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

Percentage Accurate: 99.8% → 99.8%

Time: 7.8s

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -510000:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \frac{y}{{z}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|x\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -510000.0)
   (* 0.5 x)
   (if (<= x 1.15e+36) (* 0.5 (/ y (pow z -0.5))) (* 0.5 (fabs x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -510000.0) {
		tmp = 0.5 * x;
	} else if (x <= 1.15e+36) {
		tmp = 0.5 * (y / pow(z, -0.5));
	} else {
		tmp = 0.5 * fabs(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) :: tmp
    if (x <= (-510000.0d0)) then
        tmp = 0.5d0 * x
    else if (x <= 1.15d+36) then
        tmp = 0.5d0 * (y / (z ** (-0.5d0)))
    else
        tmp = 0.5d0 * abs(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -510000.0) {
		tmp = 0.5 * x;
	} else if (x <= 1.15e+36) {
		tmp = 0.5 * (y / Math.pow(z, -0.5));
	} else {
		tmp = 0.5 * Math.abs(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -510000.0:
		tmp = 0.5 * x
	elif x <= 1.15e+36:
		tmp = 0.5 * (y / math.pow(z, -0.5))
	else:
		tmp = 0.5 * math.fabs(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -510000.0)
		tmp = Float64(0.5 * x);
	elseif (x <= 1.15e+36)
		tmp = Float64(0.5 * Float64(y / (z ^ -0.5)));
	else
		tmp = Float64(0.5 * abs(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -510000.0)
		tmp = 0.5 * x;
	elseif (x <= 1.15e+36)
		tmp = 0.5 * (y / (z ^ -0.5));
	else
		tmp = 0.5 * abs(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -510000.0], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 1.15e+36], N[(0.5 * N[(y / N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Abs[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1e5

   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 71.3%

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

   if -5.1e5 < x < 1.14999999999999998e36

   1. Initial program 99.7%

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

      1. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. add-sqr-sqrt 99.3%

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

      4. associate-*l* 99.3%

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

      5. fma-def 99.3%

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

      6. pow1/2 99.3%

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

      7. sqrt-pow1 99.4%

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

      8. metadata-eval 99.4%

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

      9. pow1/2 99.4%

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

      10. sqrt-pow1 99.3%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, \color{blue}{{z}^{\left(\frac{0.5}{2}\right)}} \cdot y, x\right) \]

      11. metadata-eval 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. fma-udef 99.3%

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

      2. associate-*r* 99.4%

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

      3. pow-prod-up 99.7%

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

      4. metadata-eval 99.7%

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

      5. pow1/2 99.7%

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

      6. *-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. flip-+ 61.6%

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

      9. unpow2 61.6%

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

      10. div-sub 61.6%

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

      11. swap-sqr 54.0%

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

      12. unpow2 54.0%

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

      13. add-sqr-sqrt 54.0%

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

      14. *-commutative 54.0%

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

      15. div-sub 54.0%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{{x}^{2} - z \cdot {y}^{2}}{x - y \cdot \sqrt{z}}} \]

      16. clear-num 53.9%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{x - y \cdot \sqrt{z}}{{x}^{2} - z \cdot {y}^{2}}}} \]

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in y around inf 76.9%

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

      1. associate-/r* 77.1%

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

      2. clear-num 77.1%

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

      3. associate-/r* 77.1%

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

      4. add-cbrt-cube 35.2%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt[3]{\left(y \cdot y\right) \cdot y}}}{1 \cdot \sqrt{\frac{1}{z}}} \]

      5. unpow2 35.2%

         \[\leadsto 0.5 \cdot \frac{\sqrt[3]{\color{blue}{{y}^{2}} \cdot y}}{1 \cdot \sqrt{\frac{1}{z}}} \]

      6. cbrt-prod 51.1%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt[3]{{y}^{2}} \cdot \sqrt[3]{y}}}{1 \cdot \sqrt{\frac{1}{z}}} \]

      7. times-frac 51.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt[3]{{y}^{2}}}{1} \cdot \frac{\sqrt[3]{y}}{\sqrt{\frac{1}{z}}}\right)} \]

      8. unpow2 51.0%

         \[\leadsto 0.5 \cdot \left(\frac{\sqrt[3]{\color{blue}{y \cdot y}}}{1} \cdot \frac{\sqrt[3]{y}}{\sqrt{\frac{1}{z}}}\right) \]

      9. cbrt-prod 75.8%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{1} \cdot \frac{\sqrt[3]{y}}{\sqrt{\frac{1}{z}}}\right) \]

      10. pow2 75.7%

          \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{y}}{\sqrt{\frac{1}{z}}}\right) \]

      11. inv-pow 75.7%

          \[\leadsto 0.5 \cdot \left(\frac{{\left(\sqrt[3]{y}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{y}}{\sqrt{\color{blue}{{z}^{-1}}}}\right) \]

      12. sqrt-pow1 75.8%

          \[\leadsto 0.5 \cdot \left(\frac{{\left(\sqrt[3]{y}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{y}}{\color{blue}{{z}^{\left(\frac{-1}{2}\right)}}}\right) \]

      13. metadata-eval 75.8%

          \[\leadsto 0.5 \cdot \left(\frac{{\left(\sqrt[3]{y}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{y}}{{z}^{\color{blue}{-0.5}}}\right) \]

   10. Applied egg-rr 75.8%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{y}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{y}}{{z}^{-0.5}}\right)} \]
   11. Step-by-step derivation

       1. /-rgt-identity 75.8%

          \[\leadsto 0.5 \cdot \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \frac{\sqrt[3]{y}}{{z}^{-0.5}}\right) \]

       2. associate-*r/ 75.8%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{{\left(\sqrt[3]{y}\right)}^{2} \cdot \sqrt[3]{y}}{{z}^{-0.5}}} \]

       3. unpow2 75.8%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)} \cdot \sqrt[3]{y}}{{z}^{-0.5}} \]

       4. rem-3cbrt-lft 77.2%

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

   12. Simplified 77.2%

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

   if 1.14999999999999998e36 < x

   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. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate-*l* 99.8%

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

      5. fma-def 99.8%

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

      6. pow1/2 99.8%

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

      7. sqrt-pow1 99.8%

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

      8. metadata-eval 99.8%

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

      9. pow1/2 99.8%

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

      10. sqrt-pow1 99.8%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, \color{blue}{{z}^{\left(\frac{0.5}{2}\right)}} \cdot y, x\right) \]

      11. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. associate-*r* 99.9%

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

      3. pow-prod-up 99.9%

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

      4. metadata-eval 99.9%

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

      5. pow1/2 99.9%

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

      6. *-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. flip-+ 40.8%

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

      9. unpow2 40.8%

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

      10. div-sub 40.8%

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

      11. swap-sqr 38.3%

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

      12. unpow2 38.3%

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

      13. add-sqr-sqrt 38.3%

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

      14. *-commutative 38.3%

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

      15. div-sub 38.3%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{{x}^{2} - z \cdot {y}^{2}}{x - y \cdot \sqrt{z}}} \]

      16. clear-num 38.1%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{x - y \cdot \sqrt{z}}{{x}^{2} - z \cdot {y}^{2}}}} \]

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in y around 0 77.9%

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

      1. remove-double-div 78.1%

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

      2. add-sqr-sqrt 77.5%

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

      3. sqrt-unprod 37.5%

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

      4. pow2 37.5%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{2}}} \]

   10. Applied egg-rr 37.5%

       \[\leadsto 0.5 \cdot \color{blue}{\sqrt{{x}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 37.5%

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

       2. rem-sqrt-square 78.1%

          \[\leadsto 0.5 \cdot \color{blue}{\left|x\right|} \]

   12. Simplified 78.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -510000:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \frac{y}{{z}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|x\right|\\ \end{array} \]

Alternative 3: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|x\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1e+23)
   (* 0.5 x)
   (if (<= x 1.95e+36) (* 0.5 (* y (sqrt z))) (* 0.5 (fabs x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e+23) {
		tmp = 0.5 * x;
	} else if (x <= 1.95e+36) {
		tmp = 0.5 * (y * sqrt(z));
	} else {
		tmp = 0.5 * fabs(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) :: tmp
    if (x <= (-1d+23)) then
        tmp = 0.5d0 * x
    else if (x <= 1.95d+36) then
        tmp = 0.5d0 * (y * sqrt(z))
    else
        tmp = 0.5d0 * abs(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e+23) {
		tmp = 0.5 * x;
	} else if (x <= 1.95e+36) {
		tmp = 0.5 * (y * Math.sqrt(z));
	} else {
		tmp = 0.5 * Math.abs(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1e+23:
		tmp = 0.5 * x
	elif x <= 1.95e+36:
		tmp = 0.5 * (y * math.sqrt(z))
	else:
		tmp = 0.5 * math.fabs(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1e+23)
		tmp = Float64(0.5 * x);
	elseif (x <= 1.95e+36)
		tmp = Float64(0.5 * Float64(y * sqrt(z)));
	else
		tmp = Float64(0.5 * abs(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1e+23)
		tmp = 0.5 * x;
	elseif (x <= 1.95e+36)
		tmp = 0.5 * (y * sqrt(z));
	else
		tmp = 0.5 * abs(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1e+23], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 1.95e+36], N[(0.5 * N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Abs[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.9999999999999992e22

   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 72.7%

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

   if -9.9999999999999992e22 < x < 1.9500000000000001e36

   1. Initial program 99.7%

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

      1. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 76.4%

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

   if 1.9500000000000001e36 < x

   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. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate-*l* 99.8%

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

      5. fma-def 99.8%

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

      6. pow1/2 99.8%

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

      7. sqrt-pow1 99.8%

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

      8. metadata-eval 99.8%

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

      9. pow1/2 99.8%

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

      10. sqrt-pow1 99.8%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, \color{blue}{{z}^{\left(\frac{0.5}{2}\right)}} \cdot y, x\right) \]

      11. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. associate-*r* 99.9%

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

      3. pow-prod-up 99.9%

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

      4. metadata-eval 99.9%

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

      5. pow1/2 99.9%

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

      6. *-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. flip-+ 40.8%

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

      9. unpow2 40.8%

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

      10. div-sub 40.8%

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

      11. swap-sqr 38.3%

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

      12. unpow2 38.3%

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

      13. add-sqr-sqrt 38.3%

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

      14. *-commutative 38.3%

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

      15. div-sub 38.3%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{{x}^{2} - z \cdot {y}^{2}}{x - y \cdot \sqrt{z}}} \]

      16. clear-num 38.1%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{x - y \cdot \sqrt{z}}{{x}^{2} - z \cdot {y}^{2}}}} \]

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in y around 0 77.9%

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

      1. remove-double-div 78.1%

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

      2. add-sqr-sqrt 77.5%

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

      3. sqrt-unprod 37.5%

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

      4. pow2 37.5%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{2}}} \]

   10. Applied egg-rr 37.5%

       \[\leadsto 0.5 \cdot \color{blue}{\sqrt{{x}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 37.5%

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

       2. rem-sqrt-square 78.1%

          \[\leadsto 0.5 \cdot \color{blue}{\left|x\right|} \]

   12. Simplified 78.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|x\right|\\ \end{array} \]

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: 50.4% 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 47.1%

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

5. Final simplification 47.1%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023318 
(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)))))