Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * sqrt(z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / 2.0d0) * (x + (y * sqrt(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * Math.sqrt(z)));
}

Python

def code(x, y, z):
	return (1.0 / 2.0) * (x + (y * math.sqrt(z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / 2.0) * Float64(x + Float64(y * sqrt(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / 2.0) * (x + (y * sqrt(z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * sqrt(z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / 2.0d0) * (x + (y * sqrt(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * Math.sqrt(z)));
}

Python

def code(x, y, z):
	return (1.0 / 2.0) * (x + (y * math.sqrt(z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / 2.0) * Float64(x + Float64(y * sqrt(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / 2.0) * (x + (y * sqrt(z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x + \frac{y}{{z}^{-0.5}}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 (+ x (/ y (pow z -0.5)))))

C

double code(double x, double y, double z) {
	return 0.5 * (x + (y / pow(z, -0.5)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * (x + (y / (z ** (-0.5d0))))
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * (x + (y / Math.pow(z, -0.5)));
}

Python

def code(x, y, z):
	return 0.5 * (x + (y / math.pow(z, -0.5)))

Julia

function code(x, y, z)
	return Float64(0.5 * Float64(x + Float64(y / (z ^ -0.5))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * (x + (y / (z ^ -0.5)));
end

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(x + N[(y / N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
2. Step-by-step derivation

   1. metadata-eval 99.8%

      \[\leadsto \color{blue}{0.5} \cdot \left(x + y \cdot \sqrt{z}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 48.3%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{\sqrt{y \cdot \sqrt{z}} \cdot \sqrt{y \cdot \sqrt{z}}}\right) \]

   2. sqrt-unprod 62.5%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{\sqrt{\left(y \cdot \sqrt{z}\right) \cdot \left(y \cdot \sqrt{z}\right)}}\right) \]

   3. pow1/2 62.5%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{{\left(\left(y \cdot \sqrt{z}\right) \cdot \left(y \cdot \sqrt{z}\right)\right)}^{0.5}}\right) \]

   4. *-commutative 62.5%

      \[\leadsto 0.5 \cdot \left(x + {\left(\color{blue}{\left(\sqrt{z} \cdot y\right)} \cdot \left(y \cdot \sqrt{z}\right)\right)}^{0.5}\right) \]

   5. *-commutative 62.5%

      \[\leadsto 0.5 \cdot \left(x + {\left(\left(\sqrt{z} \cdot y\right) \cdot \color{blue}{\left(\sqrt{z} \cdot y\right)}\right)}^{0.5}\right) \]

   6. swap-sqr 59.3%

      \[\leadsto 0.5 \cdot \left(x + {\color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \left(y \cdot y\right)\right)}}^{0.5}\right) \]

   7. add-sqr-sqrt 59.3%

      \[\leadsto 0.5 \cdot \left(x + {\left(\color{blue}{z} \cdot \left(y \cdot y\right)\right)}^{0.5}\right) \]

5. Applied egg-rr 59.3%

   \[\leadsto 0.5 \cdot \left(x + \color{blue}{{\left(z \cdot \left(y \cdot y\right)\right)}^{0.5}}\right) \]
6. Step-by-step derivation

   1. unpow1/2 59.3%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{\sqrt{z \cdot \left(y \cdot y\right)}}\right) \]

7. Simplified 59.3%

   \[\leadsto 0.5 \cdot \left(x + \color{blue}{\sqrt{z \cdot \left(y \cdot y\right)}}\right) \]
8. Step-by-step derivation

   1. sqrt-prod 63.3%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{\sqrt{z} \cdot \sqrt{y \cdot y}}\right) \]

   2. sqrt-prod 45.2%

      \[\leadsto 0.5 \cdot \left(x + \sqrt{z} \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}\right) \]

   3. pow1/2 45.2%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{{z}^{0.5}} \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)\right) \]

   4. metadata-eval 45.2%

      \[\leadsto 0.5 \cdot \left(x + {z}^{\color{blue}{\left(0.25 + 0.25\right)}} \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)\right) \]

   5. pow-prod-up 45.1%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right)} \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)\right) \]

   6. add-sqr-sqrt 99.6%

      \[\leadsto 0.5 \cdot \left(x + \left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot \color{blue}{y}\right) \]

   7. associate-*r* 99.6%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{{z}^{0.25} \cdot \left({z}^{0.25} \cdot y\right)}\right) \]

9. Applied egg-rr 99.6%

   \[\leadsto 0.5 \cdot \left(x + \color{blue}{{z}^{0.25} \cdot \left({z}^{0.25} \cdot y\right)}\right) \]
10. Step-by-step derivation

    1. associate-*r* 99.6%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot y}\right) \]

    2. pow-prod-up 99.8%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{{z}^{\left(0.25 + 0.25\right)}} \cdot y\right) \]

    3. metadata-eval 99.8%

       \[\leadsto 0.5 \cdot \left(x + {z}^{\color{blue}{0.5}} \cdot y\right) \]

    4. pow1/2 99.8%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{\sqrt{z}} \cdot y\right) \]

    5. *-commutative 99.8%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{y \cdot \sqrt{z}}\right) \]

    6. remove-double-div 99.7%

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

    7. metadata-eval 99.7%

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

    8. sqrt-div 99.7%

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

    9. div-inv 99.8%

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

    10. inv-pow 99.8%

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

    11. sqrt-pow1 99.8%

        \[\leadsto 0.5 \cdot \left(x + \frac{y}{\color{blue}{{z}^{\left(\frac{-1}{2}\right)}}}\right) \]

    12. metadata-eval 99.8%

        \[\leadsto 0.5 \cdot \left(x + \frac{y}{{z}^{\color{blue}{-0.5}}}\right) \]

11. Applied egg-rr 99.8%

    \[\leadsto 0.5 \cdot \left(x + \color{blue}{\frac{y}{{z}^{-0.5}}}\right) \]

12. Final simplification 99.8%

    \[\leadsto 0.5 \cdot \left(x + \frac{y}{{z}^{-0.5}}\right) \]

Alternative 2: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;t_0 \leq 10^{+133}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{{z}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (<= t_0 -5e-37)
     (* 0.5 t_0)
     (if (<= t_0 1e+133) (* 0.5 x) (* y (/ 0.5 (pow z -0.5)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double tmp;
	if (t_0 <= -5e-37) {
		tmp = 0.5 * t_0;
	} else if (t_0 <= 1e+133) {
		tmp = 0.5 * x;
	} else {
		tmp = y * (0.5 / pow(z, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(z)
    if (t_0 <= (-5d-37)) then
        tmp = 0.5d0 * t_0
    else if (t_0 <= 1d+133) then
        tmp = 0.5d0 * x
    else
        tmp = y * (0.5d0 / (z ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.sqrt(z);
	double tmp;
	if (t_0 <= -5e-37) {
		tmp = 0.5 * t_0;
	} else if (t_0 <= 1e+133) {
		tmp = 0.5 * x;
	} else {
		tmp = y * (0.5 / Math.pow(z, -0.5));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	tmp = 0
	if t_0 <= -5e-37:
		tmp = 0.5 * t_0
	elif t_0 <= 1e+133:
		tmp = 0.5 * x
	else:
		tmp = y * (0.5 / math.pow(z, -0.5))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	tmp = 0.0
	if (t_0 <= -5e-37)
		tmp = Float64(0.5 * t_0);
	elseif (t_0 <= 1e+133)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(y * Float64(0.5 / (z ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * sqrt(z);
	tmp = 0.0;
	if (t_0 <= -5e-37)
		tmp = 0.5 * t_0;
	elseif (t_0 <= 1e+133)
		tmp = 0.5 * x;
	else
		tmp = y * (0.5 / (z ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-37], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[t$95$0, 1e+133], N[(0.5 * x), $MachinePrecision], N[(y * N[(0.5 / N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (sqrt.f64 z)) < -4.9999999999999997e-37

   1. Initial program 99.6%

      \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 99.6%

         \[\leadsto \color{blue}{0.5} \cdot \left(x + y \cdot \sqrt{z}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]

   4. Taylor expanded in x around 0 76.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y \cdot \sqrt{z}\right)} \]

   if -4.9999999999999997e-37 < (*.f64 y (sqrt.f64 z)) < 1e133

   1. Initial program 99.9%

      \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 99.9%

         \[\leadsto \color{blue}{0.5} \cdot \left(x + y \cdot \sqrt{z}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]

   4. Taylor expanded in x around inf 76.2%

      \[\leadsto 0.5 \cdot \color{blue}{x} \]

   if 1e133 < (*.f64 y (sqrt.f64 z))

   1. Initial program 99.6%

      \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 99.6%

         \[\leadsto \color{blue}{0.5} \cdot \left(x + y \cdot \sqrt{z}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]

   4. Taylor expanded in x around 0 92.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y \cdot \sqrt{z}\right)} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 99.3%

         \[\leadsto 0.5 \cdot \left(x + \color{blue}{\sqrt{y \cdot \sqrt{z}} \cdot \sqrt{y \cdot \sqrt{z}}}\right) \]

      2. sqrt-unprod 46.4%

         \[\leadsto 0.5 \cdot \left(x + \color{blue}{\sqrt{\left(y \cdot \sqrt{z}\right) \cdot \left(y \cdot \sqrt{z}\right)}}\right) \]

      3. pow1/2 46.4%

         \[\leadsto 0.5 \cdot \left(x + \color{blue}{{\left(\left(y \cdot \sqrt{z}\right) \cdot \left(y \cdot \sqrt{z}\right)\right)}^{0.5}}\right) \]

      4. *-commutative 46.4%

         \[\leadsto 0.5 \cdot \left(x + {\left(\color{blue}{\left(\sqrt{z} \cdot y\right)} \cdot \left(y \cdot \sqrt{z}\right)\right)}^{0.5}\right) \]

      5. *-commutative 46.4%

         \[\leadsto 0.5 \cdot \left(x + {\left(\left(\sqrt{z} \cdot y\right) \cdot \color{blue}{\left(\sqrt{z} \cdot y\right)}\right)}^{0.5}\right) \]

      6. swap-sqr 46.4%

         \[\leadsto 0.5 \cdot \left(x + {\color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \left(y \cdot y\right)\right)}}^{0.5}\right) \]

      7. add-sqr-sqrt 46.4%

         \[\leadsto 0.5 \cdot \left(x + {\left(\color{blue}{z} \cdot \left(y \cdot y\right)\right)}^{0.5}\right) \]

   6. Applied egg-rr 46.4%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(z \cdot \left(y \cdot y\right)\right)}^{0.5}} \]
   7. Step-by-step derivation

      1. unpow1/2 46.4%

         \[\leadsto 0.5 \cdot \left(x + \color{blue}{\sqrt{z \cdot \left(y \cdot y\right)}}\right) \]

   8. Simplified 46.4%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{z \cdot \left(y \cdot y\right)}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 46.3%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{z \cdot \left(y \cdot y\right)}} \cdot \sqrt{0.5 \cdot \sqrt{z \cdot \left(y \cdot y\right)}}} \]

      2. sqrt-unprod 46.4%

         \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{z \cdot \left(y \cdot y\right)}\right) \cdot \left(0.5 \cdot \sqrt{z \cdot \left(y \cdot y\right)}\right)}} \]

      3. swap-sqr 46.4%

         \[\leadsto \sqrt{\color{blue}{\left(0.5 \cdot 0.5\right) \cdot \left(\sqrt{z \cdot \left(y \cdot y\right)} \cdot \sqrt{z \cdot \left(y \cdot y\right)}\right)}} \]

      4. add-sqr-sqrt 46.4%

         \[\leadsto \sqrt{\left(0.5 \cdot 0.5\right) \cdot \color{blue}{\left(z \cdot \left(y \cdot y\right)\right)}} \]

      5. *-commutative 46.4%

         \[\leadsto \sqrt{\left(0.5 \cdot 0.5\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot z\right)}} \]

      6. add-sqr-sqrt 46.4%

         \[\leadsto \sqrt{\left(0.5 \cdot 0.5\right) \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}\right)} \]

      7. swap-sqr 46.4%

         \[\leadsto \sqrt{\left(0.5 \cdot 0.5\right) \cdot \color{blue}{\left(\left(y \cdot \sqrt{z}\right) \cdot \left(y \cdot \sqrt{z}\right)\right)}} \]

      8. pow2 46.4%

         \[\leadsto \sqrt{\left(0.5 \cdot 0.5\right) \cdot \color{blue}{{\left(y \cdot \sqrt{z}\right)}^{2}}} \]

      9. remove-double-div 46.4%

         \[\leadsto \sqrt{\left(0.5 \cdot 0.5\right) \cdot {\left(y \cdot \color{blue}{\frac{1}{\frac{1}{\sqrt{z}}}}\right)}^{2}} \]

      10. metadata-eval 46.4%

          \[\leadsto \sqrt{\left(0.5 \cdot 0.5\right) \cdot {\left(y \cdot \frac{1}{\frac{\color{blue}{\sqrt{1}}}{\sqrt{z}}}\right)}^{2}} \]

      11. sqrt-div 46.4%

          \[\leadsto \sqrt{\left(0.5 \cdot 0.5\right) \cdot {\left(y \cdot \frac{1}{\color{blue}{\sqrt{\frac{1}{z}}}}\right)}^{2}} \]

      12. div-inv 46.4%

          \[\leadsto \sqrt{\left(0.5 \cdot 0.5\right) \cdot {\color{blue}{\left(\frac{y}{\sqrt{\frac{1}{z}}}\right)}}^{2}} \]

      13. clear-num 46.4%

          \[\leadsto \sqrt{\left(0.5 \cdot 0.5\right) \cdot {\color{blue}{\left(\frac{1}{\frac{\sqrt{\frac{1}{z}}}{y}}\right)}}^{2}} \]

      14. pow2 46.4%

          \[\leadsto \sqrt{\left(0.5 \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{\frac{\sqrt{\frac{1}{z}}}{y}} \cdot \frac{1}{\frac{\sqrt{\frac{1}{z}}}{y}}\right)}} \]

      15. swap-sqr 46.4%

          \[\leadsto \sqrt{\color{blue}{\left(0.5 \cdot \frac{1}{\frac{\sqrt{\frac{1}{z}}}{y}}\right) \cdot \left(0.5 \cdot \frac{1}{\frac{\sqrt{\frac{1}{z}}}{y}}\right)}} \]

      16. sqrt-unprod 92.5%

          \[\leadsto \color{blue}{\sqrt{0.5 \cdot \frac{1}{\frac{\sqrt{\frac{1}{z}}}{y}}} \cdot \sqrt{0.5 \cdot \frac{1}{\frac{\sqrt{\frac{1}{z}}}{y}}}} \]

   10. Applied egg-rr 92.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{{z}^{-0.5}}{y}}{0.5}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 92.9%

          \[\leadsto \color{blue}{\frac{1}{\frac{{z}^{-0.5}}{y}} \cdot 0.5} \]

       2. associate-*l/ 92.9%

          \[\leadsto \color{blue}{\frac{1 \cdot 0.5}{\frac{{z}^{-0.5}}{y}}} \]

       3. metadata-eval 92.9%

          \[\leadsto \frac{\color{blue}{0.5}}{\frac{{z}^{-0.5}}{y}} \]

       4. associate-/r/ 93.0%

          \[\leadsto \color{blue}{\frac{0.5}{{z}^{-0.5}} \cdot y} \]

       5. *-commutative 93.0%

          \[\leadsto \color{blue}{y \cdot \frac{0.5}{{z}^{-0.5}}} \]

   12. Simplified 93.0%

       \[\leadsto \color{blue}{y \cdot \frac{0.5}{{z}^{-0.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -5 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{elif}\;y \cdot \sqrt{z} \leq 10^{+133}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{{z}^{-0.5}}\\ \end{array} \]

Alternative 3: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -7.3 \cdot 10^{-37} \lor \neg \left(t_0 \leq 8.5 \cdot 10^{+132}\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -7.3e-37) (not (<= t_0 8.5e+132))) (* 0.5 t_0) (* 0.5 x))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double tmp;
	if ((t_0 <= -7.3e-37) || !(t_0 <= 8.5e+132)) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(z)
    if ((t_0 <= (-7.3d-37)) .or. (.not. (t_0 <= 8.5d+132))) then
        tmp = 0.5d0 * t_0
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.sqrt(z);
	double tmp;
	if ((t_0 <= -7.3e-37) || !(t_0 <= 8.5e+132)) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	tmp = 0
	if (t_0 <= -7.3e-37) or not (t_0 <= 8.5e+132):
		tmp = 0.5 * t_0
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	tmp = 0.0
	if ((t_0 <= -7.3e-37) || !(t_0 <= 8.5e+132))
		tmp = Float64(0.5 * t_0);
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * sqrt(z);
	tmp = 0.0;
	if ((t_0 <= -7.3e-37) || ~((t_0 <= 8.5e+132)))
		tmp = 0.5 * t_0;
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -7.3e-37], N[Not[LessEqual[t$95$0, 8.5e+132]], $MachinePrecision]], N[(0.5 * t$95$0), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (sqrt.f64 z)) < -7.2999999999999997e-37 or 8.49999999999999969e132 < (*.f64 y (sqrt.f64 z))

   1. Initial program 99.6%

      \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 99.6%

         \[\leadsto \color{blue}{0.5} \cdot \left(x + y \cdot \sqrt{z}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]

   4. Taylor expanded in x around 0 80.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y \cdot \sqrt{z}\right)} \]

   if -7.2999999999999997e-37 < (*.f64 y (sqrt.f64 z)) < 8.49999999999999969e132

   1. Initial program 99.9%

      \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 99.9%

         \[\leadsto \color{blue}{0.5} \cdot \left(x + y \cdot \sqrt{z}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]

   4. Taylor expanded in x around inf 76.2%

      \[\leadsto 0.5 \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -7.3 \cdot 10^{-37} \lor \neg \left(y \cdot \sqrt{z} \leq 8.5 \cdot 10^{+132}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x + y \cdot \sqrt{z}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 (+ x (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	return 0.5 * (x + (y * sqrt(z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * (x + (y * sqrt(z)))
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * (x + (y * Math.sqrt(z)));
}

Python

def code(x, y, z):
	return 0.5 * (x + (y * math.sqrt(z)))

Julia

function code(x, y, z)
	return Float64(0.5 * Float64(x + Float64(y * sqrt(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * (x + (y * sqrt(z)));
end

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
2. Step-by-step derivation

   1. metadata-eval 99.8%

      \[\leadsto \color{blue}{0.5} \cdot \left(x + y \cdot \sqrt{z}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]

4. Final simplification 99.8%

   \[\leadsto 0.5 \cdot \left(x + y \cdot \sqrt{z}\right) \]

Alternative 5: 50.2% accurate, 36.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 x))

C

double code(double x, double y, double z) {
	return 0.5 * x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * x
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * x;
}

Python

def code(x, y, z):
	return 0.5 * x

Julia

function code(x, y, z)
	return Float64(0.5 * x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * x;
end

Wolfram

code[x_, y_, z_] := N[(0.5 * x), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
2. Step-by-step derivation

   1. metadata-eval 99.8%

      \[\leadsto \color{blue}{0.5} \cdot \left(x + y \cdot \sqrt{z}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]

4. Taylor expanded in x around inf 51.9%

   \[\leadsto 0.5 \cdot \color{blue}{x} \]

5. Final simplification 51.9%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023238 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  :precision binary64
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))