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

Percentage Accurate: 99.8% → 99.8%

Time: 8.6s

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

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

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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2}\right), \color{blue}{\left(x + y \cdot \sqrt{z}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\color{blue}{x} + y \cdot \sqrt{z}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \sqrt{z}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\sqrt{z}\right)}\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(z\right)\right)\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 71.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5 \cdot y}{{z}^{-0.5}}\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{-86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 y) (pow z -0.5))))
   (if (<= y -4.7e-86) t_0 (if (<= y 1.62e+95) (* 0.5 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (0.5 * y) / pow(z, -0.5);
	double tmp;
	if (y <= -4.7e-86) {
		tmp = t_0;
	} else if (y <= 1.62e+95) {
		tmp = 0.5 * x;
	} else {
		tmp = t_0;
	}
	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 = (0.5d0 * y) / (z ** (-0.5d0))
    if (y <= (-4.7d-86)) then
        tmp = t_0
    else if (y <= 1.62d+95) then
        tmp = 0.5d0 * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (0.5 * y) / Math.pow(z, -0.5);
	double tmp;
	if (y <= -4.7e-86) {
		tmp = t_0;
	} else if (y <= 1.62e+95) {
		tmp = 0.5 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.5 * y) / math.pow(z, -0.5)
	tmp = 0
	if y <= -4.7e-86:
		tmp = t_0
	elif y <= 1.62e+95:
		tmp = 0.5 * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.5 * y) / (z ^ -0.5))
	tmp = 0.0
	if (y <= -4.7e-86)
		tmp = t_0;
	elseif (y <= 1.62e+95)
		tmp = Float64(0.5 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (0.5 * y) / (z ^ -0.5);
	tmp = 0.0;
	if (y <= -4.7e-86)
		tmp = t_0;
	elseif (y <= 1.62e+95)
		tmp = 0.5 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.5 * y), $MachinePrecision] / N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.7e-86], t$95$0, If[LessEqual[y, 1.62e+95], N[(0.5 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7000000000000001e-86 or 1.61999999999999993e95 < y

   1. Initial program 99.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2}\right), \color{blue}{\left(x + y \cdot \sqrt{z}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\color{blue}{x} + y \cdot \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \sqrt{z}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\sqrt{z}\right)}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(z\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{x - y \cdot \sqrt{z}}{x \cdot x - \left(y \cdot \sqrt{z}\right) \cdot \left(y \cdot \sqrt{z}\right)}\right)}\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{1}{\color{blue}{\frac{x \cdot x - \left(y \cdot \sqrt{z}\right) \cdot \left(y \cdot \sqrt{z}\right)}{x - y \cdot \sqrt{z}}}}\right)\right) \]

      6. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{1}{x + \color{blue}{y \cdot \sqrt{z}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \color{blue}{\left(x + y \cdot \sqrt{z}\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \sqrt{z}\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\sqrt{z}\right)}\right)\right)\right)\right) \]

      10. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \left(\sqrt{\sqrt{z} \cdot \sqrt{z}}\right)\right)\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(\left(\sqrt{z} \cdot \sqrt{z}\right)\right)\right)\right)\right)\right) \]

      12. rem-square-sqrt 99.6%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(z\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \color{blue}{\left(y \cdot \sqrt{z}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\sqrt{z}\right)}\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 78.0%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(z\right)\right)\right)\right) \]

   9. Simplified 78.0%

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

       1. associate-/r/ N/A

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

       2. metadata-eval N/A

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

       3. associate-*r* N/A

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

       4. remove-double-div N/A

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

       5. un-div-inv N/A

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

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot y\right), \color{blue}{\left(\frac{1}{\sqrt{z}}\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y\right), \left(\frac{\color{blue}{1}}{\sqrt{z}}\right)\right) \]

       8. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y\right), \left(\frac{1}{{z}^{\color{blue}{\frac{1}{2}}}}\right)\right) \]

       9. pow-flip N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y\right), \left({z}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y\right), \left({z}^{\frac{-1}{2}}\right)\right) \]

       11. pow-lowering-pow.f64 78.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y\right), \mathsf{pow.f64}\left(z, \color{blue}{\frac{-1}{2}}\right)\right) \]

   11. Applied egg-rr 78.1%

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

   if -4.7000000000000001e-86 < y < 1.61999999999999993e95

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2}\right), \color{blue}{\left(x + y \cdot \sqrt{z}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\color{blue}{x} + y \cdot \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \sqrt{z}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\sqrt{z}\right)}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(z\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

   7. Simplified 75.7%

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

Alternative 3: 71.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{-87}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{{z}^{-0.5}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.65e-87)
   (* y (* 0.5 (sqrt z)))
   (if (<= y 3.5e+95) (* 0.5 x) (/ 0.5 (/ (pow z -0.5) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.65e-87) {
		tmp = y * (0.5 * sqrt(z));
	} else if (y <= 3.5e+95) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 / (pow(z, -0.5) / y);
	}
	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 (y <= (-2.65d-87)) then
        tmp = y * (0.5d0 * sqrt(z))
    else if (y <= 3.5d+95) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 / ((z ** (-0.5d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.65e-87) {
		tmp = y * (0.5 * Math.sqrt(z));
	} else if (y <= 3.5e+95) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 / (Math.pow(z, -0.5) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.65e-87:
		tmp = y * (0.5 * math.sqrt(z))
	elif y <= 3.5e+95:
		tmp = 0.5 * x
	else:
		tmp = 0.5 / (math.pow(z, -0.5) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.65e-87)
		tmp = Float64(y * Float64(0.5 * sqrt(z)));
	elseif (y <= 3.5e+95)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 / Float64((z ^ -0.5) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.65e-87)
		tmp = y * (0.5 * sqrt(z));
	elseif (y <= 3.5e+95)
		tmp = 0.5 * x;
	else
		tmp = 0.5 / ((z ^ -0.5) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.65e-87], N[(y * N[(0.5 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e+95], N[(0.5 * x), $MachinePrecision], N[(0.5 / N[(N[Power[z, -0.5], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.64999999999999993e-87

   1. Initial program 99.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2}\right), \color{blue}{\left(x + y \cdot \sqrt{z}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\color{blue}{x} + y \cdot \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \sqrt{z}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\sqrt{z}\right)}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(z\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{2} \cdot \sqrt{z}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{z}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 71.6%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(z\right)\right)\right) \]

   7. Simplified 71.6%

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

   if -2.64999999999999993e-87 < y < 3.5e95

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2}\right), \color{blue}{\left(x + y \cdot \sqrt{z}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\color{blue}{x} + y \cdot \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \sqrt{z}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\sqrt{z}\right)}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(z\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

   7. Simplified 75.7%

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

   if 3.5e95 < y

   1. Initial program 99.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2}\right), \color{blue}{\left(x + y \cdot \sqrt{z}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\color{blue}{x} + y \cdot \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \sqrt{z}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\sqrt{z}\right)}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(z\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{x - y \cdot \sqrt{z}}{x \cdot x - \left(y \cdot \sqrt{z}\right) \cdot \left(y \cdot \sqrt{z}\right)}\right)}\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{1}{\color{blue}{\frac{x \cdot x - \left(y \cdot \sqrt{z}\right) \cdot \left(y \cdot \sqrt{z}\right)}{x - y \cdot \sqrt{z}}}}\right)\right) \]

      6. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{1}{x + \color{blue}{y \cdot \sqrt{z}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \color{blue}{\left(x + y \cdot \sqrt{z}\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \sqrt{z}\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\sqrt{z}\right)}\right)\right)\right)\right) \]

      10. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \left(\sqrt{\sqrt{z} \cdot \sqrt{z}}\right)\right)\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(\left(\sqrt{z} \cdot \sqrt{z}\right)\right)\right)\right)\right)\right) \]

      12. rem-square-sqrt 99.5%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(z\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \color{blue}{\left(y \cdot \sqrt{z}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\sqrt{z}\right)}\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 87.1%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(z\right)\right)\right)\right) \]

   9. Simplified 87.1%

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

       1. associate-/r/ N/A

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

       2. metadata-eval N/A

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

       3. pow1/2 N/A

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

       4. metadata-eval N/A

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

       5. pow-plus N/A

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

       6. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left({z}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot z\right)\right) \]

       7. pow-flip N/A

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

       8. pow1/2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. associate-*l* N/A

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

       12. associate-/r/ N/A

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

       13. un-div-inv N/A

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

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\sqrt{z}}{y \cdot z}\right)}\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{\sqrt{z}}{z \cdot \color{blue}{y}}\right)\right) \]

       16. associate-/r* N/A

           \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{\frac{\sqrt{z}}{z}}{\color{blue}{y}}\right)\right) \]

   11. Applied egg-rr 87.1%

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

Alternative 4: 71.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(0.5 \cdot \sqrt{z}\right)\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{-86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* 0.5 (sqrt z)))))
   (if (<= y -4.7e-86) t_0 (if (<= y 1.42e+103) (* 0.5 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (0.5 * sqrt(z));
	double tmp;
	if (y <= -4.7e-86) {
		tmp = t_0;
	} else if (y <= 1.42e+103) {
		tmp = 0.5 * x;
	} else {
		tmp = t_0;
	}
	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 * (0.5d0 * sqrt(z))
    if (y <= (-4.7d-86)) then
        tmp = t_0
    else if (y <= 1.42d+103) then
        tmp = 0.5d0 * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (0.5 * Math.sqrt(z));
	double tmp;
	if (y <= -4.7e-86) {
		tmp = t_0;
	} else if (y <= 1.42e+103) {
		tmp = 0.5 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (0.5 * math.sqrt(z))
	tmp = 0
	if y <= -4.7e-86:
		tmp = t_0
	elif y <= 1.42e+103:
		tmp = 0.5 * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(0.5 * sqrt(z)))
	tmp = 0.0
	if (y <= -4.7e-86)
		tmp = t_0;
	elseif (y <= 1.42e+103)
		tmp = Float64(0.5 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (0.5 * sqrt(z));
	tmp = 0.0;
	if (y <= -4.7e-86)
		tmp = t_0;
	elseif (y <= 1.42e+103)
		tmp = 0.5 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(0.5 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.7e-86], t$95$0, If[LessEqual[y, 1.42e+103], N[(0.5 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7000000000000001e-86 or 1.42e103 < y

   1. Initial program 99.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2}\right), \color{blue}{\left(x + y \cdot \sqrt{z}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\color{blue}{x} + y \cdot \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \sqrt{z}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\sqrt{z}\right)}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(z\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{2} \cdot \sqrt{z}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{z}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 78.0%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(z\right)\right)\right) \]

   7. Simplified 78.0%

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

   if -4.7000000000000001e-86 < y < 1.42e103

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2}\right), \color{blue}{\left(x + y \cdot \sqrt{z}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\color{blue}{x} + y \cdot \sqrt{z}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \sqrt{z}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\sqrt{z}\right)}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(z\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

   7. Simplified 75.7%

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

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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2}\right), \color{blue}{\left(x + y \cdot \sqrt{z}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\color{blue}{x} + y \cdot \sqrt{z}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \sqrt{z}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\sqrt{z}\right)}\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{sqrt.f64}\left(z\right)\right)\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 48.9%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

7. Simplified 48.9%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(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)))))