84.9%

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

      4. add-sqr-sqrt 86.2%

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

      5. associate--r+ 86.2%

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

      6. metadata-eval 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. associate-*r/ 86.2%

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

      2. *-rgt-identity 86.2%

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

   7. Simplified 86.2%

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

   8. Taylor expanded in x around 0 20.2%

      \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{\color{blue}{2}} \]

   9. Taylor expanded in x around inf 20.3%

      \[\leadsto \frac{\color{blue}{0.5}}{2} \]

   if -2.10000000000000015e-77 < x < 2.10000000000000015e-77

   1. Initial program 67.7%

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

      1. distribute-lft-in 67.7%

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

      2. metadata-eval 67.7%

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

      3. associate-*r/ 67.7%

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

      4. metadata-eval 67.7%

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

   3. Simplified 67.7%

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

   4. Taylor expanded in x around 0 67.7%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-77}:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]

Alternative 10: 27.7% accurate, 210.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 78.5%

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

   1. distribute-lft-in 78.5%

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

   2. metadata-eval 78.5%

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

   3. associate-*r/ 78.5%

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

   4. metadata-eval 78.5%

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

3. Simplified 78.5%

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

4. Taylor expanded in x around 0 27.2%

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

5. Final simplification 27.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023318 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))