Given's Rotation SVD example, simplified

Percentage Accurate: 98.4% → 98.4%

Time: 16.5s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. x = -3.7973225050843147e-163

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 98.3%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.3%

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

3. Simplified 98.3%

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

Alternative 2: 96.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.3%

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

3. Simplified 98.3%

   \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
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 96.7%

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

Alternative 3: 96.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 1 + \sqrt{0.5} \cdot \left(-1 + \frac{-0.5 + \frac{0.125}{x}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ 1.0 (* (sqrt 0.5) (+ -1.0 (/ (+ -0.5 (/ 0.125 x)) x)))))

C

double code(double x) {
	return 1.0 + (sqrt(0.5) * (-1.0 + ((-0.5 + (0.125 / x)) / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + (sqrt(0.5d0) * ((-1.0d0) + (((-0.5d0) + (0.125d0 / x)) / x)))
end function

Java

public static double code(double x) {
	return 1.0 + (Math.sqrt(0.5) * (-1.0 + ((-0.5 + (0.125 / x)) / x)));
}

Python

def code(x):
	return 1.0 + (math.sqrt(0.5) * (-1.0 + ((-0.5 + (0.125 / x)) / x)))

Julia

function code(x)
	return Float64(1.0 + Float64(sqrt(0.5) * Float64(-1.0 + Float64(Float64(-0.5 + Float64(0.125 / x)) / x))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (sqrt(0.5) * (-1.0 + ((-0.5 + (0.125 / x)) / x)));
end

Wolfram

code[x_] := N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[(-1.0 + N[(N[(-0.5 + N[(0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.3%

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

3. Simplified 98.3%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. pow1/2 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. distribute-rgt1-in N/A

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

   7. unpow-prod-down N/A

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

   8. distribute-lft-neg-in N/A

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

   9. fma-define N/A

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

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

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

6. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0 - \sqrt{1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}}, \sqrt{0.5}, 1\right)} \]
7. Step-by-step derivation

   1. sub0-neg N/A

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

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

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

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

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

   4. remove-double-neg N/A

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

   5. distribute-frac-neg2 N/A

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

   6. unsub-neg N/A

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

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

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

   8. metadata-eval N/A

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

   9. frac-2neg N/A

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

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

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

   11. hypot-undefine N/A

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

   12. hypot-lowering-hypot.f64 98.3%

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

8. Applied egg-rr 98.3%

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

9. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\left(\frac{\frac{1}{8}}{{x}^{2}} + \frac{3}{16} \cdot \frac{1}{{x}^{3}}\right) - \left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}, \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]
10. Step-by-step derivation

    1. associate--l+ N/A

       \[\leadsto \mathsf{fma.f64}\left(\left(\frac{\frac{1}{8}}{{x}^{2}} + \left(\frac{3}{16} \cdot \frac{1}{{x}^{3}} - \left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\frac{1}{2}}\right), 1\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{8}}{{x}^{2}}\right), \left(\frac{3}{16} \cdot \frac{1}{{x}^{3}} - \left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\frac{1}{2}}\right), 1\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \left({x}^{2}\right)\right), \left(\frac{3}{16} \cdot \frac{1}{{x}^{3}} - \left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \left(x \cdot x\right)\right), \left(\frac{3}{16} \cdot \frac{1}{{x}^{3}} - \left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{3}{16} \cdot \frac{1}{{x}^{3}} - \left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

    6. sub-neg N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{3}{16} \cdot \frac{1}{{x}^{3}} + \left(\mathsf{neg}\left(\left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(\frac{3}{16} \cdot \frac{1}{{x}^{3}}\right), \left(\mathsf{neg}\left(\left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

    8. associate-*r/ N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{3}{16} \cdot 1}{{x}^{3}}\right), \left(\mathsf{neg}\left(\left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

    9. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{3}{16}}{{x}^{3}}\right), \left(\mathsf{neg}\left(\left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

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

        \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{16}, \left({x}^{3}\right)\right), \left(\mathsf{neg}\left(\left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

    11. cube-mult N/A

        \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{16}, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{16}, \left(x \cdot {x}^{2}\right)\right), \left(\mathsf{neg}\left(\left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

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

        \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{16}, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

    14. unpow2 N/A

        \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{16}, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

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

        \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{16}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

    16. distribute-neg-in N/A

        \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{16}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

    17. metadata-eval N/A

        \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{16}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(-1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

    18. unsub-neg N/A

        \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{16}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(-1 - \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

    19. --lowering--.f64 N/A

        \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{16}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{\_.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

    20. associate-*r/ N/A

        \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{16}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{\_.f64}\left(-1, \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

    21. metadata-eval N/A

        \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{16}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{\_.f64}\left(-1, \left(\frac{\frac{1}{2}}{x}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

    22. /-lowering-/.f64 96.8%

        \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{16}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\frac{1}{2}\right), 1\right) \]

11. Simplified 96.8%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.125}{x \cdot x} + \left(\frac{0.1875}{x \cdot \left(x \cdot x\right)} + \left(-1 - \frac{0.5}{x}\right)\right)}, \sqrt{0.5}, 1\right) \]

12. Taylor expanded in x around inf

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

14. Simplified 96.8%

    \[\leadsto \color{blue}{1 + \sqrt{0.5} \cdot \left(-1 + \frac{-0.5 + \frac{0.125}{x}}{x}\right)} \]
15. 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 98.3%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.3%

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

3. Simplified 98.3%

   \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
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 96.7%

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

7. Simplified 96.7%

   \[\leadsto 1 - \sqrt{\color{blue}{0.5 + \frac{0.5}{x}}} \]
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 98.3%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.3%

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

3. Simplified 98.3%

   \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
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 96.3%

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

7. Simplified 96.3%

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

Alternative 6: 4.5% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ 1 - \left(1 + x \cdot \left(x \cdot -0.125\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (+ 1.0 (* x (* x -0.125)))))

C

double code(double x) {
	return 1.0 - (1.0 + (x * (x * -0.125)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (1.0d0 + (x * (x * (-0.125d0))))
end function

Java

public static double code(double x) {
	return 1.0 - (1.0 + (x * (x * -0.125)));
}

Python

def code(x):
	return 1.0 - (1.0 + (x * (x * -0.125)))

Julia

function code(x)
	return Float64(1.0 - Float64(1.0 + Float64(x * Float64(x * -0.125))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (1.0 + (x * (x * -0.125)));
end

Wolfram

code[x_] := N[(1.0 - N[(1.0 + N[(x * N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(1 + \frac{-1}{8} \cdot {x}^{2}\right)}\right) \]
6. Step-by-step derivation

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

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

   2. *-commutative N/A

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

   3. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right) \]

   4. associate-*l* N/A

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

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

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

   6. *-lowering-*.f64 4.3%

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

7. Simplified 4.3%

   \[\leadsto 1 - \color{blue}{\left(1 + x \cdot \left(x \cdot -0.125\right)\right)} \]
8. Add Preprocessing

Alternative 7: 4.5% accurate, 23.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{8}, \color{blue}{\left(\frac{-11}{128} \cdot {x}^{2}\right)}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{8}, \left({x}^{2} \cdot \color{blue}{\frac{-11}{128}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-11}{128}}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-11}{128}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 1.3%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-11}{128}\right)\right)\right)\right) \]

7. Simplified 1.3%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 4.3%

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

10. Simplified 4.3%

    \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.125} \]
11. Step-by-step derivation

    1. *-commutative N/A

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

    2. pow2 N/A

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

    3. metadata-eval N/A

       \[\leadsto \frac{1}{8} \cdot {x}^{\left(-1 + \color{blue}{3}\right)} \]

    4. pow-prod-up N/A

       \[\leadsto \frac{1}{8} \cdot \left({x}^{-1} \cdot \color{blue}{{x}^{3}}\right) \]

    5. inv-pow N/A

       \[\leadsto \frac{1}{8} \cdot \left(\frac{1}{x} \cdot {\color{blue}{x}}^{3}\right) \]

    6. cube-unmult N/A

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

    7. associate-*l* N/A

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

    8. div-inv N/A

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

    9. associate-*r* N/A

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

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

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

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

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

    12. /-lowering-/.f64 N/A

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

    13. *-lowering-*.f64 4.3%

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

12. Applied egg-rr 4.3%

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

13. Final simplification 4.3%

    \[\leadsto \left(x \cdot x\right) \cdot \left(x \cdot \frac{0.125}{x}\right) \]
14. Add Preprocessing

Alternative 8: 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 98.3%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.3%

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

3. Simplified 98.3%

   \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
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 9: 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 98.3%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.3%

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

3. Simplified 98.3%

   \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
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 2024164 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))