Given's Rotation SVD example, simplified

Percentage Accurate: 98.4% → 98.4%

Time: 15.3s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. x = -1.5658202485529306e-107

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: 98.4% 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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

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

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.4%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

Alternative 2: 96.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 1 - \sqrt{0.5 + \frac{\frac{\frac{0.1875}{x \cdot x} - 0.25}{x \cdot x} - -0.5}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (+ 0.5 (/ (- (/ (- (/ 0.1875 (* x x)) 0.25) (* x x)) -0.5) x)))))

C

double code(double x) {
	return 1.0 - sqrt((0.5 + (((((0.1875 / (x * x)) - 0.25) / (x * x)) - -0.5) / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - sqrt((0.5d0 + (((((0.1875d0 / (x * x)) - 0.25d0) / (x * x)) - (-0.5d0)) / x)))
end function

Java

public static double code(double x) {
	return 1.0 - Math.sqrt((0.5 + (((((0.1875 / (x * x)) - 0.25) / (x * x)) - -0.5) / x)));
}

Python

def code(x):
	return 1.0 - math.sqrt((0.5 + (((((0.1875 / (x * x)) - 0.25) / (x * x)) - -0.5) / x)))

Julia

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 + Float64(Float64(Float64(Float64(Float64(0.1875 / Float64(x * x)) - 0.25) / Float64(x * x)) - -0.5) / x))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt((0.5 + (((((0.1875 / (x * x)) - 0.25) / (x * x)) - -0.5) / x)));
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 + N[(N[(N[(N[(N[(0.1875 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

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

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.4%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\left(\frac{1}{2} + \frac{\frac{3}{16}}{{x}^{4}}\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right)\right)\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} + \frac{\frac{3}{16}}{{x}^{4}}\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right), x\right)\right)\right)\right) \]

7. Simplified 97.1%

   \[\leadsto 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 + \left(\frac{0.1875}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{-0.25}{x \cdot x}\right)}{x}}} \]
8. Step-by-step derivation

   1. frac-2neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{\mathsf{neg}\left(\left(\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)\right)\right)}{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   2. distribute-frac-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)}{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} - \frac{\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)}{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)}{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) \]

   5. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)}{x}\right)\right)\right)\right)\right) \]

   6. distribute-frac-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\left(\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)\right)\right)}{x}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)\right)\right)\right), x\right)\right)\right)\right) \]

9. Applied egg-rr 97.1%

   \[\leadsto 1 - \sqrt{\color{blue}{0.5 - \frac{-0.5 - \frac{\frac{0.1875}{x \cdot x} - 0.25}{x \cdot x}}{x}}} \]

10. Final simplification 97.1%

    \[\leadsto 1 - \sqrt{0.5 + \frac{\frac{\frac{0.1875}{x \cdot x} - 0.25}{x \cdot x} - -0.5}{x}} \]
11. Add Preprocessing

Alternative 3: 96.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 1 - \sqrt{0.5 + \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (+ 0.5 (/ (+ 0.5 (/ -0.25 (* x x))) x)))))

C

double code(double x) {
	return 1.0 - sqrt((0.5 + ((0.5 + (-0.25 / (x * x))) / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - sqrt((0.5d0 + ((0.5d0 + ((-0.25d0) / (x * x))) / x)))
end function

Java

public static double code(double x) {
	return 1.0 - Math.sqrt((0.5 + ((0.5 + (-0.25 / (x * x))) / x)));
}

Python

def code(x):
	return 1.0 - math.sqrt((0.5 + ((0.5 + (-0.25 / (x * x))) / x)))

Julia

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 + Float64(Float64(0.5 + Float64(-0.25 / Float64(x * x))) / x))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt((0.5 + ((0.5 + (-0.25 / (x * x))) / x)));
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 + N[(N[(0.5 + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

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

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.4%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\color{blue}{\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{\frac{1}{4}}{{x}^{3}}\right)}\right)\right) \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{{x}^{3}}\right)\right)\right)\right) \]

   2. unpow3 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{\left(x \cdot x\right) \cdot x}\right)\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{{x}^{2} \cdot x}\right)\right)\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{\frac{1}{4}}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{\frac{1}{4} \cdot 1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{\frac{1}{2} \cdot 1}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right), x\right)\right)\right)\right) \]

7. Simplified 97.1%

   \[\leadsto 1 - \sqrt{\color{blue}{0.5 + \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}}} \]
8. Add Preprocessing

Alternative 4: 96.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 - \sqrt{0.5 + \frac{0.5}{x}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - sqrt((0.5d0 + (0.5d0 / x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

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

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.4%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right)}\right)\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{x}\right)\right)\right)\right) \]

   4. /-lowering-/.f64 97.0%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right)\right) \]

7. Simplified 97.0%

   \[\leadsto 1 - \sqrt{\color{blue}{0.5 + \frac{0.5}{x}}} \]
8. Add Preprocessing

Alternative 5: 95.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 - \sqrt{0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (sqrt 0.5)))

C

double code(double x) {
	return 1.0 - sqrt(0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - sqrt(0.5d0)
end function

Java

public static double code(double x) {
	return 1.0 - Math.sqrt(0.5);
}

Python

def code(x):
	return 1.0 - math.sqrt(0.5)

Julia

function code(x)
	return Float64(1.0 - sqrt(0.5))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt(0.5);
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

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

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.4%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

5. Taylor expanded in x around inf

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

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

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2}}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 96.4%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right) \]

7. Simplified 96.4%

   \[\leadsto \color{blue}{1 - \sqrt{0.5}} \]
8. Add Preprocessing

Alternative 6: 4.5% accurate, 42.0× 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 98.4%

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

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

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.4%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{8}, \color{blue}{\left({x}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{8}, \left(x \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f64 4.3%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

7. Simplified 4.3%

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

8. Final simplification 4.3%

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

Alternative 7: 3.1% accurate, 210.0× 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 98.4%

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

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

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.4%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
6. Step-by-step derivation

7. Simplified 3.1%

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

   1. metadata-eval 3.1%

      \[\leadsto 0 \]

9. Applied egg-rr 3.1%

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

Reproduce

herbie shell --seed 2024159 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))