Given's Rotation SVD example, simplified

Percentage Accurate: 76.2% → 99.9%

Time: 6.3s

Alternatives: 15

Speedup: 1.9×

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: 76.2% 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: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ t_1 := 1 - t\_0\\ t_2 := {\left(\left|x\right|\right)}^{2}\\ t_3 := t\_1 \cdot \left(t\_0 - 1\right)\\ \mathbf{if}\;\left|x\right| \leq 0.0152:\\ \;\;\;\;t\_2 \cdot \left(0.125 + t\_2 \cdot \left(t\_2 \cdot \left(0.0673828125 + -0.056243896484375 \cdot t\_2\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_3 - t\_1 \cdot -0.5}{t\_3} \cdot t\_1}{1 + \sqrt{\frac{t\_0 \cdot t\_0 - 0.5 \cdot 0.5}{t\_0 - 0.5}}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0))))
        (t_1 (- 1.0 t_0))
        (t_2 (pow (fabs x) 2.0))
        (t_3 (* t_1 (- t_0 1.0))))
   (if (<= (fabs x) 0.0152)
     (*
      t_2
      (+
       0.125
       (*
        t_2
        (- (* t_2 (+ 0.0673828125 (* -0.056243896484375 t_2))) 0.0859375))))
     (/
      (* (/ (- t_3 (* t_1 -0.5)) t_3) t_1)
      (+ 1.0 (sqrt (/ (- (* t_0 t_0) (* 0.5 0.5)) (- t_0 0.5))))))))

C

double code(double x) {
	double t_0 = 0.5 / sqrt(fma(fabs(x), fabs(x), 1.0));
	double t_1 = 1.0 - t_0;
	double t_2 = pow(fabs(x), 2.0);
	double t_3 = t_1 * (t_0 - 1.0);
	double tmp;
	if (fabs(x) <= 0.0152) {
		tmp = t_2 * (0.125 + (t_2 * ((t_2 * (0.0673828125 + (-0.056243896484375 * t_2))) - 0.0859375)));
	} else {
		tmp = (((t_3 - (t_1 * -0.5)) / t_3) * t_1) / (1.0 + sqrt((((t_0 * t_0) - (0.5 * 0.5)) / (t_0 - 0.5))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0)))
	t_1 = Float64(1.0 - t_0)
	t_2 = abs(x) ^ 2.0
	t_3 = Float64(t_1 * Float64(t_0 - 1.0))
	tmp = 0.0
	if (abs(x) <= 0.0152)
		tmp = Float64(t_2 * Float64(0.125 + Float64(t_2 * Float64(Float64(t_2 * Float64(0.0673828125 + Float64(-0.056243896484375 * t_2))) - 0.0859375))));
	else
		tmp = Float64(Float64(Float64(Float64(t_3 - Float64(t_1 * -0.5)) / t_3) * t_1) / Float64(1.0 + sqrt(Float64(Float64(Float64(t_0 * t_0) - Float64(0.5 * 0.5)) / 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]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0152], N[(t$95$2 * N[(0.125 + N[(t$95$2 * N[(N[(t$95$2 * N[(0.0673828125 + N[(-0.056243896484375 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$3 - N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(0.5 * 0.5), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0152

   1. Initial program 76.2%

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

   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 49.3%

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

   4. Applied rewrites 49.3%

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

   if 0.0152 < x

   1. Initial program 76.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 77.0%

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}}} \]
   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. lift--.f64 N/A

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

      5. associate--r- N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. sub-to-mult N/A

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

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

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

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

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

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

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

      12. lower--.f64 N/A

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

      13. lower--.f64 77.0%

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

   5. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-to-fraction N/A

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

      4. div-sub N/A

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

      5. *-lft-identity N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-sub N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. metadata-eval N/A

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

      4. flip-+ N/A

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

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

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

      6. lower--.f32 N/A

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

      7. metadata-eval N/A

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

      8. associate--l- N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lift--.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-fma.f64 N/A

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

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

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

   9. Applied rewrites 51.6%

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

Alternative 2: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}\\ t_1 := \frac{0.5}{t\_0}\\ t_2 := 1 - t\_1\\ t_3 := {\left(\left|x\right|\right)}^{2}\\ t_4 := t\_2 \cdot \left(t\_1 - 1\right)\\ \mathbf{if}\;\left|x\right| \leq 0.0152:\\ \;\;\;\;t\_3 \cdot \left(0.125 + t\_3 \cdot \left(t\_3 \cdot \left(0.0673828125 + -0.056243896484375 \cdot t\_3\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_4 - t\_2 \cdot -0.5}{t\_4} \cdot t\_2}{\mathsf{fma}\left(\sqrt{\frac{1}{t\_0} - -1}, \sqrt{0.5}, 1\right)}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (fma (fabs x) (fabs x) 1.0)))
        (t_1 (/ 0.5 t_0))
        (t_2 (- 1.0 t_1))
        (t_3 (pow (fabs x) 2.0))
        (t_4 (* t_2 (- t_1 1.0))))
   (if (<= (fabs x) 0.0152)
     (*
      t_3
      (+
       0.125
       (*
        t_3
        (- (* t_3 (+ 0.0673828125 (* -0.056243896484375 t_3))) 0.0859375))))
     (/
      (* (/ (- t_4 (* t_2 -0.5)) t_4) 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 = 0.5 / t_0;
	double t_2 = 1.0 - t_1;
	double t_3 = pow(fabs(x), 2.0);
	double t_4 = t_2 * (t_1 - 1.0);
	double tmp;
	if (fabs(x) <= 0.0152) {
		tmp = t_3 * (0.125 + (t_3 * ((t_3 * (0.0673828125 + (-0.056243896484375 * t_3))) - 0.0859375)));
	} else {
		tmp = (((t_4 - (t_2 * -0.5)) / t_4) * 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 = Float64(0.5 / t_0)
	t_2 = Float64(1.0 - t_1)
	t_3 = abs(x) ^ 2.0
	t_4 = Float64(t_2 * Float64(t_1 - 1.0))
	tmp = 0.0
	if (abs(x) <= 0.0152)
		tmp = Float64(t_3 * Float64(0.125 + Float64(t_3 * Float64(Float64(t_3 * Float64(0.0673828125 + Float64(-0.056243896484375 * t_3))) - 0.0859375))));
	else
		tmp = Float64(Float64(Float64(Float64(t_4 - Float64(t_2 * -0.5)) / t_4) * 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[(0.5 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0152], N[(t$95$3 * N[(0.125 + N[(t$95$3 * N[(N[(t$95$3 * N[(0.0673828125 + N[(-0.056243896484375 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$4 - N[(t$95$2 * -0.5), $MachinePrecision]), $MachinePrecision] / t$95$4), $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 < 0.0152

   1. Initial program 76.2%

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

   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 49.3%

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

   4. Applied rewrites 49.3%

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

   if 0.0152 < x

   1. Initial program 76.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 77.0%

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}}} \]
   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. lift--.f64 N/A

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

      5. associate--r- N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. sub-to-mult N/A

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

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

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

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

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

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

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

      12. lower--.f64 N/A

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

      13. lower--.f64 77.0%

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

   5. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-to-fraction N/A

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

      4. div-sub N/A

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

      5. *-lft-identity N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-sub N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 77.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 77.0%

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

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

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

   9. Applied rewrites 77.0%

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

Alternative 3: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ t_1 := 1 - t\_0\\ t_2 := {\left(\left|x\right|\right)}^{2}\\ t_3 := t\_0 - 1\\ \mathbf{if}\;\left|x\right| \leq 0.0152:\\ \;\;\;\;t\_2 \cdot \left(0.125 + t\_2 \cdot \left(t\_2 \cdot \left(0.0673828125 + -0.056243896484375 \cdot t\_2\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_3, -0.5, t\_3 \cdot t\_1\right)}{t\_1 \cdot t\_3} \cdot t\_1}{1 + \sqrt{t\_0 - -0.5}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0))))
        (t_1 (- 1.0 t_0))
        (t_2 (pow (fabs x) 2.0))
        (t_3 (- t_0 1.0)))
   (if (<= (fabs x) 0.0152)
     (*
      t_2
      (+
       0.125
       (*
        t_2
        (- (* t_2 (+ 0.0673828125 (* -0.056243896484375 t_2))) 0.0859375))))
     (/
      (* (/ (fma t_3 -0.5 (* t_3 t_1)) (* t_1 t_3)) t_1)
      (+ 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 t_1 = 1.0 - t_0;
	double t_2 = pow(fabs(x), 2.0);
	double t_3 = t_0 - 1.0;
	double tmp;
	if (fabs(x) <= 0.0152) {
		tmp = t_2 * (0.125 + (t_2 * ((t_2 * (0.0673828125 + (-0.056243896484375 * t_2))) - 0.0859375)));
	} else {
		tmp = ((fma(t_3, -0.5, (t_3 * t_1)) / (t_1 * t_3)) * t_1) / (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)))
	t_1 = Float64(1.0 - t_0)
	t_2 = abs(x) ^ 2.0
	t_3 = Float64(t_0 - 1.0)
	tmp = 0.0
	if (abs(x) <= 0.0152)
		tmp = Float64(t_2 * Float64(0.125 + Float64(t_2 * Float64(Float64(t_2 * Float64(0.0673828125 + Float64(-0.056243896484375 * t_2))) - 0.0859375))));
	else
		tmp = Float64(Float64(Float64(fma(t_3, -0.5, Float64(t_3 * t_1)) / Float64(t_1 * t_3)) * t_1) / 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]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 - 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0152], N[(t$95$2 * N[(0.125 + N[(t$95$2 * N[(N[(t$95$2 * N[(0.0673828125 + N[(-0.056243896484375 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$3 * -0.5 + N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$1), $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 < 0.0152

   1. Initial program 76.2%

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

   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 49.3%

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

   4. Applied rewrites 49.3%

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

   if 0.0152 < x

   1. Initial program 76.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 77.0%

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}}} \]
   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. lift--.f64 N/A

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

      5. associate--r- N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. sub-to-mult N/A

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

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

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

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

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

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

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

      12. lower--.f64 N/A

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

      13. lower--.f64 77.0%

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

   5. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-to-fraction N/A

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

      4. div-sub N/A

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

      5. *-lft-identity N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-sub N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lower-fma.f64 77.0%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   9. Applied rewrites 77.0%

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

Alternative 4: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|x\right|\right)}^{2}\\ t_1 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ t_2 := \sqrt{t\_1 - -0.5} - -1\\ \mathbf{if}\;\left|x\right| \leq 0.0152:\\ \;\;\;\;t\_0 \cdot \left(0.125 + t\_0 \cdot \left(t\_0 \cdot \left(0.0673828125 + -0.056243896484375 \cdot t\_0\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1 - t\_1}{t\_2}, t\_2, -0.5\right)}{t\_2}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (fabs x) 2.0))
        (t_1 (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0))))
        (t_2 (- (sqrt (- t_1 -0.5)) -1.0)))
   (if (<= (fabs x) 0.0152)
     (*
      t_0
      (+
       0.125
       (*
        t_0
        (- (* t_0 (+ 0.0673828125 (* -0.056243896484375 t_0))) 0.0859375))))
     (/ (fma (/ (- 1.0 t_1) t_2) t_2 -0.5) t_2))))

C

double code(double x) {
	double t_0 = pow(fabs(x), 2.0);
	double t_1 = 0.5 / sqrt(fma(fabs(x), fabs(x), 1.0));
	double t_2 = sqrt((t_1 - -0.5)) - -1.0;
	double tmp;
	if (fabs(x) <= 0.0152) {
		tmp = t_0 * (0.125 + (t_0 * ((t_0 * (0.0673828125 + (-0.056243896484375 * t_0))) - 0.0859375)));
	} else {
		tmp = fma(((1.0 - t_1) / t_2), t_2, -0.5) / t_2;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = abs(x) ^ 2.0
	t_1 = Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0)))
	t_2 = Float64(sqrt(Float64(t_1 - -0.5)) - -1.0)
	tmp = 0.0
	if (abs(x) <= 0.0152)
		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(fma(Float64(Float64(1.0 - t_1) / t_2), t_2, -0.5) / t_2);
	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[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t$95$1 - -0.5), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0152], 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[(N[(1.0 - t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision] * t$95$2 + -0.5), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0152

   1. Initial program 76.2%

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

   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 49.3%

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

   4. Applied rewrites 49.3%

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

   if 0.0152 < x

   1. Initial program 76.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 77.0%

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}}} \]
   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. 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. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift--.f64 N/A

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

      6. associate--r- N/A

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

      7. div-add N/A

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

      8. add-to-fraction N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 77.0%

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

Alternative 5: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ t_1 := \sqrt{t\_0 - -0.5} - -1\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\left(0.125 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1 - t\_0}{t\_1}, t\_1, -0.5\right)}{t\_1}\\ \end{array} \]

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)) -1.0)))
   (if (<= (fabs x) 5e-12)
     (* (* 0.125 (fabs x)) (fabs x))
     (/ (fma (/ (- 1.0 t_0) t_1) t_1 -0.5) t_1))))

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)) - -1.0;
	double tmp;
	if (fabs(x) <= 5e-12) {
		tmp = (0.125 * fabs(x)) * fabs(x);
	} else {
		tmp = fma(((1.0 - t_0) / t_1), t_1, -0.5) / t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0)))
	t_1 = Float64(sqrt(Float64(t_0 - -0.5)) - -1.0)
	tmp = 0.0
	if (abs(x) <= 5e-12)
		tmp = Float64(Float64(0.125 * abs(x)) * abs(x));
	else
		tmp = Float64(fma(Float64(Float64(1.0 - t_0) / t_1), t_1, -0.5) / t_1);
	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[(N[Sqrt[N[(t$95$0 - -0.5), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-12], N[(N[(0.125 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 - t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$1 + -0.5), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999997e-12

   1. Initial program 76.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto {x}^{2} \cdot \frac{1}{8} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.8%

      \[\leadsto {x}^{2} \cdot 0.125 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{1}{8}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{8} \cdot \color{blue}{{x}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{1}{8} \cdot {x}^{\color{blue}{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{1}{8} \cdot \left(x \cdot \color{blue}{x}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot \color{blue}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot \color{blue}{x} \]

      7. lower-*.f64 50.8%

         \[\leadsto \left(0.125 \cdot x\right) \cdot x \]

   9. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\left(0.125 \cdot x\right) \cdot x} \]

   if 4.9999999999999997e-12 < x

   1. Initial program 76.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 77.0%

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}}} \]
   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. 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. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift--.f64 N/A

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

      6. associate--r- N/A

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

      7. div-add N/A

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

      8. add-to-fraction N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 77.0%

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

Alternative 6: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ t_1 := 1 - t\_0\\ t_2 := \left|x\right| \cdot \left|x\right|\\ \mathbf{if}\;\left|x\right| \leq 0.0215:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, t\_2, -0.0859375\right), t\_2, 0.125\right) \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{0.5}{t\_1}\right) \cdot t\_1}{\sqrt{t\_0 - -0.5} - -1}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0))))
        (t_1 (- 1.0 t_0))
        (t_2 (* (fabs x) (fabs x))))
   (if (<= (fabs x) 0.0215)
     (*
      (* (fma (fma 0.0673828125 t_2 -0.0859375) t_2 0.125) (fabs x))
      (fabs x))
     (/ (* (- 1.0 (/ 0.5 t_1)) t_1) (- (sqrt (- t_0 -0.5)) -1.0)))))

C

double code(double x) {
	double t_0 = 0.5 / sqrt(fma(fabs(x), fabs(x), 1.0));
	double t_1 = 1.0 - t_0;
	double t_2 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 0.0215) {
		tmp = (fma(fma(0.0673828125, t_2, -0.0859375), t_2, 0.125) * fabs(x)) * fabs(x);
	} else {
		tmp = ((1.0 - (0.5 / t_1)) * t_1) / (sqrt((t_0 - -0.5)) - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0)))
	t_1 = Float64(1.0 - t_0)
	t_2 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 0.0215)
		tmp = Float64(Float64(fma(fma(0.0673828125, t_2, -0.0859375), t_2, 0.125) * abs(x)) * abs(x));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / t_1)) * t_1) / Float64(sqrt(Float64(t_0 - -0.5)) - -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[(1.0 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0215], N[(N[(N[(N[(0.0673828125 * t$95$2 + -0.0859375), $MachinePrecision] * t$95$2 + 0.125), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(0.5 / t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$1), $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 < 0.021499999999999998

   1. Initial program 76.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 50.7%

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

   4. Applied rewrites 50.7%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 50.7%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot \color{blue}{x} \]

   if 0.021499999999999998 < x

   1. Initial program 76.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 77.0%

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}}} \]
   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. 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. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift--.f64 N/A

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

      6. associate--r- N/A

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

      7. div-add N/A

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

      8. add-to-fraction N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 77.0%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

   7. Applied rewrites 77.0%

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

Alternative 7: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}\\ t_1 := \left|x\right| \cdot \left|x\right|\\ \mathbf{if}\;\left|x\right| \leq 0.0215:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, t\_1, -0.0859375\right), t\_1, 0.125\right) \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{t\_0} - 0.5}{-1 - \sqrt{\frac{1}{t\_0} - -1} \cdot \sqrt{0.5}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (fma (fabs x) (fabs x) 1.0))) (t_1 (* (fabs x) (fabs x))))
   (if (<= (fabs x) 0.0215)
     (*
      (* (fma (fma 0.0673828125 t_1 -0.0859375) t_1 0.125) (fabs x))
      (fabs x))
     (/
      (- (/ 0.5 t_0) 0.5)
      (- -1.0 (* (sqrt (- (/ 1.0 t_0) -1.0)) (sqrt 0.5)))))))

C

double code(double x) {
	double t_0 = sqrt(fma(fabs(x), fabs(x), 1.0));
	double t_1 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 0.0215) {
		tmp = (fma(fma(0.0673828125, t_1, -0.0859375), t_1, 0.125) * fabs(x)) * fabs(x);
	} else {
		tmp = ((0.5 / t_0) - 0.5) / (-1.0 - (sqrt(((1.0 / t_0) - -1.0)) * sqrt(0.5)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(fma(abs(x), abs(x), 1.0))
	t_1 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 0.0215)
		tmp = Float64(Float64(fma(fma(0.0673828125, t_1, -0.0859375), t_1, 0.125) * abs(x)) * abs(x));
	else
		tmp = Float64(Float64(Float64(0.5 / t_0) - 0.5) / Float64(-1.0 - Float64(sqrt(Float64(Float64(1.0 / t_0) - -1.0)) * sqrt(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[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0215], N[(N[(N[(N[(0.0673828125 * t$95$1 + -0.0859375), $MachinePrecision] * t$95$1 + 0.125), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / t$95$0), $MachinePrecision] - 0.5), $MachinePrecision] / N[(-1.0 - N[(N[Sqrt[N[(N[(1.0 / t$95$0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.021499999999999998

   1. Initial program 76.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 50.7%

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

   4. Applied rewrites 50.7%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 50.7%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot \color{blue}{x} \]

   if 0.021499999999999998 < x

   1. Initial program 76.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 77.0%

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}}} \]
   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. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. associate--l- N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. sub-flip-reverse N/A

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

   5. Applied rewrites 77.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. distribute-lft-out N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. sqrt-prod N/A

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

   7. Applied rewrites 77.0%

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

Alternative 8: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ t_1 := \left|x\right| \cdot \left|x\right|\\ \mathbf{if}\;\left|x\right| \leq 0.0215:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, t\_1, -0.0859375\right), t\_1, 0.125\right) \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - 0.5}{-1 - \sqrt{t\_0 - -0.5}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0))))
        (t_1 (* (fabs x) (fabs x))))
   (if (<= (fabs x) 0.0215)
     (*
      (* (fma (fma 0.0673828125 t_1 -0.0859375) t_1 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 t_1 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 0.0215) {
		tmp = (fma(fma(0.0673828125, t_1, -0.0859375), t_1, 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)))
	t_1 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 0.0215)
		tmp = Float64(Float64(fma(fma(0.0673828125, t_1, -0.0859375), t_1, 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]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0215], N[(N[(N[(N[(0.0673828125 * t$95$1 + -0.0859375), $MachinePrecision] * t$95$1 + 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 < 0.021499999999999998

   1. Initial program 76.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 50.7%

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

   4. Applied rewrites 50.7%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 50.7%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot \color{blue}{x} \]

   if 0.021499999999999998 < x

   1. Initial program 76.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 77.0%

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}}} \]
   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. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. associate--l- N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. sub-flip-reverse N/A

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

   5. Applied rewrites 77.0%

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

Alternative 9: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\left|x\right|} - -0.5\\ t_1 := \left|x\right| \cdot \left|x\right|\\ \mathbf{if}\;\left|x\right| \leq 1.6:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, t\_1, -0.0859375\right), t\_1, 0.125\right) \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot 1 - t\_0}{1 + \sqrt{t\_0}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ 0.5 (fabs x)) -0.5)) (t_1 (* (fabs x) (fabs x))))
   (if (<= (fabs x) 1.6)
     (*
      (* (fma (fma 0.0673828125 t_1 -0.0859375) t_1 0.125) (fabs x))
      (fabs x))
     (/ (- (* 1.0 1.0) t_0) (+ 1.0 (sqrt t_0))))))

C

double code(double x) {
	double t_0 = (0.5 / fabs(x)) - -0.5;
	double t_1 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 1.6) {
		tmp = (fma(fma(0.0673828125, t_1, -0.0859375), t_1, 0.125) * fabs(x)) * fabs(x);
	} else {
		tmp = ((1.0 * 1.0) - t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(0.5 / abs(x)) - -0.5)
	t_1 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 1.6)
		tmp = Float64(Float64(fma(fma(0.0673828125, t_1, -0.0859375), t_1, 0.125) * abs(x)) * abs(x));
	else
		tmp = Float64(Float64(Float64(1.0 * 1.0) - t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1.6], N[(N[(N[(N[(0.0673828125 * t$95$1 + -0.0859375), $MachinePrecision] * t$95$1 + 0.125), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 * 1.0), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 76.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 50.7%

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

   4. Applied rewrites 50.7%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 50.7%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot \color{blue}{x} \]

   if 1.6000000000000001 < x

   1. Initial program 76.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 77.0%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 51.4%

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

   6. Applied rewrites 51.4%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{1 \cdot 1 - \left(\frac{0.5}{x} - -0.5\right)}{1 + \sqrt{\color{blue}{\frac{\frac{1}{2}}{x}} - -0.5}} \]
   8. Step-by-step derivation

      1. lower-/.f64 50.7%

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

   9. Applied rewrites 50.7%

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

Alternative 10: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\left|x\right|}\\ t_1 := \left|x\right| \cdot \left|x\right|\\ \mathbf{if}\;\left|x\right| \leq 1.6:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, t\_1, -0.0859375\right), t\_1, 0.125\right) \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - 0.5}{-1 - \sqrt{t\_0 - -0.5}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (fabs x))) (t_1 (* (fabs x) (fabs x))))
   (if (<= (fabs x) 1.6)
     (*
      (* (fma (fma 0.0673828125 t_1 -0.0859375) t_1 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 / fabs(x);
	double t_1 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 1.6) {
		tmp = (fma(fma(0.0673828125, t_1, -0.0859375), t_1, 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 / abs(x))
	t_1 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 1.6)
		tmp = Float64(Float64(fma(fma(0.0673828125, t_1, -0.0859375), t_1, 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[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1.6], N[(N[(N[(N[(0.0673828125 * t$95$1 + -0.0859375), $MachinePrecision] * t$95$1 + 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 < 1.6000000000000001

   1. Initial program 76.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 50.7%

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

   4. Applied rewrites 50.7%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 50.7%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot \color{blue}{x} \]

   if 1.6000000000000001 < x

   1. Initial program 76.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 77.0%

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}}} \]
   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. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. associate--l- N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. sub-flip-reverse N/A

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{x}} - 0.5}{-1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]
   7. Step-by-step derivation

      1. lower-/.f64 51.4%

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

   8. Applied rewrites 51.4%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 50.7%

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

   11. Applied rewrites 50.7%

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

Alternative 11: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\left|x\right|}\\ \mathbf{if}\;\left|x\right| \leq 2.2:\\ \;\;\;\;\left(0.125 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - 0.5}{-1 - \sqrt{t\_0 - -0.5}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (fabs x))))
   (if (<= (fabs x) 2.2)
     (* (* 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 / fabs(x);
	double tmp;
	if (fabs(x) <= 2.2) {
		tmp = (0.125 * fabs(x)) * fabs(x);
	} else {
		tmp = (t_0 - 0.5) / (-1.0 - sqrt((t_0 - -0.5)));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 / abs(x)
    if (abs(x) <= 2.2d0) then
        tmp = (0.125d0 * abs(x)) * abs(x)
    else
        tmp = (t_0 - 0.5d0) / ((-1.0d0) - sqrt((t_0 - (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 2.2) {
		tmp = (0.125 * Math.abs(x)) * Math.abs(x);
	} else {
		tmp = (t_0 - 0.5) / (-1.0 - Math.sqrt((t_0 - -0.5)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 2.2:
		tmp = (0.125 * math.fabs(x)) * math.fabs(x)
	else:
		tmp = (t_0 - 0.5) / (-1.0 - math.sqrt((t_0 - -0.5)))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / abs(x))
	tmp = 0.0
	if (abs(x) <= 2.2)
		tmp = Float64(Float64(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

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / abs(x);
	tmp = 0.0;
	if (abs(x) <= 2.2)
		tmp = (0.125 * abs(x)) * abs(x);
	else
		tmp = (t_0 - 0.5) / (-1.0 - sqrt((t_0 - -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2.2], N[(N[(0.125 * 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 < 2.2000000000000002

   1. Initial program 76.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto {x}^{2} \cdot \frac{1}{8} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.8%

      \[\leadsto {x}^{2} \cdot 0.125 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{1}{8}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{8} \cdot \color{blue}{{x}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{1}{8} \cdot {x}^{\color{blue}{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{1}{8} \cdot \left(x \cdot \color{blue}{x}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot \color{blue}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot \color{blue}{x} \]

      7. lower-*.f64 50.8%

         \[\leadsto \left(0.125 \cdot x\right) \cdot x \]

   9. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\left(0.125 \cdot x\right) \cdot x} \]

   if 2.2000000000000002 < x

   1. Initial program 76.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 77.0%

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}}} \]
   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. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. associate--l- N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. sub-flip-reverse N/A

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{x}} - 0.5}{-1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]
   7. Step-by-step derivation

      1. lower-/.f64 51.4%

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

   8. Applied rewrites 51.4%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 50.7%

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

   11. Applied rewrites 50.7%

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

Alternative 12: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0105:\\ \;\;\;\;\left(0.125 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}} - -0.5}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0105)
   (* (* 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) <= 0.0105) {
		tmp = (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) <= 0.0105)
		tmp = Float64(Float64(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], 0.0105], N[(N[(0.125 * 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 < 0.0105000000000000007

   1. Initial program 76.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto {x}^{2} \cdot \frac{1}{8} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.8%

      \[\leadsto {x}^{2} \cdot 0.125 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{1}{8}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{8} \cdot \color{blue}{{x}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{1}{8} \cdot {x}^{\color{blue}{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{1}{8} \cdot \left(x \cdot \color{blue}{x}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot \color{blue}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot \color{blue}{x} \]

      7. lower-*.f64 50.8%

         \[\leadsto \left(0.125 \cdot x\right) \cdot x \]

   9. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\left(0.125 \cdot x\right) \cdot x} \]

   if 0.0105000000000000007 < x

   1. Initial program 76.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 76.2%

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

   3. Applied rewrites 76.2%

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

Alternative 13: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.024:\\ \;\;\;\;\left(0.125 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;0.2928932188134525\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.024) (* (* 0.125 (fabs x)) (fabs x)) 0.2928932188134525))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.024) {
		tmp = (0.125 * fabs(x)) * fabs(x);
	} 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) <= 0.024d0) then
        tmp = (0.125d0 * abs(x)) * abs(x)
    else
        tmp = 0.2928932188134525d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.024) {
		tmp = (0.125 * Math.abs(x)) * Math.abs(x);
	} else {
		tmp = 0.2928932188134525;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.024:
		tmp = (0.125 * math.fabs(x)) * math.fabs(x)
	else:
		tmp = 0.2928932188134525
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.024)
		tmp = Float64(Float64(0.125 * abs(x)) * abs(x));
	else
		tmp = 0.2928932188134525;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.024)
		tmp = (0.125 * abs(x)) * abs(x);
	else
		tmp = 0.2928932188134525;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.024], N[(N[(0.125 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], 0.2928932188134525]

Derivation

1. Split input into 2 regimes

2. if x < 0.024

   1. Initial program 76.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto {x}^{2} \cdot \frac{1}{8} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.8%

      \[\leadsto {x}^{2} \cdot 0.125 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{1}{8}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{8} \cdot \color{blue}{{x}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{1}{8} \cdot {x}^{\color{blue}{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{1}{8} \cdot \left(x \cdot \color{blue}{x}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot \color{blue}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot \color{blue}{x} \]

      7. lower-*.f64 50.8%

         \[\leadsto \left(0.125 \cdot x\right) \cdot x \]

   9. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\left(0.125 \cdot x\right) \cdot x} \]

   if 0.024 < x

   1. Initial program 76.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 77.0%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 51.9%

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

   6. Applied rewrites 51.9%

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

   7. Evaluated real constant 51.9%

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

Alternative 14: 75.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 7.5 \cdot 10^{-80}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;0.2928932188134525\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 7.5e-80) (- 1.0 1.0) 0.2928932188134525))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 7.5e-80) {
		tmp = 1.0 - 1.0;
	} 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) <= 7.5d-80) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = 0.2928932188134525d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 7.5e-80) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 0.2928932188134525;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 7.5e-80:
		tmp = 1.0 - 1.0
	else:
		tmp = 0.2928932188134525
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 7.5e-80)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = 0.2928932188134525;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 7.5e-80)
		tmp = 1.0 - 1.0;
	else
		tmp = 0.2928932188134525;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 7.5e-80], N[(1.0 - 1.0), $MachinePrecision], 0.2928932188134525]

Derivation

1. Split input into 2 regimes

2. if x < 7.49999999999999999e-80

   1. Initial program 76.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 27.0%

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

   if 7.49999999999999999e-80 < x

   1. Initial program 76.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 77.0%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 51.9%

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

   6. Applied rewrites 51.9%

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

   7. Evaluated real constant 51.9%

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

Alternative 15: 51.9% accurate, 27.4× 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 76.2%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 77.0%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 51.9%

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

6. Applied rewrites 51.9%

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

7. Evaluated real constant 51.9%

   \[\leadsto 0.2928932188134525 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025182 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))