Given's Rotation SVD example, simplified

Percentage Accurate: 76.3% → 99.2%

Time: 4.1s

Alternatives: 9

Speedup: 1.3×

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: 76.3% 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: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.03:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right), x\_m \cdot x\_m, -0.0859375\right), x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{\sqrt{0.5}}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)} \cdot \sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.03)
   (*
    (fma
     (fma
      (fma -0.056243896484375 (* x_m x_m) 0.0673828125)
      (* x_m x_m)
      -0.0859375)
     (* x_m x_m)
     0.125)
    (* x_m x_m))
   (-
    1.0
    (sqrt (+ 0.5 (/ (sqrt 0.5) (* (sqrt (fma x_m x_m 1.0)) (sqrt 2.0))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.03) {
		tmp = fma(fma(fma(-0.056243896484375, (x_m * x_m), 0.0673828125), (x_m * x_m), -0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt((0.5 + (sqrt(0.5) / (sqrt(fma(x_m, x_m, 1.0)) * sqrt(2.0)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.03)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125), Float64(x_m * x_m), -0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(sqrt(0.5) / Float64(sqrt(fma(x_m, x_m, 1.0)) * sqrt(2.0))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.03], N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.029999999999999999

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 76.3

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

   3. Applied rewrites 76.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 50.8%

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

   if 0.029999999999999999 < x

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 76.3

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

   3. Applied rewrites 76.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{x \cdot x + 1}} \cdot \color{blue}{\sqrt{\frac{1}{4}}}} \]

      6. metadata-eval N/A

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

      7. sqrt-undiv N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. sqrt-unprod N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. lower-sqrt.f64 76.3

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

   5. Applied rewrites 76.3%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.012:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.0673828125, -0.0859375\right), x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{\sqrt{0.5}}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)} \cdot \sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.012)
   (*
    (* (fma (fma (* x_m x_m) 0.0673828125 -0.0859375) (* x_m x_m) 0.125) x_m)
    x_m)
   (-
    1.0
    (sqrt (+ 0.5 (/ (sqrt 0.5) (* (sqrt (fma x_m x_m 1.0)) (sqrt 2.0))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.012) {
		tmp = (fma(fma((x_m * x_m), 0.0673828125, -0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt((0.5 + (sqrt(0.5) / (sqrt(fma(x_m, x_m, 1.0)) * sqrt(2.0)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.012)
		tmp = Float64(Float64(fma(fma(Float64(x_m * x_m), 0.0673828125, -0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(sqrt(0.5) / Float64(sqrt(fma(x_m, x_m, 1.0)) * sqrt(2.0))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.012], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0673828125 + -0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.012

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 76.3

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

   3. Applied rewrites 76.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 52.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 52.1%

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

   if 0.012 < x

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 76.3

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

   3. Applied rewrites 76.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{x \cdot x + 1}} \cdot \color{blue}{\sqrt{\frac{1}{4}}}} \]

      6. metadata-eval N/A

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

      7. sqrt-undiv N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. sqrt-unprod N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. lower-sqrt.f64 76.3

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

   5. Applied rewrites 76.3%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.012:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.0673828125, -0.0859375\right), x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} - -0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.012)
   (*
    (* (fma (fma (* x_m x_m) 0.0673828125 -0.0859375) (* x_m x_m) 0.125) x_m)
    x_m)
   (- 1.0 (sqrt (- (/ 0.5 (sqrt (fma x_m x_m 1.0))) -0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.012) {
		tmp = (fma(fma((x_m * x_m), 0.0673828125, -0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt(((0.5 / sqrt(fma(x_m, x_m, 1.0))) - -0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.012)
		tmp = Float64(Float64(fma(fma(Float64(x_m * x_m), 0.0673828125, -0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / sqrt(fma(x_m, x_m, 1.0))) - -0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.012], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0673828125 + -0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.012

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 76.3

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

   3. Applied rewrites 76.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 52.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 52.1%

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

   if 0.012 < x

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 76.3

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

   3. Applied rewrites 76.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 76.3

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

   5. Applied rewrites 76.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-/.f64 76.3

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

   7. Applied rewrites 76.3%

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

Alternative 4: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0025:\\ \;\;\;\;\left(\mathsf{fma}\left(x\_m \cdot x\_m, -0.0859375, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} - -0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0025)
   (* (* (fma (* x_m x_m) -0.0859375 0.125) x_m) x_m)
   (- 1.0 (sqrt (- (/ 0.5 (sqrt (fma x_m x_m 1.0))) -0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0025) {
		tmp = (fma((x_m * x_m), -0.0859375, 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt(((0.5 / sqrt(fma(x_m, x_m, 1.0))) - -0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0025)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), -0.0859375, 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / sqrt(fma(x_m, x_m, 1.0))) - -0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0025], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.0859375 + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00250000000000000005

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 76.3

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

   3. Applied rewrites 76.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 50.7

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

   8. Applied rewrites 50.7%

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

   if 0.00250000000000000005 < x

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 76.3

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

   3. Applied rewrites 76.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 76.3

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

   5. Applied rewrites 76.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-/.f64 76.3

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

   7. Applied rewrites 76.3%

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

Alternative 5: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[(1.0 - N[Sqrt[N[(0.5 - N[(-0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.0859375 + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $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 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 76.3

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

   3. Applied rewrites 76.3%

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

   4. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. pow1/2 N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. pow1/2 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. mult-flip N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 49.5

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

   6. Applied rewrites 49.5%

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

   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 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 76.3

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

   3. Applied rewrites 76.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 50.7

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

   8. Applied rewrites 50.7%

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

Alternative 6: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[(1.0 - N[Sqrt[N[(0.5 - N[(-0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $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 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 76.3

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

   3. Applied rewrites 76.3%

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

   4. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. pow1/2 N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. pow1/2 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. mult-flip N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 49.5

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

   6. Applied rewrites 49.5%

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

   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 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 76.3

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

   3. Applied rewrites 76.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 52.1%

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

Alternative 7: 97.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $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 76.3%

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

   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%

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

   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 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 76.3

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

   3. Applied rewrites 76.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 52.1%

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

Alternative 8: 74.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.15 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.15e-77) 0.0 (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.15e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.15d-77) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.15e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.15e-77:
		tmp = 0.0
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.15e-77)
		tmp = 0.0;
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.15e-77)
		tmp = 0.0;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.15e-77], 0.0, N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1500000000000001e-77

   1. Initial program 76.3%

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

   2. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 28.4

         \[\leadsto 0 \]

   4. Applied rewrites 28.4%

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

   if 2.1500000000000001e-77 < x

   1. Initial program 76.3%

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

   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%

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

Alternative 9: 28.4% accurate, 27.4× speedup

Math

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

FPCore

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

C

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

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

2. Taylor expanded in x around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

      \[\leadsto 1 - 1 \]

   4. metadata-eval 28.4

      \[\leadsto 0 \]

4. Applied rewrites 28.4%

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

Reproduce

herbie shell --seed 2025134 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))