Given's Rotation SVD example, simplified

Percentage Accurate: 99.2% → 99.9%

Time: 11.3s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. x = -5.5278236200447584e-8

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: 99.2% 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

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

FPCore

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

C

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

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.cbrt(Math.pow((0.5 + t_0), 1.5)));
}

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	return Float64(Float64(0.5 - t_0) / Float64(1.0 + cbrt((Float64(0.5 + t_0) ^ 1.5))))
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[Power[N[Power[N[(0.5 + t$95$0), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.1%

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

   1. distribute-lft-in 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-*r/ 99.1%

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

   4. metadata-eval 99.1%

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

3. Simplified 99.1%

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

   1. flip-- 99.1%

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

   2. metadata-eval 99.1%

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

   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-cbrt-cube 99.9%

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

   2. pow3 99.9%

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

   3. sqrt-pow2 99.9%

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

   4. metadata-eval 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	tmp = 1.0 / ((1.0 + sqrt((0.5 + t_0))) / (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[(1.0 / N[(N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.1%

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

   1. distribute-lft-in 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-*r/ 99.1%

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

   4. metadata-eval 99.1%

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

3. Simplified 99.1%

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

   1. flip-- 99.1%

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

   2. div-inv 99.1%

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

   3. metadata-eval 99.1%

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

   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. *-commutative 99.9%

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

   2. associate-/r/ 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \frac{1}{\frac{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
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 99.1%

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

   1. distribute-lft-in 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-*r/ 99.1%

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

   4. metadata-eval 99.1%

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

3. Simplified 99.1%

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

   1. flip-- 99.1%

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

   2. metadata-eval 99.1%

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

   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: 99.2% 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 99.1%

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

   1. distribute-lft-in 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-*r/ 99.1%

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

   4. metadata-eval 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 5: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;1 + \left(-1 - {x}^{2} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.2)
   (+ 1.0 (- -1.0 (* (pow x 2.0) -0.125)))
   (/ (- 0.5 (/ 0.5 x)) (+ 1.0 (sqrt 0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = 1.0 + (-1.0 - (pow(x, 2.0) * -0.125));
	} else {
		tmp = (0.5 - (0.5 / x)) / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.2d0) then
        tmp = 1.0d0 + ((-1.0d0) - ((x ** 2.0d0) * (-0.125d0)))
    else
        tmp = (0.5d0 - (0.5d0 / x)) / (1.0d0 + sqrt(0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = 1.0 + (-1.0 - (Math.pow(x, 2.0) * -0.125));
	} else {
		tmp = (0.5 - (0.5 / x)) / (1.0 + Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.2:
		tmp = 1.0 + (-1.0 - (math.pow(x, 2.0) * -0.125))
	else:
		tmp = (0.5 - (0.5 / x)) / (1.0 + math.sqrt(0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.2)
		tmp = Float64(1.0 + Float64(-1.0 - Float64((x ^ 2.0) * -0.125)));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / x)) / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.2)
		tmp = 1.0 + (-1.0 - ((x ^ 2.0) * -0.125));
	else
		tmp = (0.5 - (0.5 / x)) / (1.0 + sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.2], N[(1.0 + N[(-1.0 - N[(N[Power[x, 2.0], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 99.4%

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

      1. distribute-lft-in 99.4%

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

      2. metadata-eval 99.4%

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

      3. associate-*r/ 99.4%

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

      4. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 66.0%

      \[\leadsto 1 - \color{blue}{\left(1 + -0.125 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto 1 - \left(1 + \color{blue}{{x}^{2} \cdot -0.125}\right) \]

   7. Simplified 66.0%

      \[\leadsto 1 - \color{blue}{\left(1 + {x}^{2} \cdot -0.125\right)} \]

   if 1.19999999999999996 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   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 100.0%

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

      4. associate--r+ 100.0%

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

      5. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

   10. Taylor expanded in x around inf 98.3%

       \[\leadsto \frac{0.5 - \frac{0.5}{x}}{\color{blue}{1 + \sqrt{0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;1 + \left(-1 - {x}^{2} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + \sqrt{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;1 + \left(-1 - {x}^{2} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55)
   (+ 1.0 (- -1.0 (* (pow x 2.0) -0.125)))
   (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = 1.0 + (-1.0 - (pow(x, 2.0) * -0.125));
	} else {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.55d0) then
        tmp = 1.0d0 + ((-1.0d0) - ((x ** 2.0d0) * (-0.125d0)))
    else
        tmp = 0.5d0 / (1.0d0 + sqrt(0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = 1.0 + (-1.0 - (Math.pow(x, 2.0) * -0.125));
	} else {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = 1.0 + (-1.0 - (math.pow(x, 2.0) * -0.125))
	else:
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = Float64(1.0 + Float64(-1.0 - Float64((x ^ 2.0) * -0.125)));
	else
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.55)
		tmp = 1.0 + (-1.0 - ((x ^ 2.0) * -0.125));
	else
		tmp = 0.5 / (1.0 + sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], N[(1.0 + N[(-1.0 - N[(N[Power[x, 2.0], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 99.4%

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

      1. distribute-lft-in 99.4%

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

      2. metadata-eval 99.4%

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

      3. associate-*r/ 99.4%

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

      4. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 66.0%

      \[\leadsto 1 - \color{blue}{\left(1 + -0.125 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto 1 - \left(1 + \color{blue}{{x}^{2} \cdot -0.125}\right) \]

   7. Simplified 66.0%

      \[\leadsto 1 - \color{blue}{\left(1 + {x}^{2} \cdot -0.125\right)} \]

   if 1.55000000000000004 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   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 100.0%

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

      5. associate--r+ 100.0%

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

      6. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 98.2%

      \[\leadsto \color{blue}{\frac{0.5}{1 + \sqrt{0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;1 + \left(-1 - {x}^{2} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-11}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e-11) 0.0 (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 1e-11) {
		tmp = 0.0;
	} else {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d-11) then
        tmp = 0.0d0
    else
        tmp = 0.5d0 / (1.0d0 + sqrt(0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e-11) {
		tmp = 0.0;
	} else {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e-11:
		tmp = 0.0
	else:
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e-11)
		tmp = 0.0;
	else
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e-11)
		tmp = 0.0;
	else
		tmp = 0.5 / (1.0 + sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e-11], 0.0, N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999939e-12

   1. Initial program 99.4%

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

      1. distribute-lft-in 99.4%

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

      2. metadata-eval 99.4%

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

      3. associate-*r/ 99.4%

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

      4. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 65.1%

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

   if 9.99999999999999939e-12 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   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 100.0%

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

      5. associate--r+ 100.0%

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

      6. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 98.2%

      \[\leadsto \color{blue}{\frac{0.5}{1 + \sqrt{0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-11}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-11}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1e-11) 0.0 (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1e-11) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d-11) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e-11) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e-11:
		tmp = 0.0
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e-11)
		tmp = 0.0;
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e-11)
		tmp = 0.0;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e-11], 0.0, N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999939e-12

   1. Initial program 99.4%

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

      1. distribute-lft-in 99.4%

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

      2. metadata-eval 99.4%

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

      3. associate-*r/ 99.4%

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

      4. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 65.1%

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

   if 9.99999999999999939e-12 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around inf 96.7%

      \[\leadsto \color{blue}{1 - \sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-11}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.3% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.25 - \frac{0.25}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) 0.0 (- 0.25 (/ 0.25 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 0.0;
	} else {
		tmp = 0.25 - (0.25 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 0.0d0
    else
        tmp = 0.25d0 - (0.25d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 0.0;
	} else {
		tmp = 0.25 - (0.25 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 0.0
	else:
		tmp = 0.25 - (0.25 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = 0.0;
	else
		tmp = Float64(0.25 - Float64(0.25 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 0.0;
	else
		tmp = 0.25 - (0.25 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], 0.0, N[(0.25 - N[(0.25 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.4%

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

      1. distribute-lft-in 99.4%

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

      2. metadata-eval 99.4%

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

      3. associate-*r/ 99.4%

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

      4. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 65.1%

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

   if 1 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   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 100.0%

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

      4. associate--r+ 100.0%

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

      5. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

   10. Taylor expanded in x around 0 22.7%

       \[\leadsto \color{blue}{0.25 - 0.25 \cdot \frac{1}{x}} \]
   11. Step-by-step derivation

       1. associate-*r/ 22.7%

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

       2. metadata-eval 22.7%

          \[\leadsto 0.25 - \frac{\color{blue}{0.25}}{x} \]

   12. Simplified 22.7%

       \[\leadsto \color{blue}{0.25 - \frac{0.25}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.25 - \frac{0.25}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.5% accurate, 34.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-11}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.18181818181818182\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1e-11) 0.0 0.18181818181818182))

C

double code(double x) {
	double tmp;
	if (x <= 1e-11) {
		tmp = 0.0;
	} else {
		tmp = 0.18181818181818182;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d-11) then
        tmp = 0.0d0
    else
        tmp = 0.18181818181818182d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e-11) {
		tmp = 0.0;
	} else {
		tmp = 0.18181818181818182;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e-11:
		tmp = 0.0
	else:
		tmp = 0.18181818181818182
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e-11)
		tmp = 0.0;
	else
		tmp = 0.18181818181818182;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e-11)
		tmp = 0.0;
	else
		tmp = 0.18181818181818182;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e-11], 0.0, 0.18181818181818182]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999939e-12

   1. Initial program 99.4%

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

      1. distribute-lft-in 99.4%

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

      2. metadata-eval 99.4%

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

      3. associate-*r/ 99.4%

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

      4. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 65.1%

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

   if 9.99999999999999939e-12 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   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 100.0%

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

      5. associate--r+ 100.0%

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

      6. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 19.5%

      \[\leadsto \frac{1}{\color{blue}{5.5 + 8 \cdot \frac{1}{{x}^{2}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 19.5%

          \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{8 \cdot 1}{{x}^{2}}}} \]

       2. metadata-eval 19.5%

          \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{8}}{{x}^{2}}} \]

   11. Simplified 19.5%

       \[\leadsto \frac{1}{\color{blue}{5.5 + \frac{8}{{x}^{2}}}} \]

   12. Taylor expanded in x around inf 19.5%

       \[\leadsto \color{blue}{0.18181818181818182} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-11}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.18181818181818182\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 11.5% accurate, 210.0× speedup

Math

\[\begin{array}{l} \\ 0.18181818181818182 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.18181818181818182)

C

double code(double x) {
	return 0.18181818181818182;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.18181818181818182d0
end function

Java

public static double code(double x) {
	return 0.18181818181818182;
}

Python

def code(x):
	return 0.18181818181818182

Julia

function code(x)
	return 0.18181818181818182
end

MATLAB

function tmp = code(x)
	tmp = 0.18181818181818182;
end

Wolfram

code[x_] := 0.18181818181818182

Derivation

1. Initial program 99.1%

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

   1. distribute-lft-in 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-*r/ 99.1%

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

   4. metadata-eval 99.1%

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

3. Simplified 99.1%

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

   1. flip-- 99.1%

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

   2. div-inv 99.1%

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

   3. metadata-eval 99.1%

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

   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. *-commutative 99.9%

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

   2. associate-/r/ 99.9%

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

8. Simplified 99.9%

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

9. Taylor expanded in x around 0 31.8%

   \[\leadsto \frac{1}{\color{blue}{5.5 + 8 \cdot \frac{1}{{x}^{2}}}} \]
10. Step-by-step derivation

    1. associate-*r/ 31.8%

       \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{8 \cdot 1}{{x}^{2}}}} \]

    2. metadata-eval 31.8%

       \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{8}}{{x}^{2}}} \]

11. Simplified 31.8%

    \[\leadsto \frac{1}{\color{blue}{5.5 + \frac{8}{{x}^{2}}}} \]

12. Taylor expanded in x around inf 11.7%

    \[\leadsto \color{blue}{0.18181818181818182} \]

13. Final simplification 11.7%

    \[\leadsto 0.18181818181818182 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))