Given's Rotation SVD example, simplified

Percentage Accurate: 76.1% → 100.0%

Time: 5.0s

Alternatives: 12

Speedup: 6.7×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0285:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(0.125 + \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(0.0673828125 + -0.056243896484375 \cdot \left(x\_m \cdot x\_m\right)\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{1}{\sqrt{{x\_m}^{2} \cdot \left(1 + \frac{1}{{x\_m}^{2}}\right)}} + 1\right) \cdot 0.5}{1 + \sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0285)
   (*
    (* x_m x_m)
    (+
     0.125
     (*
      (* x_m x_m)
      (-
       (* (* x_m x_m) (+ 0.0673828125 (* -0.056243896484375 (* x_m x_m))))
       0.0859375))))
   (/
    (-
     1.0
     (*
      (+ (/ 1.0 (sqrt (* (pow x_m 2.0) (+ 1.0 (/ 1.0 (pow x_m 2.0)))))) 1.0)
      0.5))
    (+ 1.0 (sqrt (* (+ (/ 1.0 (sqrt (fma x_m x_m 1.0))) 1.0) 0.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0285) {
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * (((x_m * x_m) * (0.0673828125 + (-0.056243896484375 * (x_m * x_m)))) - 0.0859375)));
	} else {
		tmp = (1.0 - (((1.0 / sqrt((pow(x_m, 2.0) * (1.0 + (1.0 / pow(x_m, 2.0)))))) + 1.0) * 0.5)) / (1.0 + sqrt((((1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0285)
		tmp = Float64(Float64(x_m * x_m) * Float64(0.125 + Float64(Float64(x_m * x_m) * Float64(Float64(Float64(x_m * x_m) * Float64(0.0673828125 + Float64(-0.056243896484375 * Float64(x_m * x_m)))) - 0.0859375))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(1.0 / sqrt(Float64((x_m ^ 2.0) * Float64(1.0 + Float64(1.0 / (x_m ^ 2.0)))))) + 1.0) * 0.5)) / Float64(1.0 + sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0285], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.125 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0673828125 + N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(1.0 / N[Sqrt[N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(1.0 + N[(1.0 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.028500000000000001

   1. Initial program 53.3%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. pow2 N/A

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

      18. +-commutative N/A

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

      19. pow2 N/A

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

      20. lower-fma.f64 53.3

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

   3. Applied rewrites 53.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   5. 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. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if 0.028500000000000001 < x

   1. Initial program 98.4%

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

      1. 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. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 99.9

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

   6. Applied rewrites 99.9%

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

Alternative 2: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(\frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5\\ \mathbf{if}\;x\_m \leq 0.0285:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(0.125 + \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(0.0673828125 + -0.056243896484375 \cdot \left(x\_m \cdot x\_m\right)\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{1 + \sqrt{t\_0}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (+ (/ 1.0 (sqrt (fma x_m x_m 1.0))) 1.0) 0.5)))
   (if (<= x_m 0.0285)
     (*
      (* x_m x_m)
      (+
       0.125
       (*
        (* x_m x_m)
        (-
         (* (* x_m x_m) (+ 0.0673828125 (* -0.056243896484375 (* x_m x_m))))
         0.0859375))))
     (/ (- 1.0 t_0) (+ 1.0 (sqrt t_0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = ((1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5;
	double tmp;
	if (x_m <= 0.0285) {
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * (((x_m * x_m) * (0.0673828125 + (-0.056243896484375 * (x_m * x_m)))) - 0.0859375)));
	} else {
		tmp = (1.0 - t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5)
	tmp = 0.0
	if (x_m <= 0.0285)
		tmp = Float64(Float64(x_m * x_m) * Float64(0.125 + Float64(Float64(x_m * x_m) * Float64(Float64(Float64(x_m * x_m) * Float64(0.0673828125 + Float64(-0.056243896484375 * Float64(x_m * x_m)))) - 0.0859375))));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0285], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.125 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0673828125 + N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - 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 < 0.028500000000000001

   1. Initial program 53.3%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. pow2 N/A

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

      18. +-commutative N/A

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

      19. pow2 N/A

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

      20. lower-fma.f64 53.3

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

   3. Applied rewrites 53.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   5. 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. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if 0.028500000000000001 < x

   1. Initial program 98.4%

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

      1. 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. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

Alternative 3: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 0.8:\\ \;\;\;\;\frac{0.5 - 0.5 \cdot \frac{1}{x\_m}}{1 + \sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(0.125 + \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(0.0673828125 + -0.056243896484375 \cdot \left(x\_m \cdot x\_m\right)\right) - 0.0859375\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m))))) 0.8)
   (/
    (- 0.5 (* 0.5 (/ 1.0 x_m)))
    (+ 1.0 (sqrt (* (+ (/ 1.0 (sqrt (fma x_m x_m 1.0))) 1.0) 0.5))))
   (*
    (* x_m x_m)
    (+
     0.125
     (*
      (* x_m x_m)
      (-
       (* (* x_m x_m) (+ 0.0673828125 (* -0.056243896484375 (* x_m x_m))))
       0.0859375))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m))))) <= 0.8) {
		tmp = (0.5 - (0.5 * (1.0 / x_m))) / (1.0 + sqrt((((1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5)));
	} else {
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * (((x_m * x_m) * (0.0673828125 + (-0.056243896484375 * (x_m * x_m)))) - 0.0859375)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m))))) <= 0.8)
		tmp = Float64(Float64(0.5 - Float64(0.5 * Float64(1.0 / x_m))) / Float64(1.0 + sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5))));
	else
		tmp = Float64(Float64(x_m * x_m) * Float64(0.125 + Float64(Float64(x_m * x_m) * Float64(Float64(Float64(x_m * x_m) * Float64(0.0673828125 + Float64(-0.056243896484375 * Float64(x_m * x_m)))) - 0.0859375))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[(N[(0.5 - N[(0.5 * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.125 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0673828125 + N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))) < 0.80000000000000004

   1. Initial program 98.5%

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

      1. 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. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if 0.80000000000000004 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))

   1. Initial program 53.5%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. pow2 N/A

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

      18. +-commutative N/A

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

      19. pow2 N/A

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

      20. lower-fma.f64 53.5

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

   3. Applied rewrites 53.5%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   5. 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. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 99.6

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

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(0.125 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.0673828125 + -0.056243896484375 \cdot \left(x \cdot x\right)\right) - 0.0859375\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 0.8:\\ \;\;\;\;\frac{0.5 - 0.5 \cdot \frac{1}{x\_m}}{1 + \sqrt{\left(\frac{1}{x\_m} + 1\right) \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(0.125 + \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(0.0673828125 + -0.056243896484375 \cdot \left(x\_m \cdot x\_m\right)\right) - 0.0859375\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m))))) 0.8)
   (/ (- 0.5 (* 0.5 (/ 1.0 x_m))) (+ 1.0 (sqrt (* (+ (/ 1.0 x_m) 1.0) 0.5))))
   (*
    (* x_m x_m)
    (+
     0.125
     (*
      (* x_m x_m)
      (-
       (* (* x_m x_m) (+ 0.0673828125 (* -0.056243896484375 (* x_m x_m))))
       0.0859375))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m))))) <= 0.8) {
		tmp = (0.5 - (0.5 * (1.0 / x_m))) / (1.0 + sqrt((((1.0 / x_m) + 1.0) * 0.5)));
	} else {
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * (((x_m * x_m) * (0.0673828125 + (-0.056243896484375 * (x_m * x_m)))) - 0.0859375)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.sqrt((0.5 * (1.0 + (1.0 / Math.hypot(1.0, x_m))))) <= 0.8) {
		tmp = (0.5 - (0.5 * (1.0 / x_m))) / (1.0 + Math.sqrt((((1.0 / x_m) + 1.0) * 0.5)));
	} else {
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * (((x_m * x_m) * (0.0673828125 + (-0.056243896484375 * (x_m * x_m)))) - 0.0859375)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x_m))))) <= 0.8:
		tmp = (0.5 - (0.5 * (1.0 / x_m))) / (1.0 + math.sqrt((((1.0 / x_m) + 1.0) * 0.5)))
	else:
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * (((x_m * x_m) * (0.0673828125 + (-0.056243896484375 * (x_m * x_m)))) - 0.0859375)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m))))) <= 0.8)
		tmp = Float64(Float64(0.5 - Float64(0.5 * Float64(1.0 / x_m))) / Float64(1.0 + sqrt(Float64(Float64(Float64(1.0 / x_m) + 1.0) * 0.5))));
	else
		tmp = Float64(Float64(x_m * x_m) * Float64(0.125 + Float64(Float64(x_m * x_m) * Float64(Float64(Float64(x_m * x_m) * Float64(0.0673828125 + Float64(-0.056243896484375 * Float64(x_m * x_m)))) - 0.0859375))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m))))) <= 0.8)
		tmp = (0.5 - (0.5 * (1.0 / x_m))) / (1.0 + sqrt((((1.0 / x_m) + 1.0) * 0.5)));
	else
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * (((x_m * x_m) * (0.0673828125 + (-0.056243896484375 * (x_m * x_m)))) - 0.0859375)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[(N[(0.5 - N[(0.5 * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[(1.0 / x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.125 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0673828125 + N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))) < 0.80000000000000004

   1. Initial program 98.5%

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

      1. 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. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 99.4%

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

   if 0.80000000000000004 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))

   1. Initial program 53.5%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. pow2 N/A

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

      18. +-commutative N/A

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

      19. pow2 N/A

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

      20. lower-fma.f64 53.5

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

   3. Applied rewrites 53.5%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   5. 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. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 99.6

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

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(0.125 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.0673828125 + -0.056243896484375 \cdot \left(x \cdot x\right)\right) - 0.0859375\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.03:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(0.125 + \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(0.0673828125 + -0.056243896484375 \cdot \left(x\_m \cdot x\_m\right)\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.03)
   (*
    (* x_m x_m)
    (+
     0.125
     (*
      (* x_m x_m)
      (-
       (* (* x_m x_m) (+ 0.0673828125 (* -0.056243896484375 (* x_m x_m))))
       0.0859375))))
   (- 1.0 (sqrt (+ 0.5 (* (/ 1.0 (sqrt (fma x_m x_m 1.0))) 0.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.03) {
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * (((x_m * x_m) * (0.0673828125 + (-0.056243896484375 * (x_m * x_m)))) - 0.0859375)));
	} else {
		tmp = 1.0 - sqrt((0.5 + ((1.0 / sqrt(fma(x_m, x_m, 1.0))) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.03)
		tmp = Float64(Float64(x_m * x_m) * Float64(0.125 + Float64(Float64(x_m * x_m) * Float64(Float64(Float64(x_m * x_m) * Float64(0.0673828125 + Float64(-0.056243896484375 * Float64(x_m * x_m)))) - 0.0859375))));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.03], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.125 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0673828125 + N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.029999999999999999

   1. Initial program 53.3%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. pow2 N/A

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

      18. +-commutative N/A

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

      19. pow2 N/A

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

      20. lower-fma.f64 53.3

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

   3. Applied rewrites 53.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   5. 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. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if 0.029999999999999999 < x

   1. Initial program 98.4%

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

      1. 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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 98.4

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

   3. Applied rewrites 98.4%

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

Alternative 6: 99.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0132:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(0.125 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0132)
   (*
    (* x_m x_m)
    (+ 0.125 (* (* x_m x_m) (- (* 0.0673828125 (* x_m x_m)) 0.0859375))))
   (- 1.0 (sqrt (+ 0.5 (* (/ 1.0 (sqrt (fma x_m x_m 1.0))) 0.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0132) {
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * ((0.0673828125 * (x_m * x_m)) - 0.0859375)));
	} else {
		tmp = 1.0 - sqrt((0.5 + ((1.0 / sqrt(fma(x_m, x_m, 1.0))) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0132)
		tmp = Float64(Float64(x_m * x_m) * Float64(0.125 + Float64(Float64(x_m * x_m) * Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375))));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0132], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.125 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0132

   1. Initial program 53.3%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. pow2 N/A

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

      18. +-commutative N/A

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

      19. pow2 N/A

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

      20. lower-fma.f64 53.3

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

   3. Applied rewrites 53.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   5. 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. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 100.0

          \[\leadsto \left(x \cdot x\right) \cdot \left(0.125 + \left(x \cdot x\right) \cdot \left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375\right)\right) \]

   6. Applied rewrites 100.0%

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

   if 0.0132 < x

   1. Initial program 98.4%

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

      1. 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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 98.4

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

   3. Applied rewrites 98.4%

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

Alternative 7: 98.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.3:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(0.125 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.3)
   (*
    (* x_m x_m)
    (+ 0.125 (* (* x_m x_m) (- (* 0.0673828125 (* x_m x_m)) 0.0859375))))
   (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.3) {
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * ((0.0673828125 * (x_m * x_m)) - 0.0859375)));
	} else {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.3d0) then
        tmp = (x_m * x_m) * (0.125d0 + ((x_m * x_m) * ((0.0673828125d0 * (x_m * x_m)) - 0.0859375d0)))
    else
        tmp = 0.5d0 / (1.0d0 + sqrt(0.5d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.3) {
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * ((0.0673828125 * (x_m * x_m)) - 0.0859375)));
	} else {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.3:
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * ((0.0673828125 * (x_m * x_m)) - 0.0859375)))
	else:
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.3)
		tmp = Float64(Float64(x_m * x_m) * Float64(0.125 + Float64(Float64(x_m * x_m) * Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375))));
	else
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.3)
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * ((0.0673828125 * (x_m * x_m)) - 0.0859375)));
	else
		tmp = 0.5 / (1.0 + sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.3], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.125 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000004

   1. Initial program 53.5%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. pow2 N/A

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

      18. +-commutative N/A

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

      19. pow2 N/A

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

      20. lower-fma.f64 53.5

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

   3. Applied rewrites 53.5%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   5. 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. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 99.6

          \[\leadsto \left(x \cdot x\right) \cdot \left(0.125 + \left(x \cdot x\right) \cdot \left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375\right)\right) \]

   6. Applied rewrites 99.6%

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

   if 1.30000000000000004 < x

   1. Initial program 98.5%

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

      1. 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. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

   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. lift-sqrt.f64 97.9

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

   6. Applied rewrites 97.9%

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

Alternative 8: 98.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(0.125 + -0.0859375 \cdot \left(x\_m \cdot x\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (* (* x_m x_m) (+ 0.125 (* -0.0859375 (* x_m x_m))))
   (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

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

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.1d0) then
        tmp = (x_m * x_m) * (0.125d0 + ((-0.0859375d0) * (x_m * x_m)))
    else
        tmp = 0.5d0 / (1.0d0 + sqrt(0.5d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = (x_m * x_m) * (0.125 + (-0.0859375 * (x_m * x_m)));
	} else {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.1:
		tmp = (x_m * x_m) * (0.125 + (-0.0859375 * (x_m * x_m)))
	else:
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	return tmp

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.1)
		tmp = (x_m * x_m) * (0.125 + (-0.0859375 * (x_m * x_m)));
	else
		tmp = 0.5 / (1.0 + sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.125 + N[(-0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 53.5%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. pow2 N/A

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

      18. +-commutative N/A

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

      19. pow2 N/A

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

      20. lower-fma.f64 53.5

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

   3. Applied rewrites 53.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if 1.1000000000000001 < x

   1. Initial program 98.5%

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

      1. 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. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

   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. lift-sqrt.f64 97.9

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

   6. Applied rewrites 97.9%

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

Alternative 9: 97.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(0.125 + -0.0859375 \cdot \left(x\_m \cdot x\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (* (* x_m x_m) (+ 0.125 (* -0.0859375 (* x_m x_m))))
   (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = (x_m * x_m) * (0.125 + (-0.0859375 * (x_m * x_m)));
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.1d0) then
        tmp = (x_m * x_m) * (0.125d0 + ((-0.0859375d0) * (x_m * x_m)))
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = (x_m * x_m) * (0.125 + (-0.0859375 * (x_m * x_m)));
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.1:
		tmp = (x_m * x_m) * (0.125 + (-0.0859375 * (x_m * x_m)))
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.1)
		tmp = (x_m * x_m) * (0.125 + (-0.0859375 * (x_m * x_m)));
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.125 + N[(-0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 53.5%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. pow2 N/A

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

      18. +-commutative N/A

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

      19. pow2 N/A

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

      20. lower-fma.f64 53.5

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

   3. Applied rewrites 53.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if 1.1000000000000001 < x

   1. Initial program 98.5%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 96.4%

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

Alternative 10: 97.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5:\\ \;\;\;\;0.125 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.5) (* 0.125 (* x_m x_m)) (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.5) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.5d0) then
        tmp = 0.125d0 * (x_m * x_m)
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.5) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.5:
		tmp = 0.125 * (x_m * x_m)
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.5)
		tmp = Float64(0.125 * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.5)
		tmp = 0.125 * (x_m * x_m);
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.5], N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 53.5%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. pow2 N/A

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

      18. +-commutative N/A

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

      19. pow2 N/A

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

      20. lower-fma.f64 53.5

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

   3. Applied rewrites 53.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 98.9

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

   6. Applied rewrites 98.9%

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

   if 1.5 < x

   1. Initial program 98.5%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 96.5%

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

Alternative 11: 51.4% accurate, 12.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.125 \cdot \left(x\_m \cdot x\_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* 0.125 (* x_m x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return 0.125 * (x_m * x_m);
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = 0.125d0 * (x_m * x_m)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.125 * (x_m * x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.125 * (x_m * x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(0.125 * Float64(x_m * x_m))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.125 * (x_m * x_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.1%

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

   1. lift-+.f64 N/A

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

   2. metadata-eval N/A

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

   3. lift-/.f64 N/A

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

   4. lift-hypot.f64 N/A

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

   5. metadata-eval N/A

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

   6. frac-add N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. pow2 N/A

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

   12. +-commutative N/A

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

   13. pow2 N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 N/A

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

   17. pow2 N/A

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

   18. +-commutative N/A

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

   19. pow2 N/A

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

   20. lower-fma.f64 52.6

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

3. Applied rewrites 52.6%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 51.4

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

6. Applied rewrites 51.4%

   \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right)} \]
7. Add Preprocessing

Alternative 12: 27.5% accurate, 134.0× speedup

Math

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

FPCore

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

C

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

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

2. Taylor expanded in x around 0

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

   1. sqrt-unprod N/A

      \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot 2} \]

   2. metadata-eval N/A

      \[\leadsto 1 - \sqrt{1} \]

   3. metadata-eval N/A

      \[\leadsto 1 - 1 \]

   4. metadata-eval 27.5

      \[\leadsto 0 \]

4. Applied rewrites 27.5%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))