Given's Rotation SVD example, simplified

Percentage Accurate: 75.9% → 99.9%

Time: 12.3s

Alternatives: 14

Speedup: 2.0×

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: 75.9% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)}}{t_0 + {t_0}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (hypot 1.0 x)))))
   (if (<= (hypot 1.0 x) 1.0002)
     (+
      (* -0.0859375 (pow x 4.0))
      (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0))))
     (/ (+ 0.25 (/ -0.25 (fma x x 1.0))) (+ t_0 (pow t_0 1.5))))))

C

double code(double x) {
	double t_0 = 0.5 + (0.5 / hypot(1.0, x));
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0)));
	} else {
		tmp = (0.25 + (-0.25 / fma(x, x, 1.0))) / (t_0 + pow(t_0, 1.5));
	}
	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) <= 1.0002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0))));
	else
		tmp = Float64(Float64(0.25 + Float64(-0.25 / fma(x, x, 1.0))) / Float64(t_0 + (t_0 ^ 1.5)));
	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], 1.0002], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 + N[(-0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.0002

   1. Initial program 55.2%

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

      1. distribute-lft-in 55.2%

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

      2. metadata-eval 55.2%

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

      3. associate-*r/ 55.2%

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

      4. metadata-eval 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)} \]

   if 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. 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. div-inv 98.3%

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

      3. metadata-eval 98.3%

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

      4. add-sqr-sqrt 99.8%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. un-div-inv 99.9%

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

      2. flip-- 99.8%

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

      3. associate-/l/ 99.8%

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

      4. metadata-eval 99.8%

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

      5. frac-times 99.9%

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

      6. metadata-eval 99.9%

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

      7. hypot-udef 99.9%

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

      8. hypot-udef 99.9%

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

      9. rem-square-sqrt 99.9%

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

      10. metadata-eval 99.9%

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

      11. unpow2 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-neg-frac 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. unpow2 99.9%

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

      7. fma-def 99.9%

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

   9. Simplified 99.9%

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

       1. +-commutative 99.9%

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

       2. distribute-rgt-in 99.9%

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

       3. pow1/2 99.9%

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

       4. pow-plus 99.9%

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

       5. metadata-eval 99.9%

          \[\leadsto \frac{0.25 + \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)}}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{\color{blue}{1.5}} + 1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \]

       6. *-un-lft-identity 99.9%

          \[\leadsto \frac{0.25 + \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)}}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + \color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}} \]

   11. Applied egg-rr 99.9%

       \[\leadsto \frac{0.25 + \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)}}{\color{blue}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)}}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 - t_0\right) \cdot \frac{1}{1 + \sqrt{0.5 + t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.0002)
     (+
      (* -0.0859375 (pow x 4.0))
      (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0))))
     (* (- 0.5 t_0) (/ 1.0 (+ 1.0 (sqrt (+ 0.5 t_0))))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0)));
	} else {
		tmp = (0.5 - t_0) * (1.0 / (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) <= 1.0002) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0)));
	} else {
		tmp = (0.5 - t_0) * (1.0 / (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) <= 1.0002:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0)))
	else:
		tmp = (0.5 - t_0) * (1.0 / (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) <= 1.0002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0))));
	else
		tmp = Float64(Float64(0.5 - t_0) * Float64(1.0 / 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) <= 1.0002)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0)));
	else
		tmp = (0.5 - t_0) * (1.0 / (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], 1.0002], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] * N[(1.0 / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.0002

   1. Initial program 55.2%

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

      1. distribute-lft-in 55.2%

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

      2. metadata-eval 55.2%

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

      3. associate-*r/ 55.2%

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

      4. metadata-eval 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)} \]

   if 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. 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. div-inv 98.3%

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

      3. metadata-eval 98.3%

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

      4. add-sqr-sqrt 99.8%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t_0}{1 + \sqrt{0.5 + t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.0002)
     (+
      (* -0.0859375 (pow x 4.0))
      (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0))))
     (/ (- 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) <= 1.0002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0)));
	} 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) <= 1.0002) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0)));
	} 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) <= 1.0002:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0)))
	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) <= 1.0002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0))));
	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) <= 1.0002)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0)));
	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], 1.0002], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $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 1 x) < 1.0002

   1. Initial program 55.2%

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

      1. distribute-lft-in 55.2%

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

      2. metadata-eval 55.2%

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

      3. associate-*r/ 55.2%

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

      4. metadata-eval 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)} \]

   if 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. 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 99.8%

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

      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%

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

   5. 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. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.0002)
   (+
    (* -0.0859375 (pow x 4.0))
    (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0))))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0)));
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0002) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0)));
	} else {
		tmp = 1.0 - Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.0002:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0)))
	else:
		tmp = 1.0 - math.sqrt((0.5 + (0.5 / math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0))));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0002)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0)));
	else
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.0002

   1. Initial program 55.2%

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

      1. distribute-lft-in 55.2%

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

      2. metadata-eval 55.2%

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

      3. associate-*r/ 55.2%

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

      4. metadata-eval 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)} \]

   if 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]

Alternative 5: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.0002)
   (+ (* -0.0859375 (pow x 4.0)) (* 0.125 (pow x 2.0)))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + (0.125 * pow(x, 2.0));
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0002) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + (0.125 * Math.pow(x, 2.0));
	} else {
		tmp = 1.0 - Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.0002:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + (0.125 * math.pow(x, 2.0))
	else:
		tmp = 1.0 - math.sqrt((0.5 + (0.5 / math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(0.125 * (x ^ 2.0)));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0002)
		tmp = (-0.0859375 * (x ^ 4.0)) + (0.125 * (x ^ 2.0));
	else
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.0002

   1. Initial program 55.2%

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

      1. distribute-lft-in 55.2%

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

      2. metadata-eval 55.2%

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

      3. associate-*r/ 55.2%

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

      4. metadata-eval 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}} \]

   if 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]

Alternative 6: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.00000001:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.00000001)
   (* x (* x 0.125))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.00000001) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 1.00000001) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 1.0 - Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.00000001:
		tmp = x * (x * 0.125)
	else:
		tmp = 1.0 - math.sqrt((0.5 + (0.5 / math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.00000001)
		tmp = Float64(x * Float64(x * 0.125));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.00000001)
		tmp = x * (x * 0.125);
	else
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.00000001], N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.0000000099999999

   1. Initial program 55.2%

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

      1. distribute-lft-in 55.2%

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

      2. metadata-eval 55.2%

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

      3. associate-*r/ 55.2%

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

      4. metadata-eval 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in x around 0 55.2%

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

      1. *-commutative 55.2%

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

   6. Simplified 55.2%

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

      1. add-sqr-sqrt 55.2%

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

      2. sqrt-unprod 55.2%

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

      3. pow2 55.2%

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

      4. associate--r+ 75.2%

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

      5. metadata-eval 75.2%

         \[\leadsto \sqrt{{\left(\color{blue}{0} - {x}^{2} \cdot -0.125\right)}^{2}} \]

      6. sub0-neg 75.2%

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

   8. Applied egg-rr 75.2%

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

      1. unpow2 75.2%

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

      2. distribute-rgt-neg-in 75.2%

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

      3. metadata-eval 75.2%

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

      4. distribute-rgt-neg-in 75.2%

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

      5. metadata-eval 75.2%

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

      6. swap-sqr 75.2%

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

      7. pow-sqr 75.2%

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

      8. metadata-eval 75.2%

         \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(0.125 \cdot 0.125\right)} \]

      9. metadata-eval 75.2%

         \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{0.015625}} \]

   10. Simplified 75.2%

       \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 0.015625}} \]
   11. Step-by-step derivation

       1. *-commutative 75.2%

          \[\leadsto \sqrt{\color{blue}{0.015625 \cdot {x}^{4}}} \]

       2. metadata-eval 75.2%

          \[\leadsto \sqrt{\color{blue}{\left(0.125 \cdot 0.125\right)} \cdot {x}^{4}} \]

       3. metadata-eval 75.2%

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

       4. pow-prod-up 75.2%

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

       5. swap-sqr 75.2%

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

       6. sqrt-unprod 99.5%

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

       7. add-sqr-sqrt 99.8%

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

       8. unpow2 99.8%

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

       9. associate-*r* 99.8%

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

   12. Applied egg-rr 99.8%

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

   if 1.0000000099999999 < (hypot.f64 1 x)

   1. Initial program 98.1%

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

      1. distribute-lft-in 98.1%

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

      2. metadata-eval 98.1%

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

      3. associate-*r/ 98.1%

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

      4. metadata-eval 98.1%

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

   3. Simplified 98.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.00000001:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]

Alternative 7: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = x * (x * 0.125);
	} else {
		tmp = (0.5 + (0.5 / x)) * (1.0 / (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 = x * (x * 0.125);
	} else {
		tmp = (0.5 + (0.5 / x)) * (1.0 / (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 = x * (x * 0.125)
	else:
		tmp = (0.5 + (0.5 / x)) * (1.0 / (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 * Float64(x * 0.125));
	else
		tmp = Float64(Float64(0.5 + Float64(0.5 / x)) * Float64(1.0 / 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 * (x * 0.125);
	else
		tmp = (0.5 + (0.5 / x)) * (1.0 / (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[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[Sqrt[N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 2

   1. Initial program 55.4%

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

      1. distribute-lft-in 55.4%

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

      2. metadata-eval 55.4%

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

      3. associate-*r/ 55.4%

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

      4. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in x around 0 55.0%

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

      1. *-commutative 55.0%

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

   6. Simplified 55.0%

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

      1. add-sqr-sqrt 55.0%

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

      2. sqrt-unprod 55.0%

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

      3. pow2 55.0%

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

      4. associate--r+ 74.8%

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

      5. metadata-eval 74.8%

         \[\leadsto \sqrt{{\left(\color{blue}{0} - {x}^{2} \cdot -0.125\right)}^{2}} \]

      6. sub0-neg 74.8%

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

   8. Applied egg-rr 74.8%

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

      1. unpow2 74.8%

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

      2. distribute-rgt-neg-in 74.8%

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

      3. metadata-eval 74.8%

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

      4. distribute-rgt-neg-in 74.8%

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

      5. metadata-eval 74.8%

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

      6. swap-sqr 74.8%

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

      7. pow-sqr 74.7%

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

      8. metadata-eval 74.7%

         \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(0.125 \cdot 0.125\right)} \]

      9. metadata-eval 74.7%

         \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{0.015625}} \]

   10. Simplified 74.7%

       \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 0.015625}} \]
   11. Step-by-step derivation

       1. *-commutative 74.7%

          \[\leadsto \sqrt{\color{blue}{0.015625 \cdot {x}^{4}}} \]

       2. metadata-eval 74.7%

          \[\leadsto \sqrt{\color{blue}{\left(0.125 \cdot 0.125\right)} \cdot {x}^{4}} \]

       3. metadata-eval 74.7%

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

       4. pow-prod-up 74.8%

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

       5. swap-sqr 74.8%

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

       6. sqrt-unprod 98.7%

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

       7. add-sqr-sqrt 99.0%

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

       8. unpow2 99.0%

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

       9. associate-*r* 99.0%

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

   12. Applied egg-rr 99.0%

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

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around -inf 96.7%

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

      1. associate-*r/ 96.7%

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

      2. metadata-eval 96.7%

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

   6. Simplified 96.7%

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

      1. flip-- 96.7%

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

      2. div-inv 96.7%

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

      3. metadata-eval 96.7%

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

      4. add-sqr-sqrt 98.2%

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

      5. associate--r- 98.2%

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

      6. metadata-eval 98.2%

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

      7. sub-neg 98.2%

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

      8. distribute-neg-frac 98.2%

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

      9. metadata-eval 98.2%

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

   8. Applied egg-rr 98.2%

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

4. Final simplification 98.6%

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

Alternative 8: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = x * (x * 0.125);
	} 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 = x * (x * 0.125);
	} 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 = x * (x * 0.125)
	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(x * Float64(x * 0.125));
	else
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = x * (x * 0.125);
	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[(x * N[(x * 0.125), $MachinePrecision]), $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 1 x) < 2

   1. Initial program 55.4%

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

      1. distribute-lft-in 55.4%

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

      2. metadata-eval 55.4%

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

      3. associate-*r/ 55.4%

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

      4. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in x around 0 55.0%

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

      1. *-commutative 55.0%

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

   6. Simplified 55.0%

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

      1. add-sqr-sqrt 55.0%

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

      2. sqrt-unprod 55.0%

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

      3. pow2 55.0%

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

      4. associate--r+ 74.8%

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

      5. metadata-eval 74.8%

         \[\leadsto \sqrt{{\left(\color{blue}{0} - {x}^{2} \cdot -0.125\right)}^{2}} \]

      6. sub0-neg 74.8%

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

   8. Applied egg-rr 74.8%

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

      1. unpow2 74.8%

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

      2. distribute-rgt-neg-in 74.8%

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

      3. metadata-eval 74.8%

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

      4. distribute-rgt-neg-in 74.8%

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

      5. metadata-eval 74.8%

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

      6. swap-sqr 74.8%

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

      7. pow-sqr 74.7%

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

      8. metadata-eval 74.7%

         \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(0.125 \cdot 0.125\right)} \]

      9. metadata-eval 74.7%

         \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{0.015625}} \]

   10. Simplified 74.7%

       \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 0.015625}} \]
   11. Step-by-step derivation

       1. *-commutative 74.7%

          \[\leadsto \sqrt{\color{blue}{0.015625 \cdot {x}^{4}}} \]

       2. metadata-eval 74.7%

          \[\leadsto \sqrt{\color{blue}{\left(0.125 \cdot 0.125\right)} \cdot {x}^{4}} \]

       3. metadata-eval 74.7%

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

       4. pow-prod-up 74.8%

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

       5. swap-sqr 74.8%

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

       6. sqrt-unprod 98.7%

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

       7. add-sqr-sqrt 99.0%

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

       8. unpow2 99.0%

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

       9. associate-*r* 99.0%

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

   12. Applied egg-rr 99.0%

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

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

      1. flip-- 98.4%

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

      2. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 100.0%

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

      5. associate--r+ 100.0%

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

      6. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 97.3%

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

4. Final simplification 98.1%

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

Alternative 9: 97.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 1.5))) (- 1.0 (sqrt 0.5)) (* x (* x 0.125))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 1.5)) {
		tmp = 1.0 - sqrt(0.5);
	} else {
		tmp = x * (x * 0.125);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 1.5d0))) then
        tmp = 1.0d0 - sqrt(0.5d0)
    else
        tmp = x * (x * 0.125d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 1.5)) {
		tmp = 1.0 - Math.sqrt(0.5);
	} else {
		tmp = x * (x * 0.125);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.55) or not (x <= 1.5):
		tmp = 1.0 - math.sqrt(0.5)
	else:
		tmp = x * (x * 0.125)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 1.5))
		tmp = Float64(1.0 - sqrt(0.5));
	else
		tmp = Float64(x * Float64(x * 0.125));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 1.5)))
		tmp = 1.0 - sqrt(0.5);
	else
		tmp = x * (x * 0.125);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 1.5]], $MachinePrecision]], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 1.5 < 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. Taylor expanded in x around inf 95.8%

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

   if -1.55000000000000004 < x < 1.5

   1. Initial program 55.4%

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

      1. distribute-lft-in 55.4%

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

      2. metadata-eval 55.4%

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

      3. associate-*r/ 55.4%

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

      4. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in x around 0 55.0%

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

      1. *-commutative 55.0%

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

   6. Simplified 55.0%

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

      1. add-sqr-sqrt 55.0%

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

      2. sqrt-unprod 55.0%

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

      3. pow2 55.0%

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

      4. associate--r+ 74.8%

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

      5. metadata-eval 74.8%

         \[\leadsto \sqrt{{\left(\color{blue}{0} - {x}^{2} \cdot -0.125\right)}^{2}} \]

      6. sub0-neg 74.8%

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

   8. Applied egg-rr 74.8%

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

      1. unpow2 74.8%

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

      2. distribute-rgt-neg-in 74.8%

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

      3. metadata-eval 74.8%

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

      4. distribute-rgt-neg-in 74.8%

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

      5. metadata-eval 74.8%

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

      6. swap-sqr 74.8%

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

      7. pow-sqr 74.7%

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

      8. metadata-eval 74.7%

         \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(0.125 \cdot 0.125\right)} \]

      9. metadata-eval 74.7%

         \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{0.015625}} \]

   10. Simplified 74.7%

       \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 0.015625}} \]
   11. Step-by-step derivation

       1. *-commutative 74.7%

          \[\leadsto \sqrt{\color{blue}{0.015625 \cdot {x}^{4}}} \]

       2. metadata-eval 74.7%

          \[\leadsto \sqrt{\color{blue}{\left(0.125 \cdot 0.125\right)} \cdot {x}^{4}} \]

       3. metadata-eval 74.7%

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

       4. pow-prod-up 74.8%

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

       5. swap-sqr 74.8%

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

       6. sqrt-unprod 98.7%

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

       7. add-sqr-sqrt 99.0%

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

       8. unpow2 99.0%

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

       9. associate-*r* 99.0%

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

   12. Applied egg-rr 99.0%

       \[\leadsto \color{blue}{\left(0.125 \cdot x\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \end{array} \]

Alternative 10: 60.8% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.45) 0.25 (if (<= x 1.45) (* x (* x 0.125)) 0.25)))

C

double code(double x) {
	double tmp;
	if (x <= -1.45) {
		tmp = 0.25;
	} else if (x <= 1.45) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.45d0)) then
        tmp = 0.25d0
    else if (x <= 1.45d0) then
        tmp = x * (x * 0.125d0)
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.45) {
		tmp = 0.25;
	} else if (x <= 1.45) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.45:
		tmp = 0.25
	elif x <= 1.45:
		tmp = x * (x * 0.125)
	else:
		tmp = 0.25
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.45)
		tmp = 0.25;
	elseif (x <= 1.45)
		tmp = Float64(x * Float64(x * 0.125));
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.45)
		tmp = 0.25;
	elseif (x <= 1.45)
		tmp = x * (x * 0.125);
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.45], 0.25, If[LessEqual[x, 1.45], N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision], 0.25]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999996 or 1.44999999999999996 < 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. Step-by-step derivation

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate--r+ 100.0%

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

      5. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 22.7%

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

   7. Taylor expanded in x around inf 22.7%

      \[\leadsto \color{blue}{0.25} \]

   if -1.44999999999999996 < x < 1.44999999999999996

   1. Initial program 55.4%

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

      1. distribute-lft-in 55.4%

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

      2. metadata-eval 55.4%

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

      3. associate-*r/ 55.4%

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

      4. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in x around 0 55.0%

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

      1. *-commutative 55.0%

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

   6. Simplified 55.0%

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

      1. add-sqr-sqrt 55.0%

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

      2. sqrt-unprod 55.0%

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

      3. pow2 55.0%

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

      4. associate--r+ 74.8%

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

      5. metadata-eval 74.8%

         \[\leadsto \sqrt{{\left(\color{blue}{0} - {x}^{2} \cdot -0.125\right)}^{2}} \]

      6. sub0-neg 74.8%

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

   8. Applied egg-rr 74.8%

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

      1. unpow2 74.8%

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

      2. distribute-rgt-neg-in 74.8%

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

      3. metadata-eval 74.8%

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

      4. distribute-rgt-neg-in 74.8%

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

      5. metadata-eval 74.8%

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

      6. swap-sqr 74.8%

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

      7. pow-sqr 74.7%

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

      8. metadata-eval 74.7%

         \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(0.125 \cdot 0.125\right)} \]

      9. metadata-eval 74.7%

         \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{0.015625}} \]

   10. Simplified 74.7%

       \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 0.015625}} \]
   11. Step-by-step derivation

       1. *-commutative 74.7%

          \[\leadsto \sqrt{\color{blue}{0.015625 \cdot {x}^{4}}} \]

       2. metadata-eval 74.7%

          \[\leadsto \sqrt{\color{blue}{\left(0.125 \cdot 0.125\right)} \cdot {x}^{4}} \]

       3. metadata-eval 74.7%

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

       4. pow-prod-up 74.8%

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

       5. swap-sqr 74.8%

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

       6. sqrt-unprod 98.7%

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

       7. add-sqr-sqrt 99.0%

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

       8. unpow2 99.0%

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

       9. associate-*r* 99.0%

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

   12. Applied egg-rr 99.0%

       \[\leadsto \color{blue}{\left(0.125 \cdot x\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]

Alternative 11: 60.8% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 + \frac{0.25}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.45) 0.25 (if (<= x 1.75) (* x (* x 0.125)) (+ 0.25 (/ 0.25 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.45) {
		tmp = 0.25;
	} else if (x <= 1.75) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 0.25 + (0.25 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.45d0)) then
        tmp = 0.25d0
    else if (x <= 1.75d0) then
        tmp = x * (x * 0.125d0)
    else
        tmp = 0.25d0 + (0.25d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.45) {
		tmp = 0.25;
	} else if (x <= 1.75) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 0.25 + (0.25 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.45:
		tmp = 0.25
	elif x <= 1.75:
		tmp = x * (x * 0.125)
	else:
		tmp = 0.25 + (0.25 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.45)
		tmp = 0.25;
	elseif (x <= 1.75)
		tmp = Float64(x * Float64(x * 0.125));
	else
		tmp = Float64(0.25 + Float64(0.25 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.45)
		tmp = 0.25;
	elseif (x <= 1.75)
		tmp = x * (x * 0.125);
	else
		tmp = 0.25 + (0.25 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.45], 0.25, If[LessEqual[x, 1.75], N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision], N[(0.25 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999996

   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. Step-by-step derivation

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate--r+ 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%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 22.7%

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

   7. Taylor expanded in x around inf 22.8%

      \[\leadsto \color{blue}{0.25} \]

   if -1.44999999999999996 < x < 1.75

   1. Initial program 55.4%

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

      1. distribute-lft-in 55.4%

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

      2. metadata-eval 55.4%

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

      3. associate-*r/ 55.4%

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

      4. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in x around 0 55.0%

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

      1. *-commutative 55.0%

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

   6. Simplified 55.0%

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

      1. add-sqr-sqrt 55.0%

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

      2. sqrt-unprod 55.0%

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

      3. pow2 55.0%

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

      4. associate--r+ 74.8%

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

      5. metadata-eval 74.8%

         \[\leadsto \sqrt{{\left(\color{blue}{0} - {x}^{2} \cdot -0.125\right)}^{2}} \]

      6. sub0-neg 74.8%

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

   8. Applied egg-rr 74.8%

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

      1. unpow2 74.8%

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

      2. distribute-rgt-neg-in 74.8%

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

      3. metadata-eval 74.8%

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

      4. distribute-rgt-neg-in 74.8%

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

      5. metadata-eval 74.8%

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

      6. swap-sqr 74.8%

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

      7. pow-sqr 74.7%

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

      8. metadata-eval 74.7%

         \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(0.125 \cdot 0.125\right)} \]

      9. metadata-eval 74.7%

         \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{0.015625}} \]

   10. Simplified 74.7%

       \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 0.015625}} \]
   11. Step-by-step derivation

       1. *-commutative 74.7%

          \[\leadsto \sqrt{\color{blue}{0.015625 \cdot {x}^{4}}} \]

       2. metadata-eval 74.7%

          \[\leadsto \sqrt{\color{blue}{\left(0.125 \cdot 0.125\right)} \cdot {x}^{4}} \]

       3. metadata-eval 74.7%

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

       4. pow-prod-up 74.8%

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

       5. swap-sqr 74.8%

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

       6. sqrt-unprod 98.7%

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

       7. add-sqr-sqrt 99.0%

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

       8. unpow2 99.0%

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

       9. associate-*r* 99.0%

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

   12. Applied egg-rr 99.0%

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

   if 1.75 < x

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 22.7%

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

   7. Taylor expanded in x around -inf 22.7%

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

      1. associate-*r/ 22.7%

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

      2. metadata-eval 22.7%

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

   9. Simplified 22.7%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 + \frac{0.25}{x}\\ \end{array} \]

Alternative 12: 60.8% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8:\\ \;\;\;\;0.25 - \frac{0.25}{x}\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 + \frac{0.25}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.8)
   (- 0.25 (/ 0.25 x))
   (if (<= x 1.75) (* x (* x 0.125)) (+ 0.25 (/ 0.25 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.8) {
		tmp = 0.25 - (0.25 / x);
	} else if (x <= 1.75) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 0.25 + (0.25 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.8d0)) then
        tmp = 0.25d0 - (0.25d0 / x)
    else if (x <= 1.75d0) then
        tmp = x * (x * 0.125d0)
    else
        tmp = 0.25d0 + (0.25d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.8) {
		tmp = 0.25 - (0.25 / x);
	} else if (x <= 1.75) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 0.25 + (0.25 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.8:
		tmp = 0.25 - (0.25 / x)
	elif x <= 1.75:
		tmp = x * (x * 0.125)
	else:
		tmp = 0.25 + (0.25 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.8)
		tmp = Float64(0.25 - Float64(0.25 / x));
	elseif (x <= 1.75)
		tmp = Float64(x * Float64(x * 0.125));
	else
		tmp = Float64(0.25 + Float64(0.25 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.8)
		tmp = 0.25 - (0.25 / x);
	elseif (x <= 1.75)
		tmp = x * (x * 0.125);
	else
		tmp = 0.25 + (0.25 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.8], N[(0.25 - N[(0.25 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.75], N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision], N[(0.25 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.80000000000000004

   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. Step-by-step derivation

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate--r+ 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%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 22.7%

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

   7. Taylor expanded in x around inf 22.9%

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

      1. associate-*r/ 22.9%

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

      2. metadata-eval 22.9%

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

   9. Simplified 22.9%

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

   if -1.80000000000000004 < x < 1.75

   1. Initial program 55.4%

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

      1. distribute-lft-in 55.4%

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

      2. metadata-eval 55.4%

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

      3. associate-*r/ 55.4%

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

      4. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in x around 0 55.0%

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

      1. *-commutative 55.0%

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

   6. Simplified 55.0%

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

      1. add-sqr-sqrt 55.0%

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

      2. sqrt-unprod 55.0%

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

      3. pow2 55.0%

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

      4. associate--r+ 74.8%

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

      5. metadata-eval 74.8%

         \[\leadsto \sqrt{{\left(\color{blue}{0} - {x}^{2} \cdot -0.125\right)}^{2}} \]

      6. sub0-neg 74.8%

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

   8. Applied egg-rr 74.8%

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

      1. unpow2 74.8%

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

      2. distribute-rgt-neg-in 74.8%

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

      3. metadata-eval 74.8%

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

      4. distribute-rgt-neg-in 74.8%

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

      5. metadata-eval 74.8%

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

      6. swap-sqr 74.8%

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

      7. pow-sqr 74.7%

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

      8. metadata-eval 74.7%

         \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(0.125 \cdot 0.125\right)} \]

      9. metadata-eval 74.7%

         \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{0.015625}} \]

   10. Simplified 74.7%

       \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 0.015625}} \]
   11. Step-by-step derivation

       1. *-commutative 74.7%

          \[\leadsto \sqrt{\color{blue}{0.015625 \cdot {x}^{4}}} \]

       2. metadata-eval 74.7%

          \[\leadsto \sqrt{\color{blue}{\left(0.125 \cdot 0.125\right)} \cdot {x}^{4}} \]

       3. metadata-eval 74.7%

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

       4. pow-prod-up 74.8%

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

       5. swap-sqr 74.8%

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

       6. sqrt-unprod 98.7%

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

       7. add-sqr-sqrt 99.0%

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

       8. unpow2 99.0%

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

       9. associate-*r* 99.0%

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

   12. Applied egg-rr 99.0%

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

   if 1.75 < x

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 22.7%

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

   7. Taylor expanded in x around -inf 22.7%

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

      1. associate-*r/ 22.7%

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

      2. metadata-eval 22.7%

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

   9. Simplified 22.7%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8:\\ \;\;\;\;0.25 - \frac{0.25}{x}\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 + \frac{0.25}{x}\\ \end{array} \]

Alternative 13: 37.7% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-77}:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.1e-77) 0.25 (if (<= x 2.1e-77) 0.0 0.25)))

C

double code(double x) {
	double tmp;
	if (x <= -2.1e-77) {
		tmp = 0.25;
	} else if (x <= 2.1e-77) {
		tmp = 0.0;
	} 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 = 0.25d0
    else if (x <= 2.1d-77) then
        tmp = 0.0d0
    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 = 0.25;
	} else if (x <= 2.1e-77) {
		tmp = 0.0;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.1e-77:
		tmp = 0.25
	elif x <= 2.1e-77:
		tmp = 0.0
	else:
		tmp = 0.25
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -2.1e-77], 0.25, If[LessEqual[x, 2.1e-77], 0.0, 0.25]]

Derivation

1. Split input into 2 regimes

2. if x < -2.10000000000000015e-77 or 2.10000000000000015e-77 < x

   1. Initial program 82.8%

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

      1. distribute-lft-in 82.8%

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

      2. metadata-eval 82.8%

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

      3. associate-*r/ 82.8%

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

      4. metadata-eval 82.8%

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

   3. Simplified 82.8%

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

      1. flip-- 82.8%

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

      2. metadata-eval 82.8%

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

      3. add-sqr-sqrt 84.0%

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

      4. associate--r+ 84.0%

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

      5. metadata-eval 84.0%

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

   5. Applied egg-rr 84.0%

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

   6. Taylor expanded in x around 0 20.2%

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

   7. Taylor expanded in x around inf 20.0%

      \[\leadsto \color{blue}{0.25} \]

   if -2.10000000000000015e-77 < x < 2.10000000000000015e-77

   1. Initial program 68.4%

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

      1. distribute-lft-in 68.4%

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

      2. metadata-eval 68.4%

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

      3. associate-*r/ 68.4%

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

      4. metadata-eval 68.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in x around 0 68.4%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-77}:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]

Alternative 14: 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 77.3%

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

   1. distribute-lft-in 77.3%

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

   2. metadata-eval 77.3%

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

   3. associate-*r/ 77.3%

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

   4. metadata-eval 77.3%

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

3. Simplified 77.3%

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

   1. flip-- 77.3%

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

   2. metadata-eval 77.3%

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

   3. add-sqr-sqrt 78.0%

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

   4. associate--r+ 78.1%

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

   5. metadata-eval 78.1%

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

5. Applied egg-rr 78.1%

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

6. Taylor expanded in x around 0 38.6%

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

7. Taylor expanded in x around inf 13.6%

   \[\leadsto \color{blue}{0.25} \]

8. Final simplification 13.6%

   \[\leadsto 0.25 \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))