Given's Rotation SVD example, simplified

Percentage Accurate: 75.7% → 99.9%

Time: 4.5s

Alternatives: 10

Speedup: 2.3×

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: 75.7% 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: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\\ \mathbf{if}\;x\_m \leq 0.012:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(0.125 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(0.5 - t\_0\right) \cdot 1.5}{\sqrt{t\_0 - -0.5} - -1}}{1.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 0.5 (sqrt (fma x_m x_m 1.0)))))
   (if (<= x_m 0.012)
     (*
      (* x_m x_m)
      (+ 0.125 (* (* x_m x_m) (- (* 0.0673828125 (* x_m x_m)) 0.0859375))))
     (/ (/ (* (- 0.5 t_0) 1.5) (- (sqrt (- t_0 -0.5)) -1.0)) 1.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 0.5 / sqrt(fma(x_m, x_m, 1.0));
	double tmp;
	if (x_m <= 0.012) {
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * ((0.0673828125 * (x_m * x_m)) - 0.0859375)));
	} else {
		tmp = (((0.5 - t_0) * 1.5) / (sqrt((t_0 - -0.5)) - -1.0)) / 1.5;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(0.5 / sqrt(fma(x_m, x_m, 1.0)))
	tmp = 0.0
	if (x_m <= 0.012)
		tmp = Float64(Float64(x_m * x_m) * Float64(0.125 + Float64(Float64(x_m * x_m) * Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375))));
	else
		tmp = Float64(Float64(Float64(Float64(0.5 - t_0) * 1.5) / Float64(sqrt(Float64(t_0 - -0.5)) - -1.0)) / 1.5);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.012], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.125 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 - t$95$0), $MachinePrecision] * 1.5), $MachinePrecision] / N[(N[Sqrt[N[(t$95$0 - -0.5), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.012

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. mult-flip N/A

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot \frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 76.4

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

         \[\leadsto \frac{\frac{\color{blue}{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   7. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 51.5

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

   10. Applied rewrites 51.5%

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

   if 0.012 < x

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. mult-flip N/A

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot \frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 76.4

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

         \[\leadsto \frac{\frac{\color{blue}{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   7. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\left(\frac{1}{2} - \frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \color{blue}{\frac{3}{2}}}{\sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{-1}{2}} - -1}}{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{-3}{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 74.7%

       \[\leadsto \frac{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \color{blue}{1.5}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   11. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{\left(\frac{1}{2} - \frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \frac{3}{2}}{\sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - \frac{-1}{2}} - -1}}{\color{blue}{\frac{3}{2}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 76.4%

       \[\leadsto \frac{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot 1.5}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\color{blue}{1.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\\ \mathbf{if}\;x\_m \leq 0.012:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(0.125 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(t\_0 + 0.5\right)}{\sqrt{t\_0 - -0.5} - -1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 0.5 (sqrt (fma x_m x_m 1.0)))))
   (if (<= x_m 0.012)
     (*
      (* x_m x_m)
      (+ 0.125 (* (* x_m x_m) (- (* 0.0673828125 (* x_m x_m)) 0.0859375))))
     (/ (- 1.0 (+ t_0 0.5)) (- (sqrt (- t_0 -0.5)) -1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 0.5 / sqrt(fma(x_m, x_m, 1.0));
	double tmp;
	if (x_m <= 0.012) {
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * ((0.0673828125 * (x_m * x_m)) - 0.0859375)));
	} else {
		tmp = (1.0 - (t_0 + 0.5)) / (sqrt((t_0 - -0.5)) - -1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(0.5 / sqrt(fma(x_m, x_m, 1.0)))
	tmp = 0.0
	if (x_m <= 0.012)
		tmp = Float64(Float64(x_m * x_m) * Float64(0.125 + Float64(Float64(x_m * x_m) * Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375))));
	else
		tmp = Float64(Float64(1.0 - Float64(t_0 + 0.5)) / Float64(sqrt(Float64(t_0 - -0.5)) - -1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.012], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.125 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(t$95$0 - -0.5), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.012

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. mult-flip N/A

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot \frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 76.4

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

         \[\leadsto \frac{\frac{\color{blue}{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   7. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 51.5

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

   10. Applied rewrites 51.5%

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

   if 0.012 < x

   1. Initial program 75.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 76.4

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 76.4

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

   5. Applied rewrites 76.4%

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

Alternative 3: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\\ \mathbf{if}\;x\_m \leq 0.012:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(0.125 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - 0.5}{-1 - \sqrt{t\_0 - -0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 0.5 (sqrt (fma x_m x_m 1.0)))))
   (if (<= x_m 0.012)
     (*
      (* x_m x_m)
      (+ 0.125 (* (* x_m x_m) (- (* 0.0673828125 (* x_m x_m)) 0.0859375))))
     (/ (- t_0 0.5) (- -1.0 (sqrt (- t_0 -0.5)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 0.5 / sqrt(fma(x_m, x_m, 1.0));
	double tmp;
	if (x_m <= 0.012) {
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * ((0.0673828125 * (x_m * x_m)) - 0.0859375)));
	} else {
		tmp = (t_0 - 0.5) / (-1.0 - sqrt((t_0 - -0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(0.5 / sqrt(fma(x_m, x_m, 1.0)))
	tmp = 0.0
	if (x_m <= 0.012)
		tmp = Float64(Float64(x_m * x_m) * Float64(0.125 + Float64(Float64(x_m * x_m) * Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375))));
	else
		tmp = Float64(Float64(t_0 - 0.5) / Float64(-1.0 - sqrt(Float64(t_0 - -0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.012], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.125 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - 0.5), $MachinePrecision] / N[(-1.0 - N[Sqrt[N[(t$95$0 - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.012

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. mult-flip N/A

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot \frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 76.4

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

         \[\leadsto \frac{\frac{\color{blue}{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   7. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 51.5

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

   10. Applied rewrites 51.5%

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

   if 0.012 < x

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 76.4%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(0.125 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x\_m} - 0.5}{-1 - \sqrt{\frac{0.5}{x\_m} - -0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.2)
   (*
    (* x_m x_m)
    (+ 0.125 (* (* x_m x_m) (- (* 0.0673828125 (* x_m x_m)) 0.0859375))))
   (/ (- (/ 0.5 x_m) 0.5) (- -1.0 (sqrt (- (/ 0.5 x_m) -0.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.2) {
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * ((0.0673828125 * (x_m * x_m)) - 0.0859375)));
	} else {
		tmp = ((0.5 / x_m) - 0.5) / (-1.0 - sqrt(((0.5 / x_m) - -0.5)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.2d0) then
        tmp = (x_m * x_m) * (0.125d0 + ((x_m * x_m) * ((0.0673828125d0 * (x_m * x_m)) - 0.0859375d0)))
    else
        tmp = ((0.5d0 / x_m) - 0.5d0) / ((-1.0d0) - sqrt(((0.5d0 / x_m) - (-0.5d0))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.2) {
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * ((0.0673828125 * (x_m * x_m)) - 0.0859375)));
	} else {
		tmp = ((0.5 / x_m) - 0.5) / (-1.0 - Math.sqrt(((0.5 / x_m) - -0.5)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.2:
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * ((0.0673828125 * (x_m * x_m)) - 0.0859375)))
	else:
		tmp = ((0.5 / x_m) - 0.5) / (-1.0 - math.sqrt(((0.5 / x_m) - -0.5)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.2)
		tmp = Float64(Float64(x_m * x_m) * Float64(0.125 + Float64(Float64(x_m * x_m) * Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375))));
	else
		tmp = Float64(Float64(Float64(0.5 / x_m) - 0.5) / Float64(-1.0 - sqrt(Float64(Float64(0.5 / x_m) - -0.5))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.2)
		tmp = (x_m * x_m) * (0.125 + ((x_m * x_m) * ((0.0673828125 * (x_m * x_m)) - 0.0859375)));
	else
		tmp = ((0.5 / x_m) - 0.5) / (-1.0 - sqrt(((0.5 / x_m) - -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.2], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.125 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / x$95$m), $MachinePrecision] - 0.5), $MachinePrecision] / N[(-1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. mult-flip N/A

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot \frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 76.4

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

         \[\leadsto \frac{\frac{\color{blue}{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   7. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 51.5

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

   10. Applied rewrites 51.5%

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

   if 1.19999999999999996 < x

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 50.5

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

   8. Applied rewrites 50.5%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 50.7

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

   11. Applied rewrites 50.7%

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

Alternative 5: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. mult-flip N/A

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot \frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 76.4

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

         \[\leadsto \frac{\frac{\color{blue}{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   7. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 50.1

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

   10. Applied rewrites 50.1%

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

   if 1.1000000000000001 < x

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 50.5

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

   8. Applied rewrites 50.5%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 50.7

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

   11. Applied rewrites 50.7%

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

Alternative 6: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00255:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(0.125 + -0.0859375 \cdot \left(x\_m \cdot x\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} - -0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00255)
   (* (* x_m x_m) (+ 0.125 (* -0.0859375 (* 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 <= 0.00255) {
		tmp = (x_m * x_m) * (0.125 + (-0.0859375 * (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 <= 0.00255)
		tmp = Float64(Float64(x_m * x_m) * Float64(0.125 + Float64(-0.0859375 * Float64(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, 0.00255], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.125 + N[(-0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[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 < 0.0025500000000000002

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. mult-flip N/A

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot \frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 76.4

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

         \[\leadsto \frac{\frac{\color{blue}{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   7. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 50.1

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

   10. Applied rewrites 50.1%

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

   if 0.0025500000000000002 < x

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

      \[\leadsto 1 - \color{blue}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} + 0.5}} \]
   4. 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. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval 75.7

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

   5. Applied rewrites 75.7%

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

Alternative 7: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. mult-flip N/A

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot \frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 76.4

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

         \[\leadsto \frac{\frac{\color{blue}{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   7. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 50.1

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

   10. Applied rewrites 50.1%

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

   if 1.1000000000000001 < x

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. mult-flip N/A

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot \frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 76.4

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

         \[\leadsto \frac{\frac{\color{blue}{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   7. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 50.9

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

   10. Applied rewrites 50.9%

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

Alternative 8: 98.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55:\\ \;\;\;\;0.125 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.55) (* 0.125 (* x_m x_m)) (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.55d0) then
        tmp = 0.125d0 * (x_m * x_m)
    else
        tmp = 0.5d0 / (1.0d0 + sqrt(0.5d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.55:
		tmp = 0.125 * (x_m * x_m)
	else:
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.55)
		tmp = Float64(0.125 * Float64(x_m * x_m));
	else
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.55)
		tmp = 0.125 * (x_m * x_m);
	else
		tmp = 0.5 / (1.0 + sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.55], N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. mult-flip N/A

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot \frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 76.4

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

         \[\leadsto \frac{\frac{\color{blue}{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   7. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 51.6

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

   10. Applied rewrites 51.6%

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

   if 1.55000000000000004 < x

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. mult-flip N/A

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot \frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 76.4

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

         \[\leadsto \frac{\frac{\color{blue}{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   7. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 50.9

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

   10. Applied rewrites 50.9%

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

Alternative 9: 97.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55:\\ \;\;\;\;0.125 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{0.5} - 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.55) (* 0.125 (* x_m x_m)) (* -1.0 (- (sqrt 0.5) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = -1.0 * (sqrt(0.5) - 1.0);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.55d0) then
        tmp = 0.125d0 * (x_m * x_m)
    else
        tmp = (-1.0d0) * (sqrt(0.5d0) - 1.0d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = -1.0 * (Math.sqrt(0.5) - 1.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.55:
		tmp = 0.125 * (x_m * x_m)
	else:
		tmp = -1.0 * (math.sqrt(0.5) - 1.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.55)
		tmp = Float64(0.125 * Float64(x_m * x_m));
	else
		tmp = Float64(-1.0 * Float64(sqrt(0.5) - 1.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.55)
		tmp = 0.125 * (x_m * x_m);
	else
		tmp = -1.0 * (sqrt(0.5) - 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.55], N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[0.5], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. mult-flip N/A

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot \frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 76.4

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

         \[\leadsto \frac{\frac{\color{blue}{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   7. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 51.6

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

   10. Applied rewrites 51.6%

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

   if 1.55000000000000004 < x

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 75.7

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

      11. lift-hypot.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. mult-flip N/A

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot \frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}} \]

   7. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{{\left(\frac{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - 1}\right)}^{3} - {\left(\frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - 1}\right)}^{3}} \cdot \mathsf{fma}\left(\frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - 1}, \frac{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - 1}, {\left(\frac{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - 1}\right)}^{2}\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-sqrt.f64 50.2

         \[\leadsto -1 \cdot \left(\sqrt{0.5} - 1\right) \]

   10. Applied rewrites 50.2%

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

Alternative 10: 51.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*l/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 75.7

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

   11. lift-hypot.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. metadata-eval N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f64 75.7

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

3. Applied rewrites 75.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. mult-flip N/A

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

5. Applied rewrites 76.4%

   \[\leadsto \color{blue}{\frac{\left(\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot \frac{1}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip-rev N/A

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

   4. lower-/.f64 76.4

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right) \cdot \left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 76.4

      \[\leadsto \frac{\frac{\color{blue}{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

7. Applied rewrites 76.4%

   \[\leadsto \frac{\color{blue}{\frac{\left(0.5 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5\right)}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5} - -1}}}{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -1.5} \]

8. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. pow2 N/A

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

   3. lower-*.f64 51.6

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

10. Applied rewrites 51.6%

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

Reproduce

herbie shell --seed 2025136 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))