Given's Rotation SVD example, simplified

Percentage Accurate: 76.0% → 99.9%

Time: 11.5s

Alternatives: 15

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: 76.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.005:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.056243896484375, 0.0673828125\right), -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{1 + \sqrt{t\_0}}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (sqrt (fma x x 1.0))))))
   (if (<= (hypot 1.0 x) 1.005)
     (*
      (* x x)
      (fma
       (* x x)
       (fma (* x x) (fma (* x x) -0.056243896484375 0.0673828125) -0.0859375)
       0.125))
     (/ (/ (- 0.25 (/ 0.25 (fma x x 1.0))) (+ 1.0 (sqrt t_0))) t_0))))

C

double code(double x) {
	double t_0 = 0.5 + (0.5 / sqrt(fma(x, x, 1.0)));
	double tmp;
	if (hypot(1.0, x) <= 1.005) {
		tmp = (x * x) * fma((x * x), fma((x * x), fma((x * x), -0.056243896484375, 0.0673828125), -0.0859375), 0.125);
	} else {
		tmp = ((0.25 - (0.25 / fma(x, x, 1.0))) / (1.0 + sqrt(t_0))) / t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(0.5 / sqrt(fma(x, x, 1.0))))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.005)
		tmp = Float64(Float64(x * x) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.056243896484375, 0.0673828125), -0.0859375), 0.125));
	else
		tmp = Float64(Float64(Float64(0.25 - Float64(0.25 / fma(x, x, 1.0))) / Float64(1.0 + sqrt(t_0))) / t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.005], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 - N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0049999999999999

   1. Initial program 54.3%

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

   4. Applied rewrites 54.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 1.0049999999999999 < (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 rewrites 99.8%

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

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\left(\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + -0.25\right) \cdot \frac{-1}{1 + \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}}{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}}}{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.005:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.056243896484375, 0.0673828125\right), -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot \frac{1}{t\_0 \cdot \left(1 + \sqrt{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (sqrt (fma x x 1.0))))))
   (if (<= (hypot 1.0 x) 1.005)
     (*
      (* x x)
      (fma
       (* x x)
       (fma (* x x) (fma (* x x) -0.056243896484375 0.0673828125) -0.0859375)
       0.125))
     (* (- 0.25 (/ 0.25 (fma x x 1.0))) (/ 1.0 (* t_0 (+ 1.0 (sqrt t_0))))))))

C

double code(double x) {
	double t_0 = 0.5 + (0.5 / sqrt(fma(x, x, 1.0)));
	double tmp;
	if (hypot(1.0, x) <= 1.005) {
		tmp = (x * x) * fma((x * x), fma((x * x), fma((x * x), -0.056243896484375, 0.0673828125), -0.0859375), 0.125);
	} else {
		tmp = (0.25 - (0.25 / fma(x, x, 1.0))) * (1.0 / (t_0 * (1.0 + sqrt(t_0))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(0.5 / sqrt(fma(x, x, 1.0))))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.005)
		tmp = Float64(Float64(x * x) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.056243896484375, 0.0673828125), -0.0859375), 0.125));
	else
		tmp = Float64(Float64(0.25 - Float64(0.25 / fma(x, x, 1.0))) * Float64(1.0 / Float64(t_0 * Float64(1.0 + sqrt(t_0)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.005], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 - N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$0 * N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0049999999999999

   1. Initial program 53.4%

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 1.0049999999999999 < (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%

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

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\left(\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + -0.25\right) \cdot \frac{-1}{1 + \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}}{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}} \]

   8. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot \frac{1}{\left(0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot \left(1 + \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))