Given's Rotation SVD example, simplified

Percentage Accurate: 76.5% → 99.8%

Time: 11.1s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% 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: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + -0.5), $MachinePrecision] / N[(-1.0 - N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 55.3%

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

   4. Applied egg-rr 55.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 100.0

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

   7. Simplified 100.0%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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

   4. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.056243896484375, 0.0673828125\right), -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} + -0.5}{-1 - \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (*
    (* x x)
    (fma
     (* x x)
     (fma (* x x) (fma (* x x) -0.056243896484375 0.0673828125) -0.0859375)
     0.125))
   (/
    (+ (/ 0.5 (sqrt (fma x x 1.0))) -0.5)
    (- -1.0 (sqrt (+ 0.5 (/ 0.5 x)))))))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] / N[(-1.0 - N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 55.3%

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

   4. Applied egg-rr 55.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 100.0

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

   7. Simplified 100.0%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 99.1

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

   7. Simplified 99.1%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.056243896484375, 0.0673828125\right), -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} + -0.5}{-1 - \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (*
    (* x x)
    (fma
     (* x x)
     (fma (* x x) (fma (* x x) -0.056243896484375 0.0673828125) -0.0859375)
     0.125))
   (/ (- 0.5 (/ 0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x)))))))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 55.3%

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

   4. Applied egg-rr 55.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 100.0

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

   7. Simplified 100.0%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 99.1

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

   7. Simplified 99.1%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 99.0

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

   10. Simplified 99.0%

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

       1. lift-/.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-neg.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. frac-2neg N/A

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

       9. lift-+.f64 N/A

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

       10. +-commutative N/A

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

       11. distribute-neg-in N/A

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

       12. metadata-eval N/A

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

       13. sub-neg N/A

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

       14. lift-+.f64 N/A

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

       15. distribute-neg-in N/A

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

       16. metadata-eval N/A

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

       17. mul-1-neg N/A

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

       18. lift-neg.f64 N/A

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

       19. neg-mul-1 N/A

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

   12. Applied egg-rr 99.0%

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

Alternative 4: 98.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (*
    (* x x)
    (fma
     (* x x)
     (fma (* x x) (fma (* x x) -0.056243896484375 0.0673828125) -0.0859375)
     0.125))
   (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 55.3%

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

   4. Applied egg-rr 55.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 100.0

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

   7. Simplified 100.0%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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. Simplified 97.2%

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

      1. lift-sqrt.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 98.7

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

   7. Applied egg-rr 98.7%

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

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* x (* x (fma (* x x) (fma x (* x 0.0673828125) -0.0859375) 0.125)))
   (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 55.3%

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

   4. Applied egg-rr 55.4%

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

   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. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 99.9

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

   7. Simplified 99.9%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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. Simplified 97.2%

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

      1. lift-sqrt.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 98.7

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

   7. Applied egg-rr 98.7%

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

Alternative 6: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot -0.0859375, 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (* x x) (fma x (* x -0.0859375) 0.125))
   (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (x * x) * fma(x, (x * -0.0859375), 0.125);
	} else {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(x * x) * fma(x, Float64(x * -0.0859375), 0.125));
	else
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 55.3%

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

   4. Applied egg-rr 55.4%

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

   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. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 99.7

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

   7. Simplified 99.7%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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. Simplified 97.2%

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

      1. lift-sqrt.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 98.7

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

   7. Applied egg-rr 98.7%

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

Alternative 7: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot -0.0859375, 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (* x x) (fma x (* x -0.0859375) 0.125))
   (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (x * x) * fma(x, (x * -0.0859375), 0.125);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(x * x) * fma(x, Float64(x * -0.0859375), 0.125));
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 55.3%

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

   4. Applied egg-rr 55.4%

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

   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. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 99.7

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

   7. Simplified 99.7%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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. Simplified 97.2%

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

Alternative 8: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0) (* (* x x) 0.125) (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (x * x) * 0.125;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = (x * x) * 0.125;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = (x * x) * 0.125
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(x * x) * 0.125);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = (x * x) * 0.125;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(x * x), $MachinePrecision] * 0.125), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 55.3%

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

   4. Applied egg-rr 55.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.2

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

   7. Simplified 99.2%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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. Simplified 97.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.0% accurate, 12.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x (* x 0.125)))

C

double code(double x) {
	return x * (x * 0.125);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * 0.125d0)
end function

Java

public static double code(double x) {
	return x * (x * 0.125);
}

Python

def code(x):
	return x * (x * 0.125)

Julia

function code(x)
	return Float64(x * Float64(x * 0.125))
end

MATLAB

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

Wolfram

code[x_] := N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.9%

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

3. Taylor expanded in x around 0

   \[\leadsto 1 - \sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{3}{16} \cdot {x}^{2} - \frac{1}{4}\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 1 - \sqrt{\color{blue}{{x}^{2} \cdot \left(\frac{3}{16} \cdot {x}^{2} - \frac{1}{4}\right) + 1}} \]

   2. lower-fma.f64 N/A

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{3}{16} \cdot {x}^{2} - \frac{1}{4}, 1\right)}} \]

   3. unpow2 N/A

      \[\leadsto 1 - \sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{3}{16} \cdot {x}^{2} - \frac{1}{4}, 1\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto 1 - \sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{3}{16} \cdot {x}^{2} - \frac{1}{4}, 1\right)} \]

   5. sub-neg N/A

      \[\leadsto 1 - \sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{3}{16} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 1\right)} \]

   6. *-commutative N/A

      \[\leadsto 1 - \sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{3}{16}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)} \]

   7. unpow2 N/A

      \[\leadsto 1 - \sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{3}{16} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)} \]

   8. associate-*l* N/A

      \[\leadsto 1 - \sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{3}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)} \]

   9. metadata-eval N/A

      \[\leadsto 1 - \sqrt{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{3}{16}\right) + \color{blue}{\frac{-1}{4}}, 1\right)} \]

   10. lower-fma.f64 N/A

       \[\leadsto 1 - \sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{3}{16}, \frac{-1}{4}\right)}, 1\right)} \]

   11. lower-*.f64 29.3

       \[\leadsto 1 - \sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.1875}, -0.25\right), 1\right)} \]

5. Simplified 29.3%

   \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.1875, -0.25\right), 1\right)}} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 30.8

      \[\leadsto 1 - \mathsf{fma}\left(x, \color{blue}{x \cdot -0.125}, 1\right) \]

8. Simplified 30.8%

   \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(x, x \cdot -0.125, 1\right)} \]

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. unpow2 N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 53.9

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

11. Simplified 53.9%

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

Alternative 10: 52.0% accurate, 12.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (* x x) 0.125))

C

double code(double x) {
	return (x * x) * 0.125;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * 0.125d0
end function

Java

public static double code(double x) {
	return (x * x) * 0.125;
}

Python

def code(x):
	return (x * x) * 0.125

Julia

function code(x)
	return Float64(Float64(x * x) * 0.125)
end

MATLAB

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

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * 0.125), $MachinePrecision]

Derivation

1. Initial program 75.9%

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

4. Applied egg-rr 76.7%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 53.9

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

7. Simplified 53.9%

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

8. Final simplification 53.9%

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

Alternative 11: 28.6% accurate, 134.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 75.9%

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

3. Taylor expanded in x around 0

   \[\leadsto 1 - \sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{3}{16} \cdot {x}^{2} - \frac{1}{4}\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 1 - \sqrt{\color{blue}{{x}^{2} \cdot \left(\frac{3}{16} \cdot {x}^{2} - \frac{1}{4}\right) + 1}} \]

   2. lower-fma.f64 N/A

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{3}{16} \cdot {x}^{2} - \frac{1}{4}, 1\right)}} \]

   3. unpow2 N/A

      \[\leadsto 1 - \sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{3}{16} \cdot {x}^{2} - \frac{1}{4}, 1\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto 1 - \sqrt{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{3}{16} \cdot {x}^{2} - \frac{1}{4}, 1\right)} \]

   5. sub-neg N/A

      \[\leadsto 1 - \sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{3}{16} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, 1\right)} \]

   6. *-commutative N/A

      \[\leadsto 1 - \sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{3}{16}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)} \]

   7. unpow2 N/A

      \[\leadsto 1 - \sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{3}{16} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)} \]

   8. associate-*l* N/A

      \[\leadsto 1 - \sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{3}{16}\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), 1\right)} \]

   9. metadata-eval N/A

      \[\leadsto 1 - \sqrt{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{3}{16}\right) + \color{blue}{\frac{-1}{4}}, 1\right)} \]

   10. lower-fma.f64 N/A

       \[\leadsto 1 - \sqrt{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{3}{16}, \frac{-1}{4}\right)}, 1\right)} \]

   11. lower-*.f64 29.3

       \[\leadsto 1 - \sqrt{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.1875}, -0.25\right), 1\right)} \]

5. Simplified 29.3%

   \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.1875, -0.25\right), 1\right)}} \]

6. Taylor expanded in x around 0

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

8. Simplified 29.8%

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

   1. metadata-eval 29.8

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 29.8%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))