Given's Rotation SVD example, simplified

Percentage Accurate: 76.1% → 99.8%

Time: 9.1s

Alternatives: 10

Speedup: 1.0×

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: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (sqrt (fma x x 1.0)))))
   (if (<= (hypot 1.0 x) 2.0)
     (* (* 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) <= 2.0) {
		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) <= 2.0)
		tmp = Float64(Float64(x * x) * fma(Float64(x * 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], 2.0], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.0673828125), $MachinePrecision] + -0.0859375), $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) < 2

   1. Initial program 49.1%

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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      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. lower-*.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. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. mul-1-neg N/A

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

      12. lift-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

   6. Applied rewrites 100.0%

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

Alternative 2: 98.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 53.6%

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

   4. Applied rewrites 53.6%

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

   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. lower-*.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. lower-*.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. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   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

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 96.8

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

   5. Applied rewrites 96.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 98.2

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

   10. Applied rewrites 98.2%

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

Reproduce

herbie shell --seed 2024230 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))