Given's Rotation SVD example, simplified

Percentage Accurate: 75.1% → 100.0%

Time: 4.8s

Alternatives: 12

Speedup: 1.2×

Specification

Math

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

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: 75.1% accurate, 1.0× speedup

Math

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

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: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}\\ t_1 := {\left(\left|x\right|\right)}^{2}\\ t_2 := \frac{0.5}{t\_0} - -0.5\\ \mathbf{if}\;\left|x\right| \leq 0.018:\\ \;\;\;\;\frac{\left(t\_1 \cdot \left(t\_1 \cdot \left(0.125 + t\_1 \cdot \left(0.0546875 \cdot t\_1 - 0.078125\right)\right) - 0.25\right)\right) \cdot \left(\frac{-0.5}{t\_0} - 0.5\right)}{1 + \sqrt{t\_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot 1 - t\_2}{\mathsf{fma}\left(\sqrt{\frac{1}{t\_0} - -1}, \sqrt{0.5}, 1\right)}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (sqrt (fma (fabs x) (fabs x) 1.0)))
       (t_1 (pow (fabs x) 2.0))
       (t_2 (- (/ 0.5 t_0) -0.5)))
  (if (<= (fabs x) 0.018)
    (/
     (*
      (*
       t_1
       (-
        (* t_1 (+ 0.125 (* t_1 (- (* 0.0546875 t_1) 0.078125))))
        0.25))
      (- (/ -0.5 t_0) 0.5))
     (+ 1.0 (sqrt t_2)))
    (/
     (- (* 1.0 1.0) t_2)
     (fma (sqrt (- (/ 1.0 t_0) -1.0)) (sqrt 0.5) 1.0)))))

C

double code(double x) {
	double t_0 = sqrt(fma(fabs(x), fabs(x), 1.0));
	double t_1 = pow(fabs(x), 2.0);
	double t_2 = (0.5 / t_0) - -0.5;
	double tmp;
	if (fabs(x) <= 0.018) {
		tmp = ((t_1 * ((t_1 * (0.125 + (t_1 * ((0.0546875 * t_1) - 0.078125)))) - 0.25)) * ((-0.5 / t_0) - 0.5)) / (1.0 + sqrt(t_2));
	} else {
		tmp = ((1.0 * 1.0) - t_2) / fma(sqrt(((1.0 / t_0) - -1.0)), sqrt(0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(fma(abs(x), abs(x), 1.0))
	t_1 = abs(x) ^ 2.0
	t_2 = Float64(Float64(0.5 / t_0) - -0.5)
	tmp = 0.0
	if (abs(x) <= 0.018)
		tmp = Float64(Float64(Float64(t_1 * Float64(Float64(t_1 * Float64(0.125 + Float64(t_1 * Float64(Float64(0.0546875 * t_1) - 0.078125)))) - 0.25)) * Float64(Float64(-0.5 / t_0) - 0.5)) / Float64(1.0 + sqrt(t_2)));
	else
		tmp = Float64(Float64(Float64(1.0 * 1.0) - t_2) / fma(sqrt(Float64(Float64(1.0 / t_0) - -1.0)), sqrt(0.5), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 / t$95$0), $MachinePrecision] - -0.5), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.018], N[(N[(N[(t$95$1 * N[(N[(t$95$1 * N[(0.125 + N[(t$95$1 * N[(N[(0.0546875 * t$95$1), $MachinePrecision] - 0.078125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 / t$95$0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 * 1.0), $MachinePrecision] - t$95$2), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 / t$95$0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.017999999999999999

   1. Initial program 75.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 75.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. sub-flip N/A

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

      5. +-commutative N/A

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

      6. sum-to-mult N/A

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

      7. lower-unsound-*.f64 N/A

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} \cdot {x}^{2} - \frac{5}{64}\right)\right) - \frac{1}{4}\right)\right)} \cdot \left(\frac{-0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - 0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} \cdot {x}^{2} - \frac{5}{64}\right)\right) - \frac{1}{4}\right)}\right) \cdot \left(\frac{\frac{-1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{1}{2}\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{-1}{2}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} \cdot {x}^{2} - \frac{5}{64}\right)\right)} - \frac{1}{4}\right)\right) \cdot \left(\frac{\frac{-1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{1}{2}\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{-1}{2}}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} \cdot {x}^{2} - \frac{5}{64}\right)\right) - \color{blue}{\frac{1}{4}}\right)\right) \cdot \left(\frac{\frac{-1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{1}{2}\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{-1}{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} \cdot {x}^{2} - \frac{5}{64}\right)\right) - \frac{1}{4}\right)\right) \cdot \left(\frac{\frac{-1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{1}{2}\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{-1}{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} \cdot {x}^{2} - \frac{5}{64}\right)\right) - \frac{1}{4}\right)\right) \cdot \left(\frac{\frac{-1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{1}{2}\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{-1}{2}}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} \cdot {x}^{2} - \frac{5}{64}\right)\right) - \frac{1}{4}\right)\right) \cdot \left(\frac{\frac{-1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{1}{2}\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{-1}{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} \cdot {x}^{2} - \frac{5}{64}\right)\right) - \frac{1}{4}\right)\right) \cdot \left(\frac{\frac{-1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{1}{2}\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{-1}{2}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} \cdot {x}^{2} - \frac{5}{64}\right)\right) - \frac{1}{4}\right)\right) \cdot \left(\frac{\frac{-1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{1}{2}\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{-1}{2}}} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} \cdot {x}^{2} - \frac{5}{64}\right)\right) - \frac{1}{4}\right)\right) \cdot \left(\frac{\frac{-1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{1}{2}\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{-1}{2}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} \cdot {x}^{2} - \frac{5}{64}\right)\right) - \frac{1}{4}\right)\right) \cdot \left(\frac{\frac{-1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{1}{2}\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{-1}{2}}} \]

      11. lower-pow.f64 50.5%

          \[\leadsto \frac{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0546875 \cdot {x}^{2} - 0.078125\right)\right) - 0.25\right)\right) \cdot \left(\frac{-0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - 0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]

   8. Applied rewrites 50.5%

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0546875 \cdot {x}^{2} - 0.078125\right)\right) - 0.25\right)\right)} \cdot \left(\frac{-0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - 0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]

   if 0.017999999999999999 < x

   1. Initial program 75.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 75.9%

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

      4. lower-unsound-+.f64 N/A

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

   5. Applied rewrites 75.9%

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

Alternative 2: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|x\right|\right)}^{2}\\ t_1 := \sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}\\ \mathbf{if}\;\left|x\right| \leq 0.018:\\ \;\;\;\;t\_0 \cdot \left(0.125 + t\_0 \cdot \left(t\_0 \cdot \left(0.0673828125 + -0.056243896484375 \cdot t\_0\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot 1 - \left(\frac{0.5}{t\_1} - -0.5\right)}{\mathsf{fma}\left(\sqrt{\frac{1}{t\_1} - -1}, \sqrt{0.5}, 1\right)}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (pow (fabs x) 2.0))
       (t_1 (sqrt (fma (fabs x) (fabs x) 1.0))))
  (if (<= (fabs x) 0.018)
    (*
     t_0
     (+
      0.125
      (*
       t_0
       (-
        (* t_0 (+ 0.0673828125 (* -0.056243896484375 t_0)))
        0.0859375))))
    (/
     (- (* 1.0 1.0) (- (/ 0.5 t_1) -0.5))
     (fma (sqrt (- (/ 1.0 t_1) -1.0)) (sqrt 0.5) 1.0)))))

C

double code(double x) {
	double t_0 = pow(fabs(x), 2.0);
	double t_1 = sqrt(fma(fabs(x), fabs(x), 1.0));
	double tmp;
	if (fabs(x) <= 0.018) {
		tmp = t_0 * (0.125 + (t_0 * ((t_0 * (0.0673828125 + (-0.056243896484375 * t_0))) - 0.0859375)));
	} else {
		tmp = ((1.0 * 1.0) - ((0.5 / t_1) - -0.5)) / fma(sqrt(((1.0 / t_1) - -1.0)), sqrt(0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = abs(x) ^ 2.0
	t_1 = sqrt(fma(abs(x), abs(x), 1.0))
	tmp = 0.0
	if (abs(x) <= 0.018)
		tmp = Float64(t_0 * Float64(0.125 + Float64(t_0 * Float64(Float64(t_0 * Float64(0.0673828125 + Float64(-0.056243896484375 * t_0))) - 0.0859375))));
	else
		tmp = Float64(Float64(Float64(1.0 * 1.0) - Float64(Float64(0.5 / t_1) - -0.5)) / fma(sqrt(Float64(Float64(1.0 / t_1) - -1.0)), sqrt(0.5), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.018], N[(t$95$0 * N[(0.125 + N[(t$95$0 * N[(N[(t$95$0 * N[(0.0673828125 + N[(-0.056243896484375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 * 1.0), $MachinePrecision] - N[(N[(0.5 / t$95$1), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 / t$95$1), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.017999999999999999

   1. Initial program 75.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\color{blue}{\frac{1}{8}} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right)} - \frac{11}{128}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \color{blue}{\frac{11}{128}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \]

      11. lower-pow.f64 50.5%

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

   4. Applied rewrites 50.5%

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

   if 0.017999999999999999 < x

   1. Initial program 75.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 75.9%

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

      4. lower-unsound-+.f64 N/A

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

   5. Applied rewrites 75.9%

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

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}\\ t_1 := \frac{-0.5}{t\_0} - 0.5\\ \mathbf{if}\;\left|x\right| \leq 6.3 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, \left|x\right| \cdot \left|x\right|, 0.125\right) \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\left(\frac{1}{t\_1} - -1\right) \cdot \frac{-1}{-1 - \sqrt{\frac{0.5}{t\_0} - -0.5}}\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (sqrt (fma (fabs x) (fabs x) 1.0)))
       (t_1 (- (/ -0.5 t_0) 0.5)))
  (if (<= (fabs x) 6.3e-6)
    (*
     (* (fma -0.0859375 (* (fabs x) (fabs x)) 0.125) (fabs x))
     (fabs x))
    (*
     t_1
     (*
      (- (/ 1.0 t_1) -1.0)
      (/ -1.0 (- -1.0 (sqrt (- (/ 0.5 t_0) -0.5)))))))))

C

double code(double x) {
	double t_0 = sqrt(fma(fabs(x), fabs(x), 1.0));
	double t_1 = (-0.5 / t_0) - 0.5;
	double tmp;
	if (fabs(x) <= 6.3e-6) {
		tmp = (fma(-0.0859375, (fabs(x) * fabs(x)), 0.125) * fabs(x)) * fabs(x);
	} else {
		tmp = t_1 * (((1.0 / t_1) - -1.0) * (-1.0 / (-1.0 - sqrt(((0.5 / t_0) - -0.5)))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(fma(abs(x), abs(x), 1.0))
	t_1 = Float64(Float64(-0.5 / t_0) - 0.5)
	tmp = 0.0
	if (abs(x) <= 6.3e-6)
		tmp = Float64(Float64(fma(-0.0859375, Float64(abs(x) * abs(x)), 0.125) * abs(x)) * abs(x));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(1.0 / t_1) - -1.0) * Float64(-1.0 / Float64(-1.0 - sqrt(Float64(Float64(0.5 / t_0) - -0.5))))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.5 / t$95$0), $MachinePrecision] - 0.5), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 6.3e-6], N[(N[(N[(-0.0859375 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 0.125), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(1.0 / t$95$1), $MachinePrecision] - -1.0), $MachinePrecision] * N[(-1.0 / N[(-1.0 - N[Sqrt[N[(N[(0.5 / t$95$0), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.2999999999999998e-6

   1. Initial program 75.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\color{blue}{\frac{1}{8}} + \frac{-11}{128} \cdot {x}^{2}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \color{blue}{\frac{-11}{128} \cdot {x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot \color{blue}{{x}^{2}}\right) \]

      5. lower-pow.f64 50.4%

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

   4. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot {x}^{\color{blue}{2}} \]

      4. pow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

      7. lower-*.f64 50.4%

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

      8. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      11. lower-fma.f64 50.4%

          \[\leadsto \left(\mathsf{fma}\left(-0.0859375, {x}^{2}, 0.125\right) \cdot x\right) \cdot x \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, x \cdot x, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      14. lower-*.f64 50.4%

          \[\leadsto \left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x \]

   6. Applied rewrites 50.4%

      \[\leadsto \left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot \color{blue}{x} \]

   if 6.2999999999999998e-6 < x

   1. Initial program 75.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 75.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. sub-flip N/A

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

      5. +-commutative N/A

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

      6. sum-to-mult N/A

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

      7. lower-unsound-*.f64 N/A

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 75.8%

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

Alternative 4: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ t_1 := \sqrt{t\_0 - -0.5}\\ \mathbf{if}\;\left|x\right| \leq 0.0025:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, \left|x\right| \cdot \left|x\right|, 0.125\right) \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{-1 - t\_1} - \frac{-0.5}{t\_1 - -1}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0))))
       (t_1 (sqrt (- t_0 -0.5))))
  (if (<= (fabs x) 0.0025)
    (*
     (* (fma -0.0859375 (* (fabs x) (fabs x)) 0.125) (fabs x))
     (fabs x))
    (- (/ t_0 (- -1.0 t_1)) (/ -0.5 (- t_1 -1.0))))))

C

double code(double x) {
	double t_0 = 0.5 / sqrt(fma(fabs(x), fabs(x), 1.0));
	double t_1 = sqrt((t_0 - -0.5));
	double tmp;
	if (fabs(x) <= 0.0025) {
		tmp = (fma(-0.0859375, (fabs(x) * fabs(x)), 0.125) * fabs(x)) * fabs(x);
	} else {
		tmp = (t_0 / (-1.0 - t_1)) - (-0.5 / (t_1 - -1.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0)))
	t_1 = sqrt(Float64(t_0 - -0.5))
	tmp = 0.0
	if (abs(x) <= 0.0025)
		tmp = Float64(Float64(fma(-0.0859375, Float64(abs(x) * abs(x)), 0.125) * abs(x)) * abs(x));
	else
		tmp = Float64(Float64(t_0 / Float64(-1.0 - t_1)) - Float64(-0.5 / Float64(t_1 - -1.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(t$95$0 - -0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0025], N[(N[(N[(-0.0859375 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 0.125), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision] - N[(-0.5 / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0025000000000000001

   1. Initial program 75.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\color{blue}{\frac{1}{8}} + \frac{-11}{128} \cdot {x}^{2}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \color{blue}{\frac{-11}{128} \cdot {x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot \color{blue}{{x}^{2}}\right) \]

      5. lower-pow.f64 50.4%

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

   4. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot {x}^{\color{blue}{2}} \]

      4. pow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

      7. lower-*.f64 50.4%

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

      8. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      11. lower-fma.f64 50.4%

          \[\leadsto \left(\mathsf{fma}\left(-0.0859375, {x}^{2}, 0.125\right) \cdot x\right) \cdot x \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, x \cdot x, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      14. lower-*.f64 50.4%

          \[\leadsto \left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x \]

   6. Applied rewrites 50.4%

      \[\leadsto \left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot \color{blue}{x} \]

   if 0.0025000000000000001 < x

   1. Initial program 75.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 75.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. sub-negate-rev N/A

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

      7. lift--.f64 N/A

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

      8. associate--l- N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

      11. metadata-eval N/A

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

      12. frac-2neg N/A

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

   5. Applied rewrites 75.9%

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

Alternative 5: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}\\ \mathbf{if}\;\left|x\right| \leq 0.00031:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, \left|x\right| \cdot \left|x\right|, 0.125\right) \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot 1 - \left(\frac{0.5}{t\_0} - -0.5\right)}{\mathsf{fma}\left(\sqrt{\frac{1}{t\_0} - -1}, \sqrt{0.5}, 1\right)}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (sqrt (fma (fabs x) (fabs x) 1.0))))
  (if (<= (fabs x) 0.00031)
    (*
     (* (fma -0.0859375 (* (fabs x) (fabs x)) 0.125) (fabs x))
     (fabs x))
    (/
     (- (* 1.0 1.0) (- (/ 0.5 t_0) -0.5))
     (fma (sqrt (- (/ 1.0 t_0) -1.0)) (sqrt 0.5) 1.0)))))

C

double code(double x) {
	double t_0 = sqrt(fma(fabs(x), fabs(x), 1.0));
	double tmp;
	if (fabs(x) <= 0.00031) {
		tmp = (fma(-0.0859375, (fabs(x) * fabs(x)), 0.125) * fabs(x)) * fabs(x);
	} else {
		tmp = ((1.0 * 1.0) - ((0.5 / t_0) - -0.5)) / fma(sqrt(((1.0 / t_0) - -1.0)), sqrt(0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(fma(abs(x), abs(x), 1.0))
	tmp = 0.0
	if (abs(x) <= 0.00031)
		tmp = Float64(Float64(fma(-0.0859375, Float64(abs(x) * abs(x)), 0.125) * abs(x)) * abs(x));
	else
		tmp = Float64(Float64(Float64(1.0 * 1.0) - Float64(Float64(0.5 / t_0) - -0.5)) / fma(sqrt(Float64(Float64(1.0 / t_0) - -1.0)), sqrt(0.5), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.00031], N[(N[(N[(-0.0859375 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 0.125), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 * 1.0), $MachinePrecision] - N[(N[(0.5 / t$95$0), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 / t$95$0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e-4

   1. Initial program 75.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\color{blue}{\frac{1}{8}} + \frac{-11}{128} \cdot {x}^{2}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \color{blue}{\frac{-11}{128} \cdot {x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot \color{blue}{{x}^{2}}\right) \]

      5. lower-pow.f64 50.4%

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

   4. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot {x}^{\color{blue}{2}} \]

      4. pow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

      7. lower-*.f64 50.4%

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

      8. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      11. lower-fma.f64 50.4%

          \[\leadsto \left(\mathsf{fma}\left(-0.0859375, {x}^{2}, 0.125\right) \cdot x\right) \cdot x \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, x \cdot x, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      14. lower-*.f64 50.4%

          \[\leadsto \left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x \]

   6. Applied rewrites 50.4%

      \[\leadsto \left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot \color{blue}{x} \]

   if 3.1e-4 < x

   1. Initial program 75.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 75.9%

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

      4. lower-unsound-+.f64 N/A

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

   5. Applied rewrites 75.9%

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

Alternative 6: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ \mathbf{if}\;\left|x\right| \leq 9.4 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, \left|x\right| \cdot \left|x\right|, 0.125\right) \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - 0.5}{-1 - \sqrt{t\_0 - -0.5}}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0)))))
  (if (<= (fabs x) 9.4e-5)
    (*
     (* (fma -0.0859375 (* (fabs x) (fabs x)) 0.125) (fabs x))
     (fabs x))
    (/ (- t_0 0.5) (- -1.0 (sqrt (- t_0 -0.5)))))))

C

double code(double x) {
	double t_0 = 0.5 / sqrt(fma(fabs(x), fabs(x), 1.0));
	double tmp;
	if (fabs(x) <= 9.4e-5) {
		tmp = (fma(-0.0859375, (fabs(x) * fabs(x)), 0.125) * fabs(x)) * fabs(x);
	} else {
		tmp = (t_0 - 0.5) / (-1.0 - sqrt((t_0 - -0.5)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0)))
	tmp = 0.0
	if (abs(x) <= 9.4e-5)
		tmp = Float64(Float64(fma(-0.0859375, Float64(abs(x) * abs(x)), 0.125) * abs(x)) * abs(x));
	else
		tmp = Float64(Float64(t_0 - 0.5) / Float64(-1.0 - sqrt(Float64(t_0 - -0.5))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 9.4e-5], N[(N[(N[(-0.0859375 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 0.125), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - 0.5), $MachinePrecision] / N[(-1.0 - N[Sqrt[N[(t$95$0 - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.3999999999999994e-5

   1. Initial program 75.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\color{blue}{\frac{1}{8}} + \frac{-11}{128} \cdot {x}^{2}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \color{blue}{\frac{-11}{128} \cdot {x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot \color{blue}{{x}^{2}}\right) \]

      5. lower-pow.f64 50.4%

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

   4. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot {x}^{\color{blue}{2}} \]

      4. pow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

      7. lower-*.f64 50.4%

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

      8. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      11. lower-fma.f64 50.4%

          \[\leadsto \left(\mathsf{fma}\left(-0.0859375, {x}^{2}, 0.125\right) \cdot x\right) \cdot x \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, x \cdot x, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      14. lower-*.f64 50.4%

          \[\leadsto \left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x \]

   6. Applied rewrites 50.4%

      \[\leadsto \left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot \color{blue}{x} \]

   if 9.3999999999999994e-5 < x

   1. Initial program 75.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 75.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. sub-flip N/A

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

      5. +-commutative N/A

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

      6. sum-to-mult N/A

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

      7. lower-unsound-*.f64 N/A

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 75.9%

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

Alternative 7: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\left|x\right|}\\ \mathbf{if}\;\left|x\right| \leq 1.1:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, \left|x\right| \cdot \left|x\right|, 0.125\right) \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - 0.5}{-\left(\sqrt{t\_0 - -0.5} - -1\right)}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (/ 0.5 (fabs x))))
  (if (<= (fabs x) 1.1)
    (*
     (* (fma -0.0859375 (* (fabs x) (fabs x)) 0.125) (fabs x))
     (fabs x))
    (/ (- t_0 0.5) (- (- (sqrt (- t_0 -0.5)) -1.0))))))

C

double code(double x) {
	double t_0 = 0.5 / fabs(x);
	double tmp;
	if (fabs(x) <= 1.1) {
		tmp = (fma(-0.0859375, (fabs(x) * fabs(x)), 0.125) * fabs(x)) * fabs(x);
	} else {
		tmp = (t_0 - 0.5) / -(sqrt((t_0 - -0.5)) - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / abs(x))
	tmp = 0.0
	if (abs(x) <= 1.1)
		tmp = Float64(Float64(fma(-0.0859375, Float64(abs(x) * abs(x)), 0.125) * abs(x)) * abs(x));
	else
		tmp = Float64(Float64(t_0 - 0.5) / Float64(-Float64(sqrt(Float64(t_0 - -0.5)) - -1.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1.1], N[(N[(N[(-0.0859375 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 0.125), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - 0.5), $MachinePrecision] / (-N[(N[Sqrt[N[(t$95$0 - -0.5), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 75.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\color{blue}{\frac{1}{8}} + \frac{-11}{128} \cdot {x}^{2}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \color{blue}{\frac{-11}{128} \cdot {x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot \color{blue}{{x}^{2}}\right) \]

      5. lower-pow.f64 50.4%

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

   4. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot {x}^{\color{blue}{2}} \]

      4. pow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

      7. lower-*.f64 50.4%

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

      8. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      11. lower-fma.f64 50.4%

          \[\leadsto \left(\mathsf{fma}\left(-0.0859375, {x}^{2}, 0.125\right) \cdot x\right) \cdot x \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, x \cdot x, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      14. lower-*.f64 50.4%

          \[\leadsto \left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x \]

   6. Applied rewrites 50.4%

      \[\leadsto \left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot \color{blue}{x} \]

   if 1.1000000000000001 < x

   1. Initial program 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-hypot.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval 75.1%

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

   3. Applied rewrites 75.1%

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

   4. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{\frac{1}{2}}{x}} - -0.5} \]
   5. Step-by-step derivation

      1. lower-/.f64 48.8%

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

   6. Applied rewrites 48.8%

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

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}} \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right) + 1} \]

      4. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right) - 1 \cdot 1}{\left(\mathsf{neg}\left(\sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right) - 1}} \]

      5. lower-unsound--.f32 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right) - 1 \cdot 1}{\color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right) - 1}} \]

      6. lower--.f32 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right) - 1 \cdot 1}{\color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right) - 1}} \]

      7. sub-negate N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{neg}\left(\left(1 - \left(\mathsf{neg}\left(\sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right)\right)\right)}} \]

      8. add-flip N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 49.6%

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

Alternative 8: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 6.3 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, \left|x\right| \cdot \left|x\right|, 0.125\right) \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}} - -0.5}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 6.3e-6)
  (*
   (* (fma -0.0859375 (* (fabs x) (fabs x)) 0.125) (fabs x))
   (fabs x))
  (- 1.0 (sqrt (- (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0))) -0.5)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 6.3e-6) {
		tmp = (fma(-0.0859375, (fabs(x) * fabs(x)), 0.125) * fabs(x)) * fabs(x);
	} else {
		tmp = 1.0 - sqrt(((0.5 / sqrt(fma(fabs(x), fabs(x), 1.0))) - -0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 6.3e-6)
		tmp = Float64(Float64(fma(-0.0859375, Float64(abs(x) * abs(x)), 0.125) * abs(x)) * abs(x));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0))) - -0.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 6.3e-6], N[(N[(N[(-0.0859375 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 0.125), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.2999999999999998e-6

   1. Initial program 75.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\color{blue}{\frac{1}{8}} + \frac{-11}{128} \cdot {x}^{2}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \color{blue}{\frac{-11}{128} \cdot {x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot \color{blue}{{x}^{2}}\right) \]

      5. lower-pow.f64 50.4%

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

   4. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot {x}^{\color{blue}{2}} \]

      4. pow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

      7. lower-*.f64 50.4%

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

      8. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      11. lower-fma.f64 50.4%

          \[\leadsto \left(\mathsf{fma}\left(-0.0859375, {x}^{2}, 0.125\right) \cdot x\right) \cdot x \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, x \cdot x, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      14. lower-*.f64 50.4%

          \[\leadsto \left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x \]

   6. Applied rewrites 50.4%

      \[\leadsto \left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot \color{blue}{x} \]

   if 6.2999999999999998e-6 < x

   1. Initial program 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-hypot.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval 75.1%

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

   3. Applied rewrites 75.1%

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

Alternative 9: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.016:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, \left|x\right| \cdot \left|x\right|, 0.125\right) \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;0.2928932188134525\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 0.016)
  (*
   (* (fma -0.0859375 (* (fabs x) (fabs x)) 0.125) (fabs x))
   (fabs x))
  0.2928932188134525))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.016) {
		tmp = (fma(-0.0859375, (fabs(x) * fabs(x)), 0.125) * fabs(x)) * fabs(x);
	} else {
		tmp = 0.2928932188134525;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.016)
		tmp = Float64(Float64(fma(-0.0859375, Float64(abs(x) * abs(x)), 0.125) * abs(x)) * abs(x));
	else
		tmp = 0.2928932188134525;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.016], N[(N[(N[(-0.0859375 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 0.125), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], 0.2928932188134525]

Derivation

1. Split input into 2 regimes

2. if x < 0.016

   1. Initial program 75.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\color{blue}{\frac{1}{8}} + \frac{-11}{128} \cdot {x}^{2}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \color{blue}{\frac{-11}{128} \cdot {x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot \color{blue}{{x}^{2}}\right) \]

      5. lower-pow.f64 50.4%

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

   4. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot {x}^{\color{blue}{2}} \]

      4. pow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

      7. lower-*.f64 50.4%

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

      8. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      11. lower-fma.f64 50.4%

          \[\leadsto \left(\mathsf{fma}\left(-0.0859375, {x}^{2}, 0.125\right) \cdot x\right) \cdot x \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, x \cdot x, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      14. lower-*.f64 50.4%

          \[\leadsto \left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x \]

   6. Applied rewrites 50.4%

      \[\leadsto \left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot \color{blue}{x} \]

   if 0.016 < x

   1. Initial program 75.1%

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

   2. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.1%

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

   5. Evaluated real constant 50.8%

      \[\leadsto \color{blue}{0.2928932188134525} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 8.5 \cdot 10^{-78}:\\ \;\;\;\;1 - \sqrt{0.5 - -0.5}\\ \mathbf{else}:\\ \;\;\;\;0.2928932188134525\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 8.5e-78)
  (- 1.0 (sqrt (- 0.5 -0.5)))
  0.2928932188134525))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 8.5e-78) {
		tmp = 1.0 - sqrt((0.5 - -0.5));
	} else {
		tmp = 0.2928932188134525;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) <= 8.5d-78) then
        tmp = 1.0d0 - sqrt((0.5d0 - (-0.5d0)))
    else
        tmp = 0.2928932188134525d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 8.5e-78) {
		tmp = 1.0 - Math.sqrt((0.5 - -0.5));
	} else {
		tmp = 0.2928932188134525;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 8.5e-78:
		tmp = 1.0 - math.sqrt((0.5 - -0.5))
	else:
		tmp = 0.2928932188134525
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 8.5e-78)
		tmp = Float64(1.0 - sqrt(Float64(0.5 - -0.5)));
	else
		tmp = 0.2928932188134525;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 8.5e-78)
		tmp = 1.0 - sqrt((0.5 - -0.5));
	else
		tmp = 0.2928932188134525;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 8.5e-78], N[(1.0 - N[Sqrt[N[(0.5 - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.2928932188134525]

Derivation

1. Split input into 2 regimes

2. if x < 8.4999999999999996e-78

   1. Initial program 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-hypot.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval 75.1%

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

   3. Applied rewrites 75.1%

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

   4. Taylor expanded in x around 0

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}} - -0.5} \]
   5. Step-by-step derivation

   6. Applied rewrites 27.1%

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

   if 8.4999999999999996e-78 < x

   1. Initial program 75.1%

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

   2. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.1%

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

   5. Evaluated real constant 50.8%

      \[\leadsto \color{blue}{0.2928932188134525} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 50.8% accurate, 29.6× speedup

Math

\[0.2928932188134525 \]

FPCore

(FPCore (x)
  :precision binary64
  0.2928932188134525)

C

double code(double x) {
	return 0.2928932188134525;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 0.2928932188134525d0
end function

Java

public static double code(double x) {
	return 0.2928932188134525;
}

Python

def code(x):
	return 0.2928932188134525

Julia

function code(x)
	return 0.2928932188134525
end

MATLAB

function tmp = code(x)
	tmp = 0.2928932188134525;
end

Wolfram

code[x_] := 0.2928932188134525

Derivation

1. Initial program 75.1%

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

2. Taylor expanded in x around inf

   \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
3. Step-by-step derivation

4. Applied rewrites 50.1%

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

5. Evaluated real constant 50.8%

   \[\leadsto \color{blue}{0.2928932188134525} \]
6. Add Preprocessing

Alternative 12: 50.1% accurate, 29.6× speedup

Math

\[0.2928932188134524 \]

FPCore

(FPCore (x)
  :precision binary64
  0.2928932188134524)

C

double code(double x) {
	return 0.2928932188134524;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 0.2928932188134524d0
end function

Java

public static double code(double x) {
	return 0.2928932188134524;
}

Python

def code(x):
	return 0.2928932188134524

Julia

function code(x)
	return 0.2928932188134524
end

MATLAB

function tmp = code(x)
	tmp = 0.2928932188134524;
end

Wolfram

code[x_] := 0.2928932188134524

Derivation

1. Initial program 75.1%

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

2. Taylor expanded in x around inf

   \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
3. Step-by-step derivation

4. Applied rewrites 50.1%

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

5. Evaluated real constant 50.1%

   \[\leadsto 1 - \color{blue}{0.7071067811865476} \]

6. Evaluated real constant 50.1%

   \[\leadsto \color{blue}{0.2928932188134524} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))