Given's Rotation SVD example, simplified

Percentage Accurate: 98.4% → 99.8%

Time: 17.8s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. x = -9.193628868099802e-89

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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 + t\_0\\ t_2 := {t\_1}^{0.5} + \left(t\_0 + 1.5\right)\\ t_3 := \frac{{t\_1}^{1.5}}{t\_2}\\ t_4 := \frac{1}{t\_2}\\ \frac{{t\_4}^{3} - {t\_3}^{3}}{t\_4 \cdot t\_4 + \left(t\_3 \cdot t\_3 + t\_4 \cdot t\_3\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x)))
        (t_1 (+ 0.5 t_0))
        (t_2 (+ (pow t_1 0.5) (+ t_0 1.5)))
        (t_3 (/ (pow t_1 1.5) t_2))
        (t_4 (/ 1.0 t_2)))
   (/
    (- (pow t_4 3.0) (pow t_3 3.0))
    (+ (* t_4 t_4) (+ (* t_3 t_3) (* t_4 t_3))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	double t_2 = pow(t_1, 0.5) + (t_0 + 1.5);
	double t_3 = pow(t_1, 1.5) / t_2;
	double t_4 = 1.0 / t_2;
	return (pow(t_4, 3.0) - pow(t_3, 3.0)) / ((t_4 * t_4) + ((t_3 * t_3) + (t_4 * t_3)));
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	double t_2 = Math.pow(t_1, 0.5) + (t_0 + 1.5);
	double t_3 = Math.pow(t_1, 1.5) / t_2;
	double t_4 = 1.0 / t_2;
	return (Math.pow(t_4, 3.0) - Math.pow(t_3, 3.0)) / ((t_4 * t_4) + ((t_3 * t_3) + (t_4 * t_3)));
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	t_1 = 0.5 + t_0
	t_2 = math.pow(t_1, 0.5) + (t_0 + 1.5)
	t_3 = math.pow(t_1, 1.5) / t_2
	t_4 = 1.0 / t_2
	return (math.pow(t_4, 3.0) - math.pow(t_3, 3.0)) / ((t_4 * t_4) + ((t_3 * t_3) + (t_4 * t_3)))

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 + t_0)
	t_2 = Float64((t_1 ^ 0.5) + Float64(t_0 + 1.5))
	t_3 = Float64((t_1 ^ 1.5) / t_2)
	t_4 = Float64(1.0 / t_2)
	return Float64(Float64((t_4 ^ 3.0) - (t_3 ^ 3.0)) / Float64(Float64(t_4 * t_4) + Float64(Float64(t_3 * t_3) + Float64(t_4 * t_3))))
end

MATLAB

function tmp = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	t_1 = 0.5 + t_0;
	t_2 = (t_1 ^ 0.5) + (t_0 + 1.5);
	t_3 = (t_1 ^ 1.5) / t_2;
	t_4 = 1.0 / t_2;
	tmp = ((t_4 ^ 3.0) - (t_3 ^ 3.0)) / ((t_4 * t_4) + ((t_3 * t_3) + (t_4 * t_3)));
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]}, Block[{t$95$2 = N[(N[Power[t$95$1, 0.5], $MachinePrecision] + N[(t$95$0 + 1.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[t$95$1, 1.5], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / t$95$2), $MachinePrecision]}, N[(N[(N[Power[t$95$4, 3.0], $MachinePrecision] - N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$4 * t$95$4), $MachinePrecision] + N[(N[(t$95$3 * t$95$3), $MachinePrecision] + N[(t$95$4 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 98.4%

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

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

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

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

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

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

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

   9. hypot-undefine N/A

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

   10. hypot-lowering-hypot.f64 98.4%

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

3. Simplified 98.4%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

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

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 1.5)) (+ (sqrt t_1) (+ t_0 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, 1.5)) / (sqrt(t_1) + (t_0 + 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, 1.5)) / (Math.sqrt(t_1) + (t_0 + 1.5));
}

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, 1.5)) / (math.sqrt(t_1) + (t_0 + 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(1.0 - (t_1 ^ 1.5)) / Float64(sqrt(t_1) + Float64(t_0 + 1.5)))
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 ^ 1.5)) / (sqrt(t_1) + (t_0 + 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[(1.0 - N[Power[t$95$1, 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$1], $MachinePrecision] + N[(t$95$0 + 1.5), $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. --lowering--.f64 N/A

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

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

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

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

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

   9. hypot-undefine N/A

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

   10. hypot-lowering-hypot.f64 98.4%

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

3. Simplified 98.4%

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

   1. flip3-- N/A

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

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

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

6. Applied egg-rr 99.9%

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

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

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

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

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

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

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

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

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

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

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

   6. hypot-undefine N/A

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

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

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

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right), \frac{3}{2}\right)\right), \mathsf{+.f64}\left(\left({\left(\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{3}{2} + \frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)\right) \]

8. Applied egg-rr 99.9%

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

Alternative 3: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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.pow((0.5 + t_0), 0.5));
}

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	tmp = (0.5 - t_0) / (1.0 + ((0.5 + t_0) ^ 0.5));
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[(0.5 + t$95$0), $MachinePrecision], 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. --lowering--.f64 N/A

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

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

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

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

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

   9. hypot-undefine N/A

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

   10. hypot-lowering-hypot.f64 98.4%

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

3. Simplified 98.4%

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

   1. flip-- N/A

      \[\leadsto \frac{1 \cdot 1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}} \cdot \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}}}{\color{blue}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}}}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}} \cdot \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}}} \]

   3. rem-square-sqrt N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2} - \frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}}} \]

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

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

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

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

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

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

   9. hypot-undefine N/A

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

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

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

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

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

6. Applied egg-rr 99.9%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

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

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

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

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

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

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

   9. hypot-undefine N/A

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

   10. hypot-lowering-hypot.f64 98.4%

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

3. Simplified 98.4%

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

Alternative 5: 96.9% 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. --lowering--.f64 N/A

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

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

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

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

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

   9. hypot-undefine N/A

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

   10. hypot-lowering-hypot.f64 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in x around inf

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

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

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

   2. sqrt-lowering-sqrt.f64 95.9%

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

7. Simplified 95.9%

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

   1. flip-- N/A

      \[\leadsto \frac{1 \cdot 1 - \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}{\color{blue}{1 + \sqrt{\frac{1}{2}}}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1 - \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}{1 + \sqrt{\frac{1}{2}}} \]

   3. rem-square-sqrt N/A

      \[\leadsto \frac{1 - \frac{1}{2}}{1 + \sqrt{\frac{1}{2}}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{1} + \sqrt{\frac{1}{2}}} \]

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

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

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

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

   7. sqrt-lowering-sqrt.f64 97.3%

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

9. Applied egg-rr 97.3%

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

Alternative 6: 95.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

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

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

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

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

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

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

   9. hypot-undefine N/A

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

   10. hypot-lowering-hypot.f64 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in x around inf

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

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

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

   2. sqrt-lowering-sqrt.f64 95.9%

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

7. Simplified 95.9%

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

Alternative 7: 19.3% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot 0.125\right)\\ \frac{1 + \frac{1}{\left(1 + t\_0\right) \cdot \left(-1 - t\_0\right)}}{1 - -1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x 0.125))))
   (/ (+ 1.0 (/ 1.0 (* (+ 1.0 t_0) (- -1.0 t_0)))) (- 1.0 -1.0))))

C

double code(double x) {
	double t_0 = x * (x * 0.125);
	return (1.0 + (1.0 / ((1.0 + t_0) * (-1.0 - t_0)))) / (1.0 - -1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = x * (x * 0.125d0)
    code = (1.0d0 + (1.0d0 / ((1.0d0 + t_0) * ((-1.0d0) - t_0)))) / (1.0d0 - (-1.0d0))
end function

Java

public static double code(double x) {
	double t_0 = x * (x * 0.125);
	return (1.0 + (1.0 / ((1.0 + t_0) * (-1.0 - t_0)))) / (1.0 - -1.0);
}

Python

def code(x):
	t_0 = x * (x * 0.125)
	return (1.0 + (1.0 / ((1.0 + t_0) * (-1.0 - t_0)))) / (1.0 - -1.0)

Julia

function code(x)
	t_0 = Float64(x * Float64(x * 0.125))
	return Float64(Float64(1.0 + Float64(1.0 / Float64(Float64(1.0 + t_0) * Float64(-1.0 - t_0)))) / Float64(1.0 - -1.0))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * 0.125);
	tmp = (1.0 + (1.0 / ((1.0 + t_0) * (-1.0 - t_0)))) / (1.0 - -1.0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(1.0 / N[(N[(1.0 + t$95$0), $MachinePrecision] * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.4%

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

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

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

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

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

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

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

   9. hypot-undefine N/A

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

   10. hypot-lowering-hypot.f64 98.4%

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

3. Simplified 98.4%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

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

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in x around 0

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

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

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

   2. *-commutative N/A

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 17.5%

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

9. Simplified 17.5%

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

    1. sub-neg N/A

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

    2. flip-+ N/A

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

    3. sqr-neg N/A

       \[\leadsto \frac{1 \cdot 1 - \frac{1}{1 + \left(x \cdot x\right) \cdot \frac{1}{8}} \cdot \frac{1}{1 + \left(x \cdot x\right) \cdot \frac{1}{8}}}{1 - \left(\mathsf{neg}\left(\frac{1}{1 + \left(x \cdot x\right) \cdot \frac{1}{8}}\right)\right)} \]

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

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

11. Applied egg-rr 17.5%

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

12. Taylor expanded in x around 0

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

14. Simplified 19.3%

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

15. Final simplification 19.3%

    \[\leadsto \frac{1 + \frac{1}{\left(1 + x \cdot \left(x \cdot 0.125\right)\right) \cdot \left(-1 - x \cdot \left(x \cdot 0.125\right)\right)}}{1 - -1} \]
16. Add Preprocessing

Alternative 8: 17.5% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

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

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

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

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

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

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

   9. hypot-undefine N/A

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

   10. hypot-lowering-hypot.f64 98.4%

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

3. Simplified 98.4%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

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

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in x around 0

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

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

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

   2. *-commutative N/A

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 17.5%

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

9. Simplified 17.5%

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

10. Final simplification 17.5%

    \[\leadsto 1 + \frac{1}{-1 - 0.125 \cdot \left(x \cdot x\right)} \]
11. Add Preprocessing

Alternative 9: 17.4% accurate, 210.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 98.4%

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

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

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

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

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

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

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

   9. hypot-undefine N/A

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

   10. hypot-lowering-hypot.f64 98.4%

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

3. Simplified 98.4%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

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

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in x around 0

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

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

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

   2. *-commutative N/A

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 17.5%

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

9. Simplified 17.5%

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

10. Taylor expanded in x around inf

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

12. Simplified 17.4%

    \[\leadsto \color{blue}{1} \]
13. Add Preprocessing

Alternative 10: 3.1% accurate, 210.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 98.4%

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

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

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

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

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

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

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

   9. hypot-undefine N/A

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

   10. hypot-lowering-hypot.f64 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in x around 0

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

7. Simplified 3.1%

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

   1. metadata-eval 3.1%

      \[\leadsto 0 \]

9. Applied egg-rr 3.1%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))