Given's Rotation SVD example, simplified

Percentage Accurate: 98.4% → 99.8%

Time: 22.3s

Alternatives: 9

Speedup: 0.7×

• could not determine a ground truth

  1. x = -4.389156942927111e-99

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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 + t\_0\\ \frac{1 - {t\_1}^{3}}{1 + t\_1 \cdot \left(t\_0 + 1.5\right)} \cdot \frac{1}{1 + {\left(\sqrt[3]{t\_1}\right)}^{1.5}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))) (t_1 (+ 0.5 t_0)))
   (*
    (/ (- 1.0 (pow t_1 3.0)) (+ 1.0 (* t_1 (+ t_0 1.5))))
    (/ 1.0 (+ 1.0 (pow (cbrt t_1) 1.5))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	return ((1.0 - pow(t_1, 3.0)) / (1.0 + (t_1 * (t_0 + 1.5)))) * (1.0 / (1.0 + pow(cbrt(t_1), 1.5)));
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	return ((1.0 - Math.pow(t_1, 3.0)) / (1.0 + (t_1 * (t_0 + 1.5)))) * (1.0 / (1.0 + Math.pow(Math.cbrt(t_1), 1.5)));
}

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 + t_0)
	return Float64(Float64(Float64(1.0 - (t_1 ^ 3.0)) / Float64(1.0 + Float64(t_1 * Float64(t_0 + 1.5)))) * Float64(1.0 / Float64(1.0 + (cbrt(t_1) ^ 1.5))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(t$95$0 + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 1.5], $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. div-inv 98.4%

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

   3. metadata-eval 98.4%

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

   4. add-sqr-sqrt 99.9%

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

   5. associate--r+ 99.9%

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

   6. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. metadata-eval 99.9%

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

   2. associate--r+ 99.9%

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

   3. flip3-- 99.9%

      \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 \cdot 1 + \left(\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) + 1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   4. metadata-eval 99.9%

      \[\leadsto \frac{\color{blue}{1} - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 \cdot 1 + \left(\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) + 1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   5. metadata-eval 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\color{blue}{1} + \left(\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) + 1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   6. pow2 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\color{blue}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2}} + 1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   7. *-un-lft-identity 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

8. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
9. Step-by-step derivation

   1. unpow2 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} + \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   2. distribute-lft1-in 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \color{blue}{\left(\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) + 1\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   3. +-commutative 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\color{blue}{\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 0.5\right)} + 1\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   4. associate-+l+ 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \color{blue}{\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(0.5 + 1\right)\right)} \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   5. metadata-eval 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \color{blue}{1.5}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

10. Simplified 99.9%

    \[\leadsto \color{blue}{\frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
11. Step-by-step derivation

    1. pow1/2 99.9%

       \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \color{blue}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.5}}} \]

    2. add-cube-cbrt 100.0%

       \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + {\color{blue}{\left(\left(\sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}}^{0.5}} \]

    3. pow3 100.0%

       \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + {\color{blue}{\left({\left(\sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right)}}^{0.5}} \]

    4. pow-pow 100.0%

       \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \color{blue}{{\left(\sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{\left(3 \cdot 0.5\right)}}} \]

    5. metadata-eval 100.0%

       \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + {\left(\sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{\color{blue}{1.5}}} \]

12. Applied egg-rr 100.0%

    \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \color{blue}{{\left(\sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{1.5}}} \]

13. Final simplification 100.0%

    \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \cdot \frac{1}{1 + {\left(\sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{1.5}} \]
14. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 + t\_0\\ \frac{\frac{1 - {t\_1}^{3}}{1 + t\_1 \cdot \left(t\_0 + 1.5\right)}}{1 + \sqrt{t\_1}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))) (t_1 (+ 0.5 t_0)))
   (/
    (/ (- 1.0 (pow t_1 3.0)) (+ 1.0 (* t_1 (+ t_0 1.5))))
    (+ 1.0 (sqrt t_1)))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	return ((1.0 - pow(t_1, 3.0)) / (1.0 + (t_1 * (t_0 + 1.5)))) / (1.0 + sqrt(t_1));
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	return ((1.0 - Math.pow(t_1, 3.0)) / (1.0 + (t_1 * (t_0 + 1.5)))) / (1.0 + Math.sqrt(t_1));
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	t_1 = 0.5 + t_0
	return ((1.0 - math.pow(t_1, 3.0)) / (1.0 + (t_1 * (t_0 + 1.5)))) / (1.0 + math.sqrt(t_1))

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 + t_0)
	return Float64(Float64(Float64(1.0 - (t_1 ^ 3.0)) / Float64(1.0 + Float64(t_1 * Float64(t_0 + 1.5)))) / Float64(1.0 + sqrt(t_1)))
end

MATLAB

function tmp = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	t_1 = 0.5 + t_0;
	tmp = ((1.0 - (t_1 ^ 3.0)) / (1.0 + (t_1 * (t_0 + 1.5)))) / (1.0 + sqrt(t_1));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(t$95$0 + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$1], $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. metadata-eval 99.9%

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

   2. associate--r+ 99.9%

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

   3. flip3-- 99.9%

      \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 \cdot 1 + \left(\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) + 1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   4. metadata-eval 99.9%

      \[\leadsto \frac{\color{blue}{1} - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 \cdot 1 + \left(\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) + 1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   5. metadata-eval 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\color{blue}{1} + \left(\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) + 1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   6. pow2 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\color{blue}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2}} + 1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   7. *-un-lft-identity 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
9. Step-by-step derivation

   1. unpow2 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} + \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   2. distribute-lft1-in 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \color{blue}{\left(\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) + 1\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   3. +-commutative 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\color{blue}{\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 0.5\right)} + 1\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   4. associate-+l+ 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \color{blue}{\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(0.5 + 1\right)\right)} \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   5. metadata-eval 99.9%

      \[\leadsto \frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \color{blue}{1.5}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

10. Simplified 100.0%

    \[\leadsto \frac{\color{blue}{\frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

11. Final simplification 100.0%

    \[\leadsto \frac{\frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 + \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
12. Add Preprocessing

Alternative 3: 99.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (* (- 0.5 t_0) (/ 1.0 (+ 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 / (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 / (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 / (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 / 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 / (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[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $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. div-inv 98.4%

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

   3. metadata-eval 98.4%

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

   4. add-sqr-sqrt 99.9%

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

   5. associate--r+ 99.9%

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

   6. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 4: 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. Add Preprocessing

Alternative 5: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (0.5 - (0.5 / hypot(1.0, x))) * (1.0 / (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 / (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 / (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 / 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 / (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[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $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. div-inv 98.4%

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

   3. metadata-eval 98.4%

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

   4. add-sqr-sqrt 99.9%

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

   5. associate--r+ 99.9%

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

   6. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf 98.4%

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

   1. associate-*r/ 98.4%

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

   2. metadata-eval 98.4%

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

9. Simplified 98.4%

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

Alternative 6: 97.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 / x), $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 96.6%

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

   1. associate-*r/ 98.4%

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

   2. metadata-eval 98.4%

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

7. Simplified 96.6%

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

   1. flip-- 96.6%

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

   2. div-inv 96.6%

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

   3. metadata-eval 96.6%

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

   4. add-sqr-sqrt 98.1%

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

9. Applied egg-rr 98.1%

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

    1. associate--r+ 98.1%

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

    2. metadata-eval 98.1%

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

11. Simplified 98.1%

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

12. Final simplification 98.1%

    \[\leadsto \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{x}}} \cdot \left(0.5 - \frac{0.5}{x}\right) \]
13. Add Preprocessing

Alternative 7: 97.2% 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. div-inv 98.4%

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

   3. metadata-eval 98.4%

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

   4. add-sqr-sqrt 99.9%

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

   5. associate--r+ 99.9%

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

   6. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf 97.5%

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

Alternative 8: 95.7% 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 96.0%

   \[\leadsto \color{blue}{1 - \sqrt{0.5}} \]
6. 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 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 0 3.1%

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

6. Final simplification 3.1%

   \[\leadsto 0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024099 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))