Given's Rotation SVD example, simplified

Percentage Accurate: 76.4% → 99.9%

Time: 10.5s

Alternatives: 10

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.001:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0673828125, -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, x, 1\right)}}, 0.5, -0.5\right)}{-1 - \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.001)
   (* (* x x) (fma x (* x (fma x (* x 0.0673828125) -0.0859375)) 0.125))
   (/
    (fma (sqrt (/ 1.0 (fma x x 1.0))) 0.5 -0.5)
    (- -1.0 (sqrt (+ 0.5 (/ 0.5 (sqrt (fma x x 1.0)))))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.001) {
		tmp = (x * x) * fma(x, (x * fma(x, (x * 0.0673828125), -0.0859375)), 0.125);
	} else {
		tmp = fma(sqrt((1.0 / fma(x, x, 1.0))), 0.5, -0.5) / (-1.0 - sqrt((0.5 + (0.5 / sqrt(fma(x, x, 1.0))))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.001)
		tmp = Float64(Float64(x * x) * fma(x, Float64(x * fma(x, Float64(x * 0.0673828125), -0.0859375)), 0.125));
	else
		tmp = Float64(fma(sqrt(Float64(1.0 / fma(x, x, 1.0))), 0.5, -0.5) / Float64(-1.0 - sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(x, x, 1.0)))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.001], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision]), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] / N[(-1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0009999999999999

   1. Initial program 47.4%

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

   4. Applied egg-rr 47.4%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 100.0

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

   7. Simplified 100.0%

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

   if 1.0009999999999999 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied egg-rr 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      4. sqrt-div N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. accelerator-lowering-fma.f64 99.9

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

   6. Applied egg-rr 99.9%

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

      1. unsub-neg N/A

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

      2. +-commutative N/A

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

      3. --lowering--.f64 N/A

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

      4. +-commutative N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. accelerator-lowering-fma.f64 99.9

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

   8. Applied egg-rr 99.9%

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

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.001:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0673828125, -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 + t\_0}{-1 - \sqrt{0.5 + t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (sqrt (fma x x 1.0)))))
   (if (<= (hypot 1.0 x) 1.001)
     (* (* x x) (fma x (* x (fma x (* x 0.0673828125) -0.0859375)) 0.125))
     (/ (+ -0.5 t_0) (- -1.0 (sqrt (+ 0.5 t_0)))))))

C

double code(double x) {
	double t_0 = 0.5 / sqrt(fma(x, x, 1.0));
	double tmp;
	if (hypot(1.0, x) <= 1.001) {
		tmp = (x * x) * fma(x, (x * fma(x, (x * 0.0673828125), -0.0859375)), 0.125);
	} else {
		tmp = (-0.5 + t_0) / (-1.0 - sqrt((0.5 + t_0)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / sqrt(fma(x, x, 1.0)))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.001)
		tmp = Float64(Float64(x * x) * fma(x, Float64(x * fma(x, Float64(x * 0.0673828125), -0.0859375)), 0.125));
	else
		tmp = Float64(Float64(-0.5 + t_0) / Float64(-1.0 - sqrt(Float64(0.5 + t_0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.001], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision]), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 + t$95$0), $MachinePrecision] / N[(-1.0 - N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0009999999999999

   1. Initial program 47.4%

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

   4. Applied egg-rr 47.4%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 100.0

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

   7. Simplified 100.0%

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

   if 1.0009999999999999 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied egg-rr 99.9%

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

      1. unsub-neg N/A

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

      2. --lowering--.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. rem-square-sqrt N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. sqr-neg N/A

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.001:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0673828125, -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{-1 - \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.001:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0673828125, -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, x, 1\right)}}, 0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.001)
   (* (* x x) (fma x (* x (fma x (* x 0.0673828125) -0.0859375)) 0.125))
   (- 1.0 (sqrt (fma (sqrt (/ 1.0 (fma x x 1.0))) 0.5 0.5)))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.001) {
		tmp = (x * x) * fma(x, (x * fma(x, (x * 0.0673828125), -0.0859375)), 0.125);
	} else {
		tmp = 1.0 - sqrt(fma(sqrt((1.0 / fma(x, x, 1.0))), 0.5, 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.001)
		tmp = Float64(Float64(x * x) * fma(x, Float64(x * fma(x, Float64(x * 0.0673828125), -0.0859375)), 0.125));
	else
		tmp = Float64(1.0 - sqrt(fma(sqrt(Float64(1.0 / fma(x, x, 1.0))), 0.5, 0.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.001], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision]), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[Sqrt[N[(1.0 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0009999999999999

   1. Initial program 47.4%

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

   4. Applied egg-rr 47.4%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 100.0

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

   7. Simplified 100.0%

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

   if 1.0009999999999999 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 98.4%

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

Alternative 4: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.001:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0673828125, -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.001)
   (* (* x x) (fma x (* x (fma x (* x 0.0673828125) -0.0859375)) 0.125))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (sqrt (fma x x 1.0))))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.001) {
		tmp = (x * x) * fma(x, (x * fma(x, (x * 0.0673828125), -0.0859375)), 0.125);
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / sqrt(fma(x, x, 1.0)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.001)
		tmp = Float64(Float64(x * x) * fma(x, Float64(x * fma(x, Float64(x * 0.0673828125), -0.0859375)), 0.125));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(x, x, 1.0))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.001], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision]), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0009999999999999

   1. Initial program 47.4%

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

   4. Applied egg-rr 47.4%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 100.0

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

   7. Simplified 100.0%

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

   if 1.0009999999999999 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. accelerator-lowering-fma.f64 98.4

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

   4. Applied egg-rr 98.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.001:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0673828125, -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.2%

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

   4. Applied egg-rr 48.2%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 98.9

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

   7. Simplified 98.9%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 95.8%

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

      1. flip-- N/A

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

      2. metadata-eval N/A

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

      3. rem-square-sqrt N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 97.3

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

   7. Applied egg-rr 97.3%

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

Alternative 6: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.2%

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

   4. Applied egg-rr 48.2%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 98.8

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

   7. Simplified 98.8%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 95.8%

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

      1. flip-- N/A

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

      2. metadata-eval N/A

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

      3. rem-square-sqrt N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 97.3

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

   7. Applied egg-rr 97.3%

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

Alternative 7: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.2%

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

   4. Applied egg-rr 48.2%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 98.8

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

   7. Simplified 98.8%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 95.8%

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

Alternative 8: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.2%

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

   4. Applied egg-rr 48.2%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 98.1

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

   7. Simplified 98.1%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 95.8%

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

4. Final simplification 96.9%

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

Alternative 9: 50.9% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

4. Applied egg-rr 76.1%

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

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 47.4

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

7. Simplified 47.4%

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

8. Final simplification 47.4%

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

Alternative 10: 27.4% accurate, 134.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 75.3%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. metadata-eval N/A

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

   4. +-lowering-+.f64 N/A

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

   5. associate-*l/ N/A

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

   6. metadata-eval N/A

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

   7. /-lowering-/.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. sqrt-lowering-sqrt.f64 N/A

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

   10. rem-square-sqrt N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

   13. accelerator-lowering-fma.f64 75.3

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

4. Applied egg-rr 75.3%

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

5. Taylor expanded in x around 0

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

7. Simplified 22.8%

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

   1. metadata-eval 22.8

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

9. Applied egg-rr 22.8%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))