Given's Rotation SVD example, simplified

Percentage Accurate: 75.7% → 99.9%

Time: 21.9s

Alternatives: 15

Speedup: 1.0×

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: 75.7% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 1 + \sqrt{t\_0}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.001:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(0.25 + {x}^{2} \cdot \left(0.15625 \cdot {x}^{2} - 0.1875\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25}{-0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} - \frac{0.25}{t\_0 \cdot \mathsf{fma}\left(x, x, 1\right)}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (hypot 1.0 x)))) (t_1 (+ 1.0 (sqrt t_0))))
   (if (<= (hypot 1.0 x) 1.001)
     (/
      (*
       (pow x 2.0)
       (+ 0.25 (* (pow x 2.0) (- (* 0.15625 (pow x 2.0)) 0.1875))))
      t_1)
     (/
      (-
       (/ -0.25 (+ -0.5 (/ -0.5 (hypot 1.0 x))))
       (/ 0.25 (* t_0 (fma x x 1.0))))
      t_1))))

C

double code(double x) {
	double t_0 = 0.5 + (0.5 / hypot(1.0, x));
	double t_1 = 1.0 + sqrt(t_0);
	double tmp;
	if (hypot(1.0, x) <= 1.001) {
		tmp = (pow(x, 2.0) * (0.25 + (pow(x, 2.0) * ((0.15625 * pow(x, 2.0)) - 0.1875)))) / t_1;
	} else {
		tmp = ((-0.25 / (-0.5 + (-0.5 / hypot(1.0, x)))) - (0.25 / (t_0 * fma(x, x, 1.0)))) / t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(0.5 / hypot(1.0, x)))
	t_1 = Float64(1.0 + sqrt(t_0))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.001)
		tmp = Float64(Float64((x ^ 2.0) * Float64(0.25 + Float64((x ^ 2.0) * Float64(Float64(0.15625 * (x ^ 2.0)) - 0.1875)))) / t_1);
	else
		tmp = Float64(Float64(Float64(-0.25 / Float64(-0.5 + Float64(-0.5 / hypot(1.0, x)))) - Float64(0.25 / Float64(t_0 * fma(x, x, 1.0)))) / t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.001], N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.25 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.15625 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.1875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(-0.25 / N[(-0.5 + N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.25 / N[(t$95$0 * N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0009999999999999

   1. Initial program 49.7%

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

      1. distribute-lft-in 49.7%

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

      2. metadata-eval 49.7%

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

      3. associate-*r/ 49.7%

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

      4. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

      1. flip-- 49.7%

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

      2. metadata-eval 49.7%

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

      3. add-sqr-sqrt 49.7%

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

      4. associate--r+ 49.7%

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

      5. metadata-eval 49.7%

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

   6. Applied egg-rr 49.7%

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

   7. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(0.25 + {x}^{2} \cdot \left(0.15625 \cdot {x}^{2} - 0.1875\right)\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   if 1.0009999999999999 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

      1. distribute-lft-in 98.3%

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

      2. metadata-eval 98.3%

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

      3. associate-*r/ 98.3%

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

      4. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. flip-- 98.3%

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

      2. metadata-eval 98.3%

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

      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. flip-- 99.9%

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

      2. metadata-eval 99.9%

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

      3. frac-times 99.9%

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

      4. metadata-eval 99.9%

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

      5. hypot-undefine 99.9%

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

      6. hypot-undefine 99.9%

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

      7. rem-square-sqrt 99.9%

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

      8. metadata-eval 99.9%

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

      9. unpow2 99.9%

         \[\leadsto \frac{\frac{0.25 - \frac{0.25}{1 + \color{blue}{{x}^{2}}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\frac{0.25 - \frac{0.25}{1 + {x}^{2}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   9. Step-by-step derivation

      1. div-sub 99.9%

         \[\leadsto \frac{\color{blue}{\frac{0.25}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} - \frac{\frac{0.25}{1 + {x}^{2}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      2. frac-2neg 99.9%

         \[\leadsto \frac{\color{blue}{\frac{-0.25}{-\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}} - \frac{\frac{0.25}{1 + {x}^{2}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      3. metadata-eval 99.9%

         \[\leadsto \frac{\frac{\color{blue}{-0.25}}{-\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} - \frac{\frac{0.25}{1 + {x}^{2}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      4. distribute-neg-in 99.9%

         \[\leadsto \frac{\frac{-0.25}{\color{blue}{\left(-0.5\right) + \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}} - \frac{\frac{0.25}{1 + {x}^{2}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{\frac{-0.25}{\color{blue}{-0.5} + \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} - \frac{\frac{0.25}{1 + {x}^{2}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      6. distribute-neg-frac 99.9%

         \[\leadsto \frac{\frac{-0.25}{-0.5 + \color{blue}{\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}} - \frac{\frac{0.25}{1 + {x}^{2}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      7. metadata-eval 99.9%

         \[\leadsto \frac{\frac{-0.25}{-0.5 + \frac{\color{blue}{-0.5}}{\mathsf{hypot}\left(1, x\right)}} - \frac{\frac{0.25}{1 + {x}^{2}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      8. associate-/l/ 99.9%

         \[\leadsto \frac{\frac{-0.25}{-0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} - \color{blue}{\frac{0.25}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 + {x}^{2}\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      9. +-commutative 99.9%

         \[\leadsto \frac{\frac{-0.25}{-0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} - \frac{0.25}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \color{blue}{\left({x}^{2} + 1\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      10. unpow2 99.9%

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

      11. fma-define 99.9%

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

   10. Applied egg-rr 99.9%

       \[\leadsto \frac{\color{blue}{\frac{-0.25}{-0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} - \frac{0.25}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \mathsf{fma}\left(x, x, 1\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 1 + \sqrt{t\_0}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.001:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(0.25 + {x}^{2} \cdot \left(0.15625 \cdot {x}^{2} - 0.1875\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} + 0.25}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (hypot 1.0 x)))) (t_1 (+ 1.0 (sqrt t_0))))
   (if (<= (hypot 1.0 x) 1.001)
     (/
      (*
       (pow x 2.0)
       (+ 0.25 (* (pow x 2.0) (- (* 0.15625 (pow x 2.0)) 0.1875))))
      t_1)
     (/ (/ (+ (/ -0.25 (fma x x 1.0)) 0.25) t_0) t_1))))

C

double code(double x) {
	double t_0 = 0.5 + (0.5 / hypot(1.0, x));
	double t_1 = 1.0 + sqrt(t_0);
	double tmp;
	if (hypot(1.0, x) <= 1.001) {
		tmp = (pow(x, 2.0) * (0.25 + (pow(x, 2.0) * ((0.15625 * pow(x, 2.0)) - 0.1875)))) / t_1;
	} else {
		tmp = (((-0.25 / fma(x, x, 1.0)) + 0.25) / t_0) / t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(0.5 / hypot(1.0, x)))
	t_1 = Float64(1.0 + sqrt(t_0))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.001)
		tmp = Float64(Float64((x ^ 2.0) * Float64(0.25 + Float64((x ^ 2.0) * Float64(Float64(0.15625 * (x ^ 2.0)) - 0.1875)))) / t_1);
	else
		tmp = Float64(Float64(Float64(Float64(-0.25 / fma(x, x, 1.0)) + 0.25) / t_0) / t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.001], N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.25 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.15625 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.1875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(N[(-0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] + 0.25), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0009999999999999

   1. Initial program 49.7%

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

      1. distribute-lft-in 49.7%

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

      2. metadata-eval 49.7%

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

      3. associate-*r/ 49.7%

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

      4. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

      1. flip-- 49.7%

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

      2. metadata-eval 49.7%

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

      3. add-sqr-sqrt 49.7%

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

      4. associate--r+ 49.7%

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

      5. metadata-eval 49.7%

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

   6. Applied egg-rr 49.7%

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

   7. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(0.25 + {x}^{2} \cdot \left(0.15625 \cdot {x}^{2} - 0.1875\right)\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   if 1.0009999999999999 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

      1. distribute-lft-in 98.3%

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

      2. metadata-eval 98.3%

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

      3. associate-*r/ 98.3%

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

      4. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. flip-- 98.3%

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

      2. metadata-eval 98.3%

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

      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. flip-- 99.9%

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

      2. metadata-eval 99.9%

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

      3. frac-times 99.9%

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

      4. metadata-eval 99.9%

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

      5. hypot-undefine 99.9%

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

      6. hypot-undefine 99.9%

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

      7. rem-square-sqrt 99.9%

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

      8. metadata-eval 99.9%

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

      9. unpow2 99.9%

         \[\leadsto \frac{\frac{0.25 - \frac{0.25}{1 + \color{blue}{{x}^{2}}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\frac{0.25 - \frac{0.25}{1 + {x}^{2}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   9. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \frac{\frac{\color{blue}{0.25 + \left(-\frac{0.25}{1 + {x}^{2}}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      2. +-commutative 99.9%

         \[\leadsto \frac{\frac{\color{blue}{\left(-\frac{0.25}{1 + {x}^{2}}\right) + 0.25}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      3. distribute-neg-frac 99.9%

         \[\leadsto \frac{\frac{\color{blue}{\frac{-0.25}{1 + {x}^{2}}} + 0.25}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      4. metadata-eval 99.9%

         \[\leadsto \frac{\frac{\frac{\color{blue}{-0.25}}{1 + {x}^{2}} + 0.25}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      5. +-commutative 99.9%

         \[\leadsto \frac{\frac{\frac{-0.25}{\color{blue}{{x}^{2} + 1}} + 0.25}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      6. unpow2 99.9%

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

      7. fma-define 99.9%

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

   10. Applied egg-rr 99.9%

       \[\leadsto \frac{\frac{\color{blue}{\frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} + 0.25}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.001:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} + 0.25}{t\_0}}{1 + \sqrt{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (hypot 1.0 x)))))
   (if (<= (hypot 1.0 x) 1.001)
     (*
      (pow x 2.0)
      (+ 0.125 (* (pow x 2.0) (- (* 0.0673828125 (pow x 2.0)) 0.0859375))))
     (/ (/ (+ (/ -0.25 (fma x x 1.0)) 0.25) t_0) (+ 1.0 (sqrt t_0))))))

C

double code(double x) {
	double t_0 = 0.5 + (0.5 / hypot(1.0, x));
	double tmp;
	if (hypot(1.0, x) <= 1.001) {
		tmp = pow(x, 2.0) * (0.125 + (pow(x, 2.0) * ((0.0673828125 * pow(x, 2.0)) - 0.0859375)));
	} else {
		tmp = (((-0.25 / fma(x, x, 1.0)) + 0.25) / t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(0.5 / hypot(1.0, x)))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.001)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64((x ^ 2.0) * Float64(Float64(0.0673828125 * (x ^ 2.0)) - 0.0859375))));
	else
		tmp = Float64(Float64(Float64(Float64(-0.25 / fma(x, x, 1.0)) + 0.25) / t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.001], N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.125 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.0673828125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] + 0.25), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0009999999999999

   1. Initial program 49.7%

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

      1. distribute-lft-in 49.7%

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

      2. metadata-eval 49.7%

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

      3. associate-*r/ 49.7%

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

      4. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)} \]

   if 1.0009999999999999 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

      1. distribute-lft-in 98.3%

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

      2. metadata-eval 98.3%

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

      3. associate-*r/ 98.3%

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

      4. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. flip-- 98.3%

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

      2. metadata-eval 98.3%

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

      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. flip-- 99.9%

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

      2. metadata-eval 99.9%

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

      3. frac-times 99.9%

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

      4. metadata-eval 99.9%

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

      5. hypot-undefine 99.9%

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

      6. hypot-undefine 99.9%

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

      7. rem-square-sqrt 99.9%

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

      8. metadata-eval 99.9%

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

      9. unpow2 99.9%

         \[\leadsto \frac{\frac{0.25 - \frac{0.25}{1 + \color{blue}{{x}^{2}}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\frac{0.25 - \frac{0.25}{1 + {x}^{2}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   9. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \frac{\frac{\color{blue}{0.25 + \left(-\frac{0.25}{1 + {x}^{2}}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      2. +-commutative 99.9%

         \[\leadsto \frac{\frac{\color{blue}{\left(-\frac{0.25}{1 + {x}^{2}}\right) + 0.25}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      3. distribute-neg-frac 99.9%

         \[\leadsto \frac{\frac{\color{blue}{\frac{-0.25}{1 + {x}^{2}}} + 0.25}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      4. metadata-eval 99.9%

         \[\leadsto \frac{\frac{\frac{\color{blue}{-0.25}}{1 + {x}^{2}} + 0.25}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      5. +-commutative 99.9%

         \[\leadsto \frac{\frac{\frac{-0.25}{\color{blue}{{x}^{2} + 1}} + 0.25}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      6. unpow2 99.9%

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

      7. fma-define 99.9%

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

   10. Applied egg-rr 99.9%

       \[\leadsto \frac{\frac{\color{blue}{\frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)} + 0.25}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.001:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t\_0}{1 + \sqrt{0.5 + t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.001)
     (*
      (pow x 2.0)
      (+ 0.125 (* (pow x 2.0) (- (* 0.0673828125 (pow x 2.0)) 0.0859375))))
     (/ (- 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);
	double tmp;
	if (hypot(1.0, x) <= 1.001) {
		tmp = pow(x, 2.0) * (0.125 + (pow(x, 2.0) * ((0.0673828125 * pow(x, 2.0)) - 0.0859375)));
	} else {
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double tmp;
	if (Math.hypot(1.0, x) <= 1.001) {
		tmp = Math.pow(x, 2.0) * (0.125 + (Math.pow(x, 2.0) * ((0.0673828125 * Math.pow(x, 2.0)) - 0.0859375)));
	} else {
		tmp = (0.5 - t_0) / (1.0 + Math.sqrt((0.5 + t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	tmp = 0
	if math.hypot(1.0, x) <= 1.001:
		tmp = math.pow(x, 2.0) * (0.125 + (math.pow(x, 2.0) * ((0.0673828125 * math.pow(x, 2.0)) - 0.0859375)))
	else:
		tmp = (0.5 - t_0) / (1.0 + math.sqrt((0.5 + t_0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.001)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64((x ^ 2.0) * Float64(Float64(0.0673828125 * (x ^ 2.0)) - 0.0859375))));
	else
		tmp = Float64(Float64(0.5 - t_0) / Float64(1.0 + sqrt(Float64(0.5 + t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.001)
		tmp = (x ^ 2.0) * (0.125 + ((x ^ 2.0) * ((0.0673828125 * (x ^ 2.0)) - 0.0859375)));
	else
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.001], N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.125 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.0673828125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $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. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0009999999999999

   1. Initial program 49.7%

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

      1. distribute-lft-in 49.7%

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

      2. metadata-eval 49.7%

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

      3. associate-*r/ 49.7%

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

      4. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)} \]

   if 1.0009999999999999 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

      1. distribute-lft-in 98.3%

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

      2. metadata-eval 98.3%

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

      3. associate-*r/ 98.3%

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

      4. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. flip-- 98.3%

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

      2. metadata-eval 98.3%

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

      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%

      \[\leadsto \color{blue}{\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + 0.5 \cdot \frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (*
    (pow x 2.0)
    (+ 0.125 (* (pow x 2.0) (- (* 0.0673828125 (pow x 2.0)) 0.0859375))))
   (/ (- 0.5 (/ 0.5 (hypot 1.0 x))) (+ 1.0 (sqrt (+ 0.5 (* 0.5 (/ 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = pow(x, 2.0) * (0.125 + (pow(x, 2.0) * ((0.0673828125 * pow(x, 2.0)) - 0.0859375)));
	} else {
		tmp = (0.5 - (0.5 / hypot(1.0, x))) / (1.0 + sqrt((0.5 + (0.5 * (1.0 / x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = Math.pow(x, 2.0) * (0.125 + (Math.pow(x, 2.0) * ((0.0673828125 * Math.pow(x, 2.0)) - 0.0859375)));
	} else {
		tmp = (0.5 - (0.5 / Math.hypot(1.0, x))) / (1.0 + Math.sqrt((0.5 + (0.5 * (1.0 / x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = math.pow(x, 2.0) * (0.125 + (math.pow(x, 2.0) * ((0.0673828125 * math.pow(x, 2.0)) - 0.0859375)))
	else:
		tmp = (0.5 - (0.5 / math.hypot(1.0, x))) / (1.0 + math.sqrt((0.5 + (0.5 * (1.0 / x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64((x ^ 2.0) * Float64(Float64(0.0673828125 * (x ^ 2.0)) - 0.0859375))));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / hypot(1.0, x))) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = (x ^ 2.0) * (0.125 + ((x ^ 2.0) * ((0.0673828125 * (x ^ 2.0)) - 0.0859375)));
	else
		tmp = (0.5 - (0.5 / hypot(1.0, x))) / (1.0 + sqrt((0.5 + (0.5 * (1.0 / x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.125 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.0673828125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.9%

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

      1. distribute-lft-in 49.9%

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

      2. metadata-eval 49.9%

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

      3. associate-*r/ 49.9%

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

      4. metadata-eval 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) 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-- 98.5%

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

      2. metadata-eval 98.5%

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

      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%

      \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{\color{blue}{0.5 + 0.5 \cdot \frac{1}{x}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + 0.5 \cdot \frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (pow x 2.0) (+ 0.125 (* -0.0859375 (pow x 2.0))))
   (/ (- 0.5 (/ 0.5 (hypot 1.0 x))) (+ 1.0 (sqrt (+ 0.5 (* 0.5 (/ 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = pow(x, 2.0) * (0.125 + (-0.0859375 * pow(x, 2.0)));
	} else {
		tmp = (0.5 - (0.5 / hypot(1.0, x))) / (1.0 + sqrt((0.5 + (0.5 * (1.0 / x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = Math.pow(x, 2.0) * (0.125 + (-0.0859375 * Math.pow(x, 2.0)));
	} else {
		tmp = (0.5 - (0.5 / Math.hypot(1.0, x))) / (1.0 + Math.sqrt((0.5 + (0.5 * (1.0 / x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = math.pow(x, 2.0) * (0.125 + (-0.0859375 * math.pow(x, 2.0)))
	else:
		tmp = (0.5 - (0.5 / math.hypot(1.0, x))) / (1.0 + math.sqrt((0.5 + (0.5 * (1.0 / x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64(-0.0859375 * (x ^ 2.0))));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / hypot(1.0, x))) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = (x ^ 2.0) * (0.125 + (-0.0859375 * (x ^ 2.0)));
	else
		tmp = (0.5 - (0.5 / hypot(1.0, x))) / (1.0 + sqrt((0.5 + (0.5 * (1.0 / x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.125 + N[(-0.0859375 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.9%

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

      1. distribute-lft-in 49.9%

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

      2. metadata-eval 49.9%

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

      3. associate-*r/ 49.9%

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

      4. metadata-eval 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in x around 0 99.5%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) 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-- 98.5%

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

      2. metadata-eval 98.5%

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

      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%

      \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{\color{blue}{0.5 + 0.5 \cdot \frac{1}{x}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (pow x 2.0) (+ 0.125 (* -0.0859375 (pow x 2.0))))
   (/ (- 0.5 (/ 0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = pow(x, 2.0) * (0.125 + (-0.0859375 * pow(x, 2.0)));
	} else {
		tmp = (0.5 - (0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = Math.pow(x, 2.0) * (0.125 + (-0.0859375 * Math.pow(x, 2.0)));
	} else {
		tmp = (0.5 - (0.5 / x)) / (1.0 + Math.sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = math.pow(x, 2.0) * (0.125 + (-0.0859375 * math.pow(x, 2.0)))
	else:
		tmp = (0.5 - (0.5 / x)) / (1.0 + math.sqrt((0.5 + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64(-0.0859375 * (x ^ 2.0))));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / x)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = (x ^ 2.0) * (0.125 + (-0.0859375 * (x ^ 2.0)));
	else
		tmp = (0.5 - (0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.125 + N[(-0.0859375 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.9%

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

      1. distribute-lft-in 49.9%

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

      2. metadata-eval 49.9%

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

      3. associate-*r/ 49.9%

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

      4. metadata-eval 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in x around 0 99.5%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) 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 97.7%

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

      1. flip-- 97.7%

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

      2. metadata-eval 97.7%

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

      3. add-sqr-sqrt 99.2%

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

      4. associate--r+ 99.2%

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

      5. metadata-eval 99.2%

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

   7. Applied egg-rr 99.2%

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

Alternative 8: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(0.125 * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.9%

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

      1. distribute-lft-in 49.9%

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

      2. metadata-eval 49.9%

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

      3. associate-*r/ 49.9%

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

      4. metadata-eval 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in x around 0 49.4%

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

      1. +-commutative 49.4%

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

      2. associate--r+ 49.4%

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

      3. add-exp-log 49.4%

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

      4. cancel-sign-sub-inv 49.4%

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

      5. metadata-eval 49.4%

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

      6. log1p-undefine 49.4%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)}} - 1 \]

      7. expm1-undefine 98.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)\right)} \]

      8. expm1-log1p-u 98.9%

         \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]

      9. unpow2 98.9%

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

      10. associate-*r* 98.9%

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

   7. Applied egg-rr 98.9%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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 97.7%

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

      1. flip-- 97.7%

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

      2. metadata-eval 97.7%

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

      3. add-sqr-sqrt 99.2%

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

      4. associate--r+ 99.2%

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

      5. metadata-eval 99.2%

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

   7. Applied egg-rr 99.2%

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

Alternative 9: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(0.125 * x), $MachinePrecision] * x), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.9%

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

      1. distribute-lft-in 49.9%

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

      2. metadata-eval 49.9%

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

      3. associate-*r/ 49.9%

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

      4. metadata-eval 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in x around 0 49.4%

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

      1. +-commutative 49.4%

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

      2. associate--r+ 49.4%

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

      3. add-exp-log 49.4%

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

      4. cancel-sign-sub-inv 49.4%

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

      5. metadata-eval 49.4%

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

      6. log1p-undefine 49.4%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)}} - 1 \]

      7. expm1-undefine 98.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)\right)} \]

      8. expm1-log1p-u 98.9%

         \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]

      9. unpow2 98.9%

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

      10. associate-*r* 98.9%

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

   7. Applied egg-rr 98.9%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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 97.7%

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

Alternative 10: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(0.125 * x) * x);
	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 (hypot(1.0, x) <= 2.0)
		tmp = (0.125 * x) * x;
	else
		tmp = 0.5 / (1.0 + sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(0.125 * x), $MachinePrecision] * x), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.9%

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

      1. distribute-lft-in 49.9%

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

      2. metadata-eval 49.9%

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

      3. associate-*r/ 49.9%

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

      4. metadata-eval 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in x around 0 49.4%

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

      1. +-commutative 49.4%

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

      2. associate--r+ 49.4%

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

      3. add-exp-log 49.4%

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

      4. cancel-sign-sub-inv 49.4%

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

      5. metadata-eval 49.4%

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

      6. log1p-undefine 49.4%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)}} - 1 \]

      7. expm1-undefine 98.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)\right)} \]

      8. expm1-log1p-u 98.9%

         \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]

      9. unpow2 98.9%

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

      10. associate-*r* 98.9%

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

   7. Applied egg-rr 98.9%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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-- 98.5%

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

      2. metadata-eval 98.5%

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

      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 97.9%

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

Alternative 11: 74.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = (0.125 * x) * x;
	} 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 <= 1.55d0) then
        tmp = (0.125d0 * x) * x
    else
        tmp = 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 = (0.125 * x) * x;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 65.6%

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

      1. distribute-lft-in 65.6%

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

      2. metadata-eval 65.6%

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

      3. associate-*r/ 65.6%

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

      4. metadata-eval 65.6%

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

   3. Simplified 65.6%

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

   5. Taylor expanded in x around 0 34.8%

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

      1. +-commutative 34.8%

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

      2. associate--r+ 34.8%

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

      3. add-exp-log 34.8%

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

      4. cancel-sign-sub-inv 34.8%

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

      5. metadata-eval 34.8%

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

      6. log1p-undefine 34.8%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)}} - 1 \]

      7. expm1-undefine 68.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)\right)} \]

      8. expm1-log1p-u 68.2%

         \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]

      9. unpow2 68.2%

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

      10. associate-*r* 68.3%

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

   7. Applied egg-rr 68.3%

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

   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. Taylor expanded in x around inf 94.9%

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

Alternative 12: 56.1% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\left(0.125 \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8) (* (* 0.125 x) x) (+ 0.25 (* 0.25 (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = (0.125 * x) * x;
	} else {
		tmp = 0.25 + (0.25 * (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.8d0) then
        tmp = (0.125d0 * x) * x
    else
        tmp = 0.25d0 + (0.25d0 * (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = (0.125 * x) * x;
	} else {
		tmp = 0.25 + (0.25 * (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.8:
		tmp = (0.125 * x) * x
	else:
		tmp = 0.25 + (0.25 * (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = Float64(Float64(0.125 * x) * x);
	else
		tmp = Float64(0.25 + Float64(0.25 * Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = (0.125 * x) * x;
	else
		tmp = 0.25 + (0.25 * (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[(N[(0.125 * x), $MachinePrecision] * x), $MachinePrecision], N[(0.25 + N[(0.25 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 65.6%

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

      1. distribute-lft-in 65.6%

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

      2. metadata-eval 65.6%

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

      3. associate-*r/ 65.6%

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

      4. metadata-eval 65.6%

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

   3. Simplified 65.6%

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

   5. Taylor expanded in x around 0 34.8%

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

      1. +-commutative 34.8%

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

      2. associate--r+ 34.8%

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

      3. add-exp-log 34.8%

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

      4. cancel-sign-sub-inv 34.8%

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

      5. metadata-eval 34.8%

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

      6. log1p-undefine 34.8%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)}} - 1 \]

      7. expm1-undefine 68.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)\right)} \]

      8. expm1-log1p-u 68.2%

         \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]

      9. unpow2 68.2%

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

      10. associate-*r* 68.3%

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

   7. Applied egg-rr 68.3%

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

   if 1.80000000000000004 < 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-- 98.5%

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

      2. metadata-eval 98.5%

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

      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 0 22.7%

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

   8. Taylor expanded in x around -inf 22.8%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot \frac{1}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.1% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\left(0.125 \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.4) (* (* 0.125 x) x) 0.25))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = (0.125 * x) * x;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = (0.125d0 * x) * x
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = (0.125 * x) * x;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = (0.125 * x) * x
	else:
		tmp = 0.25
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(Float64(0.125 * x) * x);
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = (0.125 * x) * x;
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], N[(N[(0.125 * x), $MachinePrecision] * x), $MachinePrecision], 0.25]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 65.6%

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

      1. distribute-lft-in 65.6%

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

      2. metadata-eval 65.6%

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

      3. associate-*r/ 65.6%

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

      4. metadata-eval 65.6%

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

   3. Simplified 65.6%

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

   5. Taylor expanded in x around 0 34.8%

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

      1. +-commutative 34.8%

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

      2. associate--r+ 34.8%

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

      3. add-exp-log 34.8%

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

      4. cancel-sign-sub-inv 34.8%

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

      5. metadata-eval 34.8%

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

      6. log1p-undefine 34.8%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)}} - 1 \]

      7. expm1-undefine 68.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)\right)} \]

      8. expm1-log1p-u 68.2%

         \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]

      9. unpow2 68.2%

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

      10. associate-*r* 68.3%

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

   7. Applied egg-rr 68.3%

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

   if 1.3999999999999999 < 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-- 98.5%

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

      2. metadata-eval 98.5%

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

      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 0 22.7%

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

   8. Taylor expanded in x around inf 22.8%

      \[\leadsto \color{blue}{0.25} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 32.2% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-77}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.1e-77) (- 1.0 1.0) 0.25))

C

double code(double x) {
	double tmp;
	if (x <= 2.1e-77) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.1d-77) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.1e-77) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.1e-77:
		tmp = 1.0 - 1.0
	else:
		tmp = 0.25
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.1e-77)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.1e-77)
		tmp = 1.0 - 1.0;
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.1e-77], N[(1.0 - 1.0), $MachinePrecision], 0.25]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000015e-77

   1. Initial program 70.8%

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

      1. distribute-lft-in 70.8%

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

      2. metadata-eval 70.8%

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

      3. associate-*r/ 70.8%

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

      4. metadata-eval 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in x around 0 36.4%

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

   if 2.10000000000000015e-77 < x

   1. Initial program 79.6%

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

      1. distribute-lft-in 79.6%

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

      2. metadata-eval 79.6%

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

      3. associate-*r/ 79.6%

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

      4. metadata-eval 79.6%

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

   3. Simplified 79.6%

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

      1. flip-- 79.6%

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

      2. metadata-eval 79.6%

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

      3. add-sqr-sqrt 80.9%

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

      4. associate--r+ 80.9%

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

      5. metadata-eval 80.9%

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

   6. Applied egg-rr 80.9%

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

   7. Taylor expanded in x around 0 19.5%

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

   8. Taylor expanded in x around inf 19.4%

      \[\leadsto \color{blue}{0.25} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 13.5% accurate, 210.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.25)

C

double code(double x) {
	return 0.25;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.25d0
end function

Java

public static double code(double x) {
	return 0.25;
}

Python

def code(x):
	return 0.25

Julia

function code(x)
	return 0.25
end

MATLAB

function tmp = code(x)
	tmp = 0.25;
end

Wolfram

code[x_] := 0.25

Derivation

1. Initial program 73.4%

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

   1. distribute-lft-in 73.4%

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

   2. metadata-eval 73.4%

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

   3. associate-*r/ 73.4%

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

   4. metadata-eval 73.4%

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

3. Simplified 73.4%

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

   1. flip-- 73.4%

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

   2. metadata-eval 73.4%

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

   3. add-sqr-sqrt 74.2%

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

   4. associate--r+ 74.2%

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

   5. metadata-eval 74.2%

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

6. Applied egg-rr 74.2%

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

7. Taylor expanded in x around 0 36.5%

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

8. Taylor expanded in x around inf 13.3%

   \[\leadsto \color{blue}{0.25} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024076 -o generate:simplify
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))