Given's Rotation SVD example, simplified

Percentage Accurate: 98.4% → 99.9%

Time: 20.0s

Alternatives: 8

Speedup: 0.7×

• could not determine a ground truth

  1. x = -2.5426008075914415e-226

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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \frac{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{t\_0}}{1 + \sqrt{t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (hypot 1.0 x)))))
   (/ (/ (- 0.25 (/ 0.25 (fma x x 1.0))) t_0) (+ 1.0 (sqrt t_0)))))

C

double code(double x) {
	double t_0 = 0.5 + (0.5 / hypot(1.0, x));
	return ((0.25 - (0.25 / fma(x, x, 1.0))) / t_0) / (1.0 + sqrt(t_0));
}

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(0.5 / hypot(1.0, x)))
	return Float64(Float64(Float64(0.25 - Float64(0.25 / fma(x, x, 1.0))) / t_0) / Float64(1.0 + sqrt(t_0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(0.25 - N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $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. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. flip-- 98.4%

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

   2. metadata-eval 98.4%

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

   3. add-sqr-sqrt 99.9%

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

   4. associate--r+ 99.9%

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

   5. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. add-sqr-sqrt 99.9%

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

   2. sqrt-unprod 99.9%

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

   3. frac-times 99.9%

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

   4. metadata-eval 99.9%

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

   5. hypot-undefine 99.9%

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

   6. hypot-undefine 99.9%

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

   7. rem-square-sqrt 99.9%

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

   8. metadata-eval 99.9%

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

   9. +-commutative 99.9%

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

   10. fma-define 99.9%

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

8. Applied egg-rr 99.9%

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

   1. sqrt-div 99.9%

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

   2. metadata-eval 99.9%

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

   3. fma-undefine 99.9%

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

   4. +-commutative 99.9%

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

   5. hypot-1-def 99.9%

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

   6. flip-- 99.9%

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

   7. metadata-eval 99.9%

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

   8. frac-times 99.9%

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

   9. metadata-eval 99.9%

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

   10. hypot-undefine 99.9%

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

   11. hypot-undefine 99.9%

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

   12. rem-square-sqrt 99.9%

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

   13. metadata-eval 99.9%

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

   14. +-commutative 99.9%

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

   15. fma-undefine 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

    \[\leadsto \frac{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
12. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 - \sqrt{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \end{array} \]

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(0.5 - sqrt(Float64(0.25 / fma(x, x, 1.0)))) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x))))))
end

Wolfram

code[x_] := N[(N[(0.5 - N[Sqrt[N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $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. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. flip-- 98.4%

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

   2. metadata-eval 98.4%

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

   3. add-sqr-sqrt 99.9%

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

   4. associate--r+ 99.9%

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

   5. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. add-sqr-sqrt 99.9%

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

   2. sqrt-unprod 99.9%

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

   3. frac-times 99.9%

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

   4. metadata-eval 99.9%

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

   5. hypot-undefine 99.9%

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

   6. hypot-undefine 99.9%

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

   7. rem-square-sqrt 99.9%

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

   8. metadata-eval 99.9%

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

   9. +-commutative 99.9%

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

   10. fma-define 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \frac{0.5 - \sqrt{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
10. Add Preprocessing

Alternative 3: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \frac{0.5 - t\_0}{1 + \sqrt{0.5 + t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (/ (- 0.5 t_0) (+ 1.0 (sqrt (+ 0.5 t_0))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	return (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	return (0.5 - t_0) / (1.0 + Math.sqrt((0.5 + t_0)));
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	return (0.5 - t_0) / (1.0 + math.sqrt((0.5 + t_0)))

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	return Float64(Float64(0.5 - t_0) / Float64(1.0 + sqrt(Float64(0.5 + t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. 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. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. flip-- 98.4%

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

   2. metadata-eval 98.4%

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

   3. add-sqr-sqrt 99.9%

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

   4. associate--r+ 99.9%

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

   5. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 4: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(0.5 - N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $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. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. flip-- 98.4%

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

   2. metadata-eval 98.4%

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

   3. add-sqr-sqrt 99.9%

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

   4. associate--r+ 99.9%

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

   5. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf 97.8%

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

8. Final simplification 97.8%

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

Alternative 5: 97.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. 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. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

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 96.1%

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

   1. flip-- 96.1%

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

   2. metadata-eval 96.1%

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

   3. add-sqr-sqrt 97.6%

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

   4. associate--r+ 97.6%

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

   5. metadata-eval 97.6%

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

7. Applied egg-rr 97.6%

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

8. Final simplification 97.6%

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

Alternative 6: 97.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.5 / N[(1.0 + N[Sqrt[0.5], $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. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. flip-- 98.4%

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

   2. metadata-eval 98.4%

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

   3. add-sqr-sqrt 99.9%

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

   4. associate--r+ 99.9%

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

   5. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf 97.0%

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

8. Final simplification 97.0%

   \[\leadsto \frac{0.5}{1 + \sqrt{0.5}} \]
9. Add Preprocessing

Alternative 7: 95.9% 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. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

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 95.5%

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

6. Final simplification 95.5%

   \[\leadsto 1 - \sqrt{0.5} \]
7. Add Preprocessing

Alternative 8: 17.4% accurate, 210.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

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. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

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 96.1%

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

6. Taylor expanded in x around 0 17.3%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 17.3%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024062 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))