Given's Rotation SVD example, simplified

Percentage Accurate: 75.4% → 99.9%

Time: 4.9s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|x\right|\right)}^{2}\\ t_1 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ \mathbf{if}\;\left|x\right| \leq 0.0105:\\ \;\;\;\;t\_0 \cdot \left(0.125 + t\_0 \cdot \left(t\_0 \cdot \left(0.0673828125 + -0.056243896484375 \cdot t\_0\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 - \sqrt{t\_1 - -0.5}} \cdot \left(0.5 - t\_1\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (pow (fabs x) 2.0))
       (t_1 (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0)))))
  (if (<= (fabs x) 0.0105)
    (*
     t_0
     (+
      0.125
      (*
       t_0
       (-
        (* t_0 (+ 0.0673828125 (* -0.056243896484375 t_0)))
        0.0859375))))
    (* (/ -1.0 (- -1.0 (sqrt (- t_1 -0.5)))) (- 0.5 t_1)))))

C

double code(double x) {
	double t_0 = pow(fabs(x), 2.0);
	double t_1 = 0.5 / sqrt(fma(fabs(x), fabs(x), 1.0));
	double tmp;
	if (fabs(x) <= 0.0105) {
		tmp = t_0 * (0.125 + (t_0 * ((t_0 * (0.0673828125 + (-0.056243896484375 * t_0))) - 0.0859375)));
	} else {
		tmp = (-1.0 / (-1.0 - sqrt((t_1 - -0.5)))) * (0.5 - t_1);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = abs(x) ^ 2.0
	t_1 = Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0)))
	tmp = 0.0
	if (abs(x) <= 0.0105)
		tmp = Float64(t_0 * Float64(0.125 + Float64(t_0 * Float64(Float64(t_0 * Float64(0.0673828125 + Float64(-0.056243896484375 * t_0))) - 0.0859375))));
	else
		tmp = Float64(Float64(-1.0 / Float64(-1.0 - sqrt(Float64(t_1 - -0.5)))) * Float64(0.5 - t_1));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0105], N[(t$95$0 * N[(0.125 + N[(t$95$0 * N[(N[(t$95$0 * N[(0.0673828125 + N[(-0.056243896484375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(-1.0 - N[Sqrt[N[(t$95$1 - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.010500000000000001

   1. Initial program 75.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 50.9%

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

   4. Applied rewrites 50.9%

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

   if 0.010500000000000001 < x

   1. Initial program 75.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

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

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

   3. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sum-to-mult-rev N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

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

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

      8. lower--.f32 N/A

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

      9. lift-neg.f64 N/A

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

      10. add-flip N/A

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

      11. lift-+.f64 N/A

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

   5. Applied rewrites 76.2%

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

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ t_1 := {\left(\left|x\right|\right)}^{2}\\ \mathbf{if}\;\left|x\right| \leq 1.55:\\ \;\;\;\;t\_1 \cdot \left(0.125 + t\_1 \cdot \left(0.0673828125 \cdot t\_1 - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 - \sqrt{t\_0 - -0.5}} \cdot \left(0.5 - t\_0\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0))))
       (t_1 (pow (fabs x) 2.0)))
  (if (<= (fabs x) 1.55)
    (* t_1 (+ 0.125 (* t_1 (- (* 0.0673828125 t_1) 0.0859375))))
    (* (/ -1.0 (- -1.0 (sqrt (- t_0 -0.5)))) (- 0.5 t_0)))))

C

double code(double x) {
	double t_0 = 0.5 / sqrt(fma(fabs(x), fabs(x), 1.0));
	double t_1 = pow(fabs(x), 2.0);
	double tmp;
	if (fabs(x) <= 1.55) {
		tmp = t_1 * (0.125 + (t_1 * ((0.0673828125 * t_1) - 0.0859375)));
	} else {
		tmp = (-1.0 / (-1.0 - sqrt((t_0 - -0.5)))) * (0.5 - t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0)))
	t_1 = abs(x) ^ 2.0
	tmp = 0.0
	if (abs(x) <= 1.55)
		tmp = Float64(t_1 * Float64(0.125 + Float64(t_1 * Float64(Float64(0.0673828125 * t_1) - 0.0859375))));
	else
		tmp = Float64(Float64(-1.0 / Float64(-1.0 - sqrt(Float64(t_0 - -0.5)))) * Float64(0.5 - t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1.55], N[(t$95$1 * N[(0.125 + N[(t$95$1 * N[(N[(0.0673828125 * t$95$1), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(-1.0 - N[Sqrt[N[(t$95$0 - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55

   1. Initial program 75.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 50.8%

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

   4. Applied rewrites 50.8%

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

   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 52.3%

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

   7. Applied rewrites 52.3%

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

   if 1.55 < x

   1. Initial program 75.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

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

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

   3. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sum-to-mult-rev N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

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

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

      8. lower--.f32 N/A

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

      9. lift-neg.f64 N/A

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

      10. add-flip N/A

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

      11. lift-+.f64 N/A

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

   5. Applied rewrites 76.2%

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

Alternative 3: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ \mathbf{if}\;\left|x\right| \leq 0.000165:\\ \;\;\;\;\mathsf{fma}\left(0.125 \cdot \left|x\right|, \left|x\right|, \left(\left(-0.0859375 \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 - \sqrt{t\_0 - -0.5}} \cdot \left(0.5 - t\_0\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0)))))
  (if (<= (fabs x) 0.000165)
    (fma
     (* 0.125 (fabs x))
     (fabs x)
     (* (* (* -0.0859375 (* (fabs x) (fabs x))) (fabs x)) (fabs x)))
    (* (/ -1.0 (- -1.0 (sqrt (- t_0 -0.5)))) (- 0.5 t_0)))))

C

double code(double x) {
	double t_0 = 0.5 / sqrt(fma(fabs(x), fabs(x), 1.0));
	double tmp;
	if (fabs(x) <= 0.000165) {
		tmp = fma((0.125 * fabs(x)), fabs(x), (((-0.0859375 * (fabs(x) * fabs(x))) * fabs(x)) * fabs(x)));
	} else {
		tmp = (-1.0 / (-1.0 - sqrt((t_0 - -0.5)))) * (0.5 - t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0)))
	tmp = 0.0
	if (abs(x) <= 0.000165)
		tmp = fma(Float64(0.125 * abs(x)), abs(x), Float64(Float64(Float64(-0.0859375 * Float64(abs(x) * abs(x))) * abs(x)) * abs(x)));
	else
		tmp = Float64(Float64(-1.0 / Float64(-1.0 - sqrt(Float64(t_0 - -0.5)))) * Float64(0.5 - t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.000165], N[(N[(0.125 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(N[(N[(-0.0859375 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(-1.0 - N[Sqrt[N[(t$95$0 - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65e-4

   1. Initial program 75.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 50.8%

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 50.9%

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 50.9%

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

   6. Applied rewrites 50.9%

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

   if 1.65e-4 < x

   1. Initial program 75.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

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

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

   3. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sum-to-mult-rev N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

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

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

      8. lower--.f32 N/A

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

      9. lift-neg.f64 N/A

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

      10. add-flip N/A

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

      11. lift-+.f64 N/A

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

   5. Applied rewrites 76.2%

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

Alternative 4: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ \mathbf{if}\;\left|x\right| \leq 0.000165:\\ \;\;\;\;\mathsf{fma}\left(0.125 \cdot \left|x\right|, \left|x\right|, \left(\left(-0.0859375 \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - 0.5}{-1 - \sqrt{t\_0 - -0.5}}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0)))))
  (if (<= (fabs x) 0.000165)
    (fma
     (* 0.125 (fabs x))
     (fabs x)
     (* (* (* -0.0859375 (* (fabs x) (fabs x))) (fabs x)) (fabs x)))
    (/ (- t_0 0.5) (- -1.0 (sqrt (- t_0 -0.5)))))))

C

double code(double x) {
	double t_0 = 0.5 / sqrt(fma(fabs(x), fabs(x), 1.0));
	double tmp;
	if (fabs(x) <= 0.000165) {
		tmp = fma((0.125 * fabs(x)), fabs(x), (((-0.0859375 * (fabs(x) * fabs(x))) * fabs(x)) * fabs(x)));
	} else {
		tmp = (t_0 - 0.5) / (-1.0 - sqrt((t_0 - -0.5)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0)))
	tmp = 0.0
	if (abs(x) <= 0.000165)
		tmp = fma(Float64(0.125 * abs(x)), abs(x), Float64(Float64(Float64(-0.0859375 * Float64(abs(x) * abs(x))) * abs(x)) * abs(x)));
	else
		tmp = Float64(Float64(t_0 - 0.5) / Float64(-1.0 - sqrt(Float64(t_0 - -0.5))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.000165], N[(N[(0.125 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(N[(N[(-0.0859375 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - 0.5), $MachinePrecision] / N[(-1.0 - N[Sqrt[N[(t$95$0 - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65e-4

   1. Initial program 75.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 50.8%

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 50.9%

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 50.9%

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

   6. Applied rewrites 50.9%

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

   if 1.65e-4 < x

   1. Initial program 75.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

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

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

   3. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sum-to-mult-rev N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

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

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

      8. lower--.f32 N/A

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

      9. lift-neg.f64 N/A

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

      10. add-flip N/A

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

      11. lift-+.f64 N/A

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

   5. Applied rewrites 76.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lower--.f64 76.2%

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

   7. Applied rewrites 76.2%

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

Alternative 5: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0017:\\ \;\;\;\;\mathsf{fma}\left(0.125 \cdot \left|x\right|, \left|x\right|, \left(\left(-0.0859375 \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}} - -0.5}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 0.0017)
  (fma
   (* 0.125 (fabs x))
   (fabs x)
   (* (* (* -0.0859375 (* (fabs x) (fabs x))) (fabs x)) (fabs x)))
  (- 1.0 (sqrt (- (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0))) -0.5)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0017) {
		tmp = fma((0.125 * fabs(x)), fabs(x), (((-0.0859375 * (fabs(x) * fabs(x))) * fabs(x)) * fabs(x)));
	} else {
		tmp = 1.0 - sqrt(((0.5 / sqrt(fma(fabs(x), fabs(x), 1.0))) - -0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0017)
		tmp = fma(Float64(0.125 * abs(x)), abs(x), Float64(Float64(Float64(-0.0859375 * Float64(abs(x) * abs(x))) * abs(x)) * abs(x)));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0))) - -0.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0017], N[(N[(0.125 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(N[(N[(-0.0859375 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0016999999999999999

   1. Initial program 75.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 50.8%

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 50.9%

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 50.9%

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

   6. Applied rewrites 50.9%

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

   if 0.0016999999999999999 < x

   1. Initial program 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-hypot.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval 75.4%

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

   3. Applied rewrites 75.4%

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

Alternative 6: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0017:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, \left|x\right| \cdot \left|x\right|, 0.125\right) \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}} - -0.5}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 0.0017)
  (*
   (* (fma -0.0859375 (* (fabs x) (fabs x)) 0.125) (fabs x))
   (fabs x))
  (- 1.0 (sqrt (- (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0))) -0.5)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0017) {
		tmp = (fma(-0.0859375, (fabs(x) * fabs(x)), 0.125) * fabs(x)) * fabs(x);
	} else {
		tmp = 1.0 - sqrt(((0.5 / sqrt(fma(fabs(x), fabs(x), 1.0))) - -0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0017)
		tmp = Float64(Float64(fma(-0.0859375, Float64(abs(x) * abs(x)), 0.125) * abs(x)) * abs(x));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0))) - -0.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0017], N[(N[(N[(-0.0859375 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 0.125), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0016999999999999999

   1. Initial program 75.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 50.8%

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 50.9%

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 50.9%

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. lower-*.f64 50.9%

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

   6. Applied rewrites 50.9%

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

   if 0.0016999999999999999 < x

   1. Initial program 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-hypot.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval 75.4%

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

   3. Applied rewrites 75.4%

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

Alternative 7: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \leq 0.8:\\ \;\;\;\;0.2928932188134525\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x))))) 0.8)
  0.2928932188134525
  (* (* (fma -0.0859375 (* x x) 0.125) x) x)))

C

double code(double x) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x))))) <= 0.8) {
		tmp = 0.2928932188134525;
	} else {
		tmp = (fma(-0.0859375, (x * x), 0.125) * x) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))) <= 0.8)
		tmp = 0.2928932188134525;
	else
		tmp = Float64(Float64(fma(-0.0859375, Float64(x * x), 0.125) * x) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], 0.2928932188134525, N[(N[(N[(-0.0859375 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

   3. Applied rewrites 76.2%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 50.4%

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

   6. Applied rewrites 50.4%

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

   7. Evaluated real constant 50.4%

      \[\leadsto 0.2928932188134525 \]

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

   1. Initial program 75.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 50.8%

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 50.9%

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 50.9%

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. lower-*.f64 50.9%

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

   6. Applied rewrites 50.9%

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

Alternative 8: 74.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5.5 \cdot 10^{-79}:\\ \;\;\;\;1 - \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;0.2928932188134525\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 5.5e-79) (- 1.0 (sqrt 1.0)) 0.2928932188134525))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 5.5e-79) {
		tmp = 1.0 - sqrt(1.0);
	} else {
		tmp = 0.2928932188134525;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) <= 5.5d-79) then
        tmp = 1.0d0 - sqrt(1.0d0)
    else
        tmp = 0.2928932188134525d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 5.5e-79) {
		tmp = 1.0 - Math.sqrt(1.0);
	} else {
		tmp = 0.2928932188134525;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 5.5e-79:
		tmp = 1.0 - math.sqrt(1.0)
	else:
		tmp = 0.2928932188134525
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 5.5e-79)
		tmp = Float64(1.0 - sqrt(1.0));
	else
		tmp = 0.2928932188134525;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 5.5e-79)
		tmp = 1.0 - sqrt(1.0);
	else
		tmp = 0.2928932188134525;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 5.5e-79], N[(1.0 - N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision], 0.2928932188134525]

Derivation

1. Split input into 2 regimes

2. if x < 5.4999999999999997e-79

   1. Initial program 75.4%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 49.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 27.7%

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

   if 5.4999999999999997e-79 < x

   1. Initial program 75.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

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

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

   3. Applied rewrites 76.2%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 50.4%

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

   6. Applied rewrites 50.4%

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

   7. Evaluated real constant 50.4%

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

Alternative 9: 50.4% accurate, 29.6× speedup

Math

\[0.2928932188134525 \]

FPCore

(FPCore (x)
  :precision binary64
  0.2928932188134525)

C

double code(double x) {
	return 0.2928932188134525;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x):
	return 0.2928932188134525

Julia

function code(x)
	return 0.2928932188134525
end

MATLAB

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

Wolfram

code[x_] := 0.2928932188134525

Derivation

1. Initial program 75.4%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

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

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

3. Applied rewrites 76.2%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 50.4%

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

6. Applied rewrites 50.4%

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

7. Evaluated real constant 50.4%

   \[\leadsto 0.2928932188134525 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025254 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))