Given's Rotation SVD example, simplified

Percentage Accurate: 75.6% → 99.5%

Time: 8.4s

Alternatives: 10

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: 75.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 0.5 - \frac{-0.5}{x\_m}\\ t_1 := \sqrt{t\_0} + 1\\ \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(0.125 \cdot x\_m, x\_m, \left(\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375\right) \cdot {x\_m}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{t\_1}^{-1} - \frac{t\_0}{t\_1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ -0.5 x_m))) (t_1 (+ (sqrt t_0) 1.0)))
   (if (<= x_m 1.1)
     (fma
      (* 0.125 x_m)
      x_m
      (*
       (-
        (* (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) x_m) x_m)
        0.0859375)
       (pow x_m 4.0)))
     (- (pow t_1 -1.0) (/ t_0 t_1)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 0.5 - (-0.5 / x_m);
	double t_1 = sqrt(t_0) + 1.0;
	double tmp;
	if (x_m <= 1.1) {
		tmp = fma((0.125 * x_m), x_m, ((((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375) * pow(x_m, 4.0)));
	} else {
		tmp = pow(t_1, -1.0) - (t_0 / t_1);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(0.5 - Float64(-0.5 / x_m))
	t_1 = Float64(sqrt(t_0) + 1.0)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = fma(Float64(0.125 * x_m), x_m, Float64(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375) * (x_m ^ 4.0)));
	else
		tmp = Float64((t_1 ^ -1.0) - Float64(t_0 / t_1));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(0.5 - N[(-0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 1.1], N[(N[(0.125 * x$95$m), $MachinePrecision] * x$95$m + N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[t$95$1, -1.0], $MachinePrecision] - N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 68.7%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 68.7

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

   4. Applied rewrites 68.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 69.1%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, \left(x \cdot x\right) \cdot 0.125\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 73.3%

       \[\leadsto \mathsf{fma}\left(0.125 \cdot x, \color{blue}{x}, \left(\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375\right) \cdot {x}^{4}\right) \]

   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. Add Preprocessing
   3. 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. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 98.5

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

   7. Applied rewrites 98.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   9. Applied rewrites 100.0%

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

Alternative 2: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(0.125 \cdot x\_m, x\_m, \left(\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375\right) \cdot {x\_m}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (fma
    (* 0.125 x_m)
    x_m
    (*
     (-
      (* (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) x_m) x_m)
      0.0859375)
     (pow x_m 4.0)))
   (/ 0.5 (+ (sqrt 0.5) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = fma((0.125 * x_m), x_m, ((((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375) * pow(x_m, 4.0)));
	} else {
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = fma(Float64(0.125 * x_m), x_m, Float64(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375) * (x_m ^ 4.0)));
	else
		tmp = Float64(0.5 / Float64(sqrt(0.5) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(0.125 * x$95$m), $MachinePrecision] * x$95$m + N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 68.7%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 68.7

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

   4. Applied rewrites 68.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 69.1%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, \left(x \cdot x\right) \cdot 0.125\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 73.3%

       \[\leadsto \mathsf{fma}\left(0.125 \cdot x, \color{blue}{x}, \left(\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375\right) \cdot {x}^{4}\right) \]

   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. Add Preprocessing
   3. 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. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 99.4

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

   9. Applied rewrites 99.4%

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

Alternative 3: 98.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375\right) \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, \left(x\_m \cdot x\_m\right) \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (fma
    (*
     (-
      (* (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) x_m) x_m)
      0.0859375)
     (* x_m x_m))
    (* x_m x_m)
    (* (* x_m x_m) 0.125))
   (/ 0.5 (+ (sqrt 0.5) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = fma(((((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375) * (x_m * x_m)), (x_m * x_m), ((x_m * x_m) * 0.125));
	} else {
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = fma(Float64(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375) * Float64(x_m * x_m)), Float64(x_m * x_m), Float64(Float64(x_m * x_m) * 0.125));
	else
		tmp = Float64(0.5 / Float64(sqrt(0.5) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 68.7%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 68.7

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

   4. Applied rewrites 68.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 69.1%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, \left(x \cdot x\right) \cdot 0.125\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 73.3%

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

   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. Add Preprocessing
   3. 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. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 99.4

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

   9. Applied rewrites 99.4%

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

Alternative 4: 98.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000004

   1. Initial program 68.7%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 68.7

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

   4. Applied rewrites 68.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, \color{blue}{x \cdot x}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      14. lower-*.f64 74.0

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

   7. Applied rewrites 74.0%

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

   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. Add Preprocessing
   3. 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. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 99.4

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

   9. Applied rewrites 99.4%

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

Alternative 5: 98.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 68.7%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 68.7

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

   4. Applied rewrites 68.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 73.2

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

   7. Applied rewrites 73.2%

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

   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. Add Preprocessing
   3. 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. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 99.4

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

   9. Applied rewrites 99.4%

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

Alternative 6: 98.0% accurate, 4.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (* (* (fma -0.0859375 (* x_m x_m) 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 <= 1.1) {
		tmp = (fma(-0.0859375, (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - 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(fma(-0.0859375, Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(N[(-0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $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 68.7%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 68.7

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

   4. Applied rewrites 68.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 73.2

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

   7. Applied rewrites 73.2%

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

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.9%

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

Alternative 7: 97.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.52:\\ \;\;\;\;\left(0.125 \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.52) (* (* 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 <= 1.52) {
		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 <= 1.52d0) 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 <= 1.52) {
		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 <= 1.52:
		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 <= 1.52)
		tmp = Float64(Float64(0.125 * 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 <= 1.52)
		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, 1.52], N[(N[(0.125 * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.52

   1. Initial program 68.7%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 68.7

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

   4. Applied rewrites 68.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 73.2

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

   7. Applied rewrites 73.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 73.8%

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

   if 1.52 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.9%

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

Alternative 8: 51.2% accurate, 12.2× speedup

Math

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

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(Float64(0.125 * 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[(N[(0.125 * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
2. Add Preprocessing
3. 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. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. fp-cancel-sign-sub-inv N/A

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

   6. lower--.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 76.4

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 54.5

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

7. Applied rewrites 54.5%

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

8. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 55.9%

    \[\leadsto \left(0.125 \cdot x\right) \cdot x \]
11. Add Preprocessing

Alternative 9: 51.2% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

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 = (x_m * x_m) * 0.125d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
2. Add Preprocessing
3. 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. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. fp-cancel-sign-sub-inv N/A

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

   6. lower--.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 76.4

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 55.9

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

7. Applied rewrites 55.9%

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

Alternative 10: 26.9% accurate, 33.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 - 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (- 1.0 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 - 1.0;
}

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 = 1.0d0 - 1.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0 - 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0 - 1.0

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 - 1.0)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0 - 1.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
2. Add Preprocessing
3. 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. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. fp-cancel-sign-sub-inv N/A

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

   6. lower--.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 76.4

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 32.4%

   \[\leadsto 1 - \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024354 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))