Given's Rotation SVD example, simplified

Percentage Accurate: 98.4% → 99.9%

Time: 16.6s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. x = -4.3010013284805526e-234

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}\\ t_1 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \frac{e^{\log \left(\frac{t\_0 + -0.25}{-0.5 - t\_1}\right)}}{1 + \frac{\sqrt{0.25 - t\_0}}{\sqrt{0.5 - t\_1}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.25 (fma x x 1.0))) (t_1 (/ 0.5 (hypot 1.0 x))))
   (/
    (exp (log (/ (+ t_0 -0.25) (- -0.5 t_1))))
    (+ 1.0 (/ (sqrt (- 0.25 t_0)) (sqrt (- 0.5 t_1)))))))

C

double code(double x) {
	double t_0 = 0.25 / fma(x, x, 1.0);
	double t_1 = 0.5 / hypot(1.0, x);
	return exp(log(((t_0 + -0.25) / (-0.5 - t_1)))) / (1.0 + (sqrt((0.25 - t_0)) / sqrt((0.5 - t_1))));
}

Julia

function code(x)
	t_0 = Float64(0.25 / fma(x, x, 1.0))
	t_1 = Float64(0.5 / hypot(1.0, x))
	return Float64(exp(log(Float64(Float64(t_0 + -0.25) / Float64(-0.5 - t_1)))) / Float64(1.0 + Float64(sqrt(Float64(0.25 - t_0)) / sqrt(Float64(0.5 - t_1)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(N[Exp[N[Log[N[(N[(t$95$0 + -0.25), $MachinePrecision] / N[(-0.5 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(N[Sqrt[N[(0.25 - t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. flip-- 98.4%

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

   2. metadata-eval 98.4%

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

   3. add-sqr-sqrt 99.9%

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

   4. associate--r+ 99.9%

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

   5. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. flip-- 99.9%

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

   2. div-inv 99.9%

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

   3. metadata-eval 99.9%

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

   4. frac-times 99.9%

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

   5. metadata-eval 99.9%

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

   6. pow2 99.9%

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

8. Applied egg-rr 99.9%

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

   1. associate-*r/ 99.9%

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

   2. *-rgt-identity 99.9%

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

   3. times-frac 99.9%

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

   4. metadata-eval 99.9%

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

   5. metadata-eval 99.9%

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

   6. times-frac 99.9%

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

   7. *-commutative 99.9%

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

   8. *-commutative 99.9%

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

   9. neg-mul-1 99.9%

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

   10. neg-sub0 99.9%

       \[\leadsto \frac{\frac{\color{blue}{0 - \left(0.25 - \frac{0.25}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{2}}\right)}}{-1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   11. neg-mul-1 99.9%

       \[\leadsto \frac{\frac{0 - \left(0.25 - \frac{0.25}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{2}}\right)}{\color{blue}{-\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

10. Simplified 99.9%

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

    1. flip-+ 99.9%

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

    2. sqrt-div 99.9%

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

    3. metadata-eval 99.9%

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

    4. frac-times 99.9%

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

    5. metadata-eval 99.9%

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

    6. hypot-undefine 99.9%

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

    7. hypot-undefine 99.9%

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

    8. rem-square-sqrt 99.9%

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

    9. metadata-eval 99.9%

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

    10. +-commutative 99.9%

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

    11. fma-define 99.9%

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

12. Applied egg-rr 99.9%

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

    1. add-exp-log 100.0%

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

    2. +-commutative 100.0%

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

    3. +-commutative 100.0%

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

    4. unpow2 100.0%

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

    5. fma-undefine 100.0%

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

14. Applied egg-rr 100.0%

    \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + -0.25}{-0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
15. Add Preprocessing

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}\\ t_1 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ {\left({\left(\frac{t\_0 + -0.25}{\left(-0.5 - t\_1\right) \cdot \left(1 + \sqrt{\frac{0.25 - t\_0}{0.5 - t\_1}}\right)}\right)}^{3}\right)}^{0.3333333333333333} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.25 (fma x x 1.0))) (t_1 (/ 0.5 (hypot 1.0 x))))
   (pow
    (pow
     (/
      (+ t_0 -0.25)
      (* (- -0.5 t_1) (+ 1.0 (sqrt (/ (- 0.25 t_0) (- 0.5 t_1))))))
     3.0)
    0.3333333333333333)))

C

double code(double x) {
	double t_0 = 0.25 / fma(x, x, 1.0);
	double t_1 = 0.5 / hypot(1.0, x);
	return pow(pow(((t_0 + -0.25) / ((-0.5 - t_1) * (1.0 + sqrt(((0.25 - t_0) / (0.5 - t_1)))))), 3.0), 0.3333333333333333);
}

Julia

function code(x)
	t_0 = Float64(0.25 / fma(x, x, 1.0))
	t_1 = Float64(0.5 / hypot(1.0, x))
	return (Float64(Float64(t_0 + -0.25) / Float64(Float64(-0.5 - t_1) * Float64(1.0 + sqrt(Float64(Float64(0.25 - t_0) / Float64(0.5 - t_1)))))) ^ 3.0) ^ 0.3333333333333333
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, N[Power[N[Power[N[(N[(t$95$0 + -0.25), $MachinePrecision] / N[(N[(-0.5 - t$95$1), $MachinePrecision] * N[(1.0 + N[Sqrt[N[(N[(0.25 - t$95$0), $MachinePrecision] / N[(0.5 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]]

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. flip-- 98.4%

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

   2. metadata-eval 98.4%

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

   3. add-sqr-sqrt 99.9%

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

   4. associate--r+ 99.9%

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

   5. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. flip-- 99.9%

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

   2. div-inv 99.9%

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

   3. metadata-eval 99.9%

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

   4. frac-times 99.9%

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

   5. metadata-eval 99.9%

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

   6. pow2 99.9%

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

8. Applied egg-rr 99.9%

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

   1. associate-*r/ 99.9%

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

   2. *-rgt-identity 99.9%

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

   3. times-frac 99.9%

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

   4. metadata-eval 99.9%

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

   5. metadata-eval 99.9%

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

   6. times-frac 99.9%

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

   7. *-commutative 99.9%

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

   8. *-commutative 99.9%

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

   9. neg-mul-1 99.9%

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

   10. neg-sub0 99.9%

       \[\leadsto \frac{\frac{\color{blue}{0 - \left(0.25 - \frac{0.25}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{2}}\right)}}{-1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   11. neg-mul-1 99.9%

       \[\leadsto \frac{\frac{0 - \left(0.25 - \frac{0.25}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{2}}\right)}{\color{blue}{-\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

10. Simplified 99.9%

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

    1. flip-+ 99.9%

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

    2. sqrt-div 99.9%

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

    3. metadata-eval 99.9%

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

    4. frac-times 99.9%

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

    5. metadata-eval 99.9%

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

    6. hypot-undefine 99.9%

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

    7. hypot-undefine 99.9%

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

    8. rem-square-sqrt 99.9%

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

    9. metadata-eval 99.9%

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

    10. +-commutative 99.9%

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

    11. fma-define 99.9%

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

12. Applied egg-rr 99.9%

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

    1. add-cbrt-cube 98.5%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\frac{-0.25 + \frac{0.25}{1 + {x}^{2}}}{-0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \cdot \frac{\frac{-0.25 + \frac{0.25}{1 + {x}^{2}}}{-0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right) \cdot \frac{\frac{-0.25 + \frac{0.25}{1 + {x}^{2}}}{-0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}}} \]

    2. pow1/3 99.9%

       \[\leadsto \color{blue}{{\left(\left(\frac{\frac{-0.25 + \frac{0.25}{1 + {x}^{2}}}{-0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \cdot \frac{\frac{-0.25 + \frac{0.25}{1 + {x}^{2}}}{-0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right) \cdot \frac{\frac{-0.25 + \frac{0.25}{1 + {x}^{2}}}{-0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right)}^{0.3333333333333333}} \]

14. Applied egg-rr 99.9%

    \[\leadsto \color{blue}{{\left({\left(\frac{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + -0.25}{\left(1 + \sqrt{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\right) \cdot \left(-0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

15. Final simplification 99.9%

    \[\leadsto {\left({\left(\frac{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + -0.25}{\left(-0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 + \sqrt{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\right)}\right)}^{3}\right)}^{0.3333333333333333} \]
16. Add Preprocessing

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ \frac{\frac{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + -0.25}{-0.5 + t\_0}}{1 + \sqrt{\frac{0.25 + \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.5 + t\_0}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x))))
   (/
    (/ (+ (/ 0.25 (fma x x 1.0)) -0.25) (+ -0.5 t_0))
    (+ 1.0 (sqrt (/ (+ 0.25 (/ -0.25 (fma x x 1.0))) (+ 0.5 t_0)))))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	return (((0.25 / fma(x, x, 1.0)) + -0.25) / (-0.5 + t_0)) / (1.0 + sqrt(((0.25 + (-0.25 / fma(x, x, 1.0))) / (0.5 + t_0))));
}

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	return Float64(Float64(Float64(Float64(0.25 / fma(x, x, 1.0)) + -0.25) / Float64(-0.5 + t_0)) / Float64(1.0 + sqrt(Float64(Float64(0.25 + Float64(-0.25 / fma(x, x, 1.0))) / Float64(0.5 + t_0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] + -0.25), $MachinePrecision] / N[(-0.5 + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(0.25 + N[(-0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. flip-- 98.4%

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

   2. metadata-eval 98.4%

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

   3. add-sqr-sqrt 99.9%

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

   4. associate--r+ 99.9%

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

   5. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. flip-- 99.9%

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

   2. div-inv 99.9%

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

   3. metadata-eval 99.9%

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

   4. frac-times 99.9%

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

   5. metadata-eval 99.9%

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

   6. pow2 99.9%

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

8. Applied egg-rr 99.9%

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

   1. associate-*r/ 99.9%

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

   2. *-rgt-identity 99.9%

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

   3. times-frac 99.9%

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

   4. metadata-eval 99.9%

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

   5. metadata-eval 99.9%

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

   6. times-frac 99.9%

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

   7. *-commutative 99.9%

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

   8. *-commutative 99.9%

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

   9. neg-mul-1 99.9%

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

   10. neg-sub0 99.9%

       \[\leadsto \frac{\frac{\color{blue}{0 - \left(0.25 - \frac{0.25}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{2}}\right)}}{-1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   11. neg-mul-1 99.9%

       \[\leadsto \frac{\frac{0 - \left(0.25 - \frac{0.25}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{2}}\right)}{\color{blue}{-\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

10. Simplified 99.9%

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

    1. flip-+ 99.9%

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

    2. sqrt-div 99.9%

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

    3. metadata-eval 99.9%

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

    4. frac-times 99.9%

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

    5. metadata-eval 99.9%

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

    6. hypot-undefine 99.9%

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

    7. hypot-undefine 99.9%

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

    8. rem-square-sqrt 99.9%

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

    9. metadata-eval 99.9%

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

    10. +-commutative 99.9%

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

    11. fma-define 99.9%

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

12. Applied egg-rr 99.9%

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

    1. *-un-lft-identity 99.9%

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

    2. associate-/l/ 99.9%

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

    3. +-commutative 99.9%

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

    4. +-commutative 99.9%

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

    5. unpow2 99.9%

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

    6. fma-undefine 99.9%

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

    7. sqrt-undiv 99.9%

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

14. Applied egg-rr 99.9%

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

    1. associate-*r/ 99.9%

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

    2. times-frac 99.9%

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

    3. *-commutative 99.9%

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

    4. associate-*r/ 99.9%

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

    5. *-rgt-identity 99.9%

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

    6. +-commutative 99.9%

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

    7. sub-neg 99.9%

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

    8. distribute-neg-frac 99.9%

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

    9. metadata-eval 99.9%

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

16. Simplified 99.9%

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

17. Final simplification 99.9%

    \[\leadsto \frac{\frac{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + -0.25}{-0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{0.25 + \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
18. Add Preprocessing

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 - \sqrt{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (- 0.5 (sqrt (/ 0.25 (fma x x 1.0))))
  (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))

C

double code(double x) {
	return (0.5 - sqrt((0.25 / fma(x, x, 1.0)))) / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x)))));
}

Julia

function code(x)
	return Float64(Float64(0.5 - sqrt(Float64(0.25 / fma(x, x, 1.0)))) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x))))))
end

Wolfram

code[x_] := N[(N[(0.5 - N[Sqrt[N[(0.25 / N[(x * x + 1.0), $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]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. flip-- 98.4%

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

   2. metadata-eval 98.4%

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

   3. add-sqr-sqrt 99.9%

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

   4. associate--r+ 99.9%

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

   5. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. add-sqr-sqrt 99.9%

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

   2. sqrt-unprod 99.9%

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

   3. frac-times 99.9%

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

   4. metadata-eval 99.9%

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

   5. hypot-undefine 99.9%

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

   6. hypot-undefine 99.9%

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

   7. rem-square-sqrt 99.9%

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

   8. metadata-eval 99.9%

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

   9. +-commutative 99.9%

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

   10. fma-define 99.9%

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

8. Applied egg-rr 99.9%

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

Alternative 5: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. flip-- 98.4%

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

   2. metadata-eval 98.4%

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

   3. add-sqr-sqrt 99.9%

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

   4. associate--r+ 99.9%

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

   5. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 6: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 1 - \sqrt{0.5 + \sqrt{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (+ 0.5 (sqrt (/ 0.25 (fma x x 1.0)))))))

C

double code(double x) {
	return 1.0 - sqrt((0.5 + sqrt((0.25 / fma(x, x, 1.0)))));
}

Julia

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 + sqrt(Float64(0.25 / fma(x, x, 1.0))))))
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 + N[Sqrt[N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. add-sqr-sqrt 99.9%

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

   2. sqrt-unprod 99.9%

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

   3. frac-times 99.9%

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

   4. metadata-eval 99.9%

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

   5. hypot-undefine 99.9%

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

   6. hypot-undefine 99.9%

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

   7. rem-square-sqrt 99.9%

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

   8. metadata-eval 99.9%

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

   9. +-commutative 99.9%

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

   10. fma-define 99.9%

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

6. Applied egg-rr 98.4%

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

Alternative 7: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

Alternative 8: 97.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. flip-- 98.4%

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

   2. metadata-eval 98.4%

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

   3. add-sqr-sqrt 99.9%

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

   4. associate--r+ 99.9%

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

   5. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf 97.2%

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

8. Taylor expanded in x around inf 96.9%

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

   1. associate-*r/ 96.9%

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

   2. metadata-eval 96.9%

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

10. Simplified 96.9%

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

Alternative 9: 97.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. flip-- 98.4%

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

   2. div-inv 98.4%

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

   3. metadata-eval 98.4%

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

   4. add-sqr-sqrt 99.9%

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

   5. associate--r+ 99.9%

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

   6. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf 96.6%

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

Alternative 10: 95.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in x around inf 95.1%

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

Alternative 11: 4.5% accurate, 23.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in x around 0 4.5%

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

   1. *-commutative 4.5%

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

7. Simplified 4.5%

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

   1. unpow2 4.5%

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

9. Applied egg-rr 4.5%

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

10. Final simplification 4.5%

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

Alternative 12: 4.5% accurate, 42.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x (* x 0.125)))

C

double code(double x) {
	return x * (x * 0.125);
}

Fortran

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

Java

public static double code(double x) {
	return x * (x * 0.125);
}

Python

def code(x):
	return x * (x * 0.125)

Julia

function code(x)
	return Float64(x * Float64(x * 0.125))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * 0.125);
end

Wolfram

code[x_] := N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in x around 0 4.5%

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

   1. *-commutative 4.5%

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

7. Simplified 4.5%

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

   1. +-commutative 4.5%

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

   2. associate--r+ 4.5%

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

   3. add-exp-log 4.5%

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

   4. *-commutative 4.5%

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

   5. cancel-sign-sub-inv 4.5%

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

   6. metadata-eval 4.5%

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

   7. log1p-undefine 4.5%

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

   8. expm1-undefine 4.5%

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

   9. expm1-log1p-u 4.5%

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

   10. unpow2 4.5%

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

   11. associate-*r* 4.5%

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

9. Applied egg-rr 4.5%

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

10. Final simplification 4.5%

    \[\leadsto x \cdot \left(x \cdot 0.125\right) \]
11. Add Preprocessing

Alternative 13: 3.1% accurate, 210.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r/ 98.4%

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

   4. metadata-eval 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in x around 0 3.1%

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

   1. metadata-eval 3.1%

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 3.1%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))