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

Percentage Accurate: 99.8% → 99.8%

Time: 6.4s

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. 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 2: 70.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ t_1 := 0.5 \cdot \left(x - z \cdot \left(y \cdot \frac{y}{-x}\right)\right)\\ t_2 := 0.5 \cdot \left(x - z \cdot \left(y \cdot \frac{y}{x}\right)\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-92}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3500000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (* y (sqrt z))))
        (t_1 (* 0.5 (- x (* z (* y (/ y (- x)))))))
        (t_2 (* 0.5 (- x (* z (* y (/ y x)))))))
   (if (<= x -1.4e-150)
     t_1
     (if (<= x 3.5e-124)
       t_0
       (if (<= x 7.8e-92)
         t_2
         (if (<= x 3500000000000.0)
           t_0
           (if (<= x 8.5e+28) t_1 (if (<= x 2.05e+90) t_0 t_2))))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 * (y * sqrt(z));
	double t_1 = 0.5 * (x - (z * (y * (y / -x))));
	double t_2 = 0.5 * (x - (z * (y * (y / x))));
	double tmp;
	if (x <= -1.4e-150) {
		tmp = t_1;
	} else if (x <= 3.5e-124) {
		tmp = t_0;
	} else if (x <= 7.8e-92) {
		tmp = t_2;
	} else if (x <= 3500000000000.0) {
		tmp = t_0;
	} else if (x <= 8.5e+28) {
		tmp = t_1;
	} else if (x <= 2.05e+90) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * (y * sqrt(z))
    t_1 = 0.5d0 * (x - (z * (y * (y / -x))))
    t_2 = 0.5d0 * (x - (z * (y * (y / x))))
    if (x <= (-1.4d-150)) then
        tmp = t_1
    else if (x <= 3.5d-124) then
        tmp = t_0
    else if (x <= 7.8d-92) then
        tmp = t_2
    else if (x <= 3500000000000.0d0) then
        tmp = t_0
    else if (x <= 8.5d+28) then
        tmp = t_1
    else if (x <= 2.05d+90) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.5 * (y * Math.sqrt(z));
	double t_1 = 0.5 * (x - (z * (y * (y / -x))));
	double t_2 = 0.5 * (x - (z * (y * (y / x))));
	double tmp;
	if (x <= -1.4e-150) {
		tmp = t_1;
	} else if (x <= 3.5e-124) {
		tmp = t_0;
	} else if (x <= 7.8e-92) {
		tmp = t_2;
	} else if (x <= 3500000000000.0) {
		tmp = t_0;
	} else if (x <= 8.5e+28) {
		tmp = t_1;
	} else if (x <= 2.05e+90) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 * (y * math.sqrt(z))
	t_1 = 0.5 * (x - (z * (y * (y / -x))))
	t_2 = 0.5 * (x - (z * (y * (y / x))))
	tmp = 0
	if x <= -1.4e-150:
		tmp = t_1
	elif x <= 3.5e-124:
		tmp = t_0
	elif x <= 7.8e-92:
		tmp = t_2
	elif x <= 3500000000000.0:
		tmp = t_0
	elif x <= 8.5e+28:
		tmp = t_1
	elif x <= 2.05e+90:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 * Float64(y * sqrt(z)))
	t_1 = Float64(0.5 * Float64(x - Float64(z * Float64(y * Float64(y / Float64(-x))))))
	t_2 = Float64(0.5 * Float64(x - Float64(z * Float64(y * Float64(y / x)))))
	tmp = 0.0
	if (x <= -1.4e-150)
		tmp = t_1;
	elseif (x <= 3.5e-124)
		tmp = t_0;
	elseif (x <= 7.8e-92)
		tmp = t_2;
	elseif (x <= 3500000000000.0)
		tmp = t_0;
	elseif (x <= 8.5e+28)
		tmp = t_1;
	elseif (x <= 2.05e+90)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.5 * (y * sqrt(z));
	t_1 = 0.5 * (x - (z * (y * (y / -x))));
	t_2 = 0.5 * (x - (z * (y * (y / x))));
	tmp = 0.0;
	if (x <= -1.4e-150)
		tmp = t_1;
	elseif (x <= 3.5e-124)
		tmp = t_0;
	elseif (x <= 7.8e-92)
		tmp = t_2;
	elseif (x <= 3500000000000.0)
		tmp = t_0;
	elseif (x <= 8.5e+28)
		tmp = t_1;
	elseif (x <= 2.05e+90)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 * N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x - N[(z * N[(y * N[(y / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(x - N[(z * N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e-150], t$95$1, If[LessEqual[x, 3.5e-124], t$95$0, If[LessEqual[x, 7.8e-92], t$95$2, If[LessEqual[x, 3500000000000.0], t$95$0, If[LessEqual[x, 8.5e+28], t$95$1, If[LessEqual[x, 2.05e+90], t$95$0, t$95$2]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.39999999999999998e-150 or 3.5e12 < x < 8.49999999999999954e28

   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. flip-+ 51.1%

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

      2. div-inv 51.0%

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

      3. *-commutative 51.0%

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

      4. *-commutative 51.0%

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

      5. swap-sqr 48.6%

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

      6. add-sqr-sqrt 48.6%

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

   5. Applied egg-rr 48.6%

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

   6. Taylor expanded in x around inf 42.8%

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

   7. Taylor expanded in x around 0 69.0%

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

      1. +-commutative 69.0%

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

      2. mul-1-neg 69.0%

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

      3. unsub-neg 69.0%

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

      4. unpow2 69.0%

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

      5. associate-/l* 69.6%

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

      6. associate-/r/ 70.2%

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

   9. Simplified 70.2%

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

       1. frac-2neg 70.2%

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

       2. neg-sub0 70.2%

          \[\leadsto 0.5 \cdot \left(x - \frac{\color{blue}{0 - y \cdot y}}{-x} \cdot z\right) \]

       3. metadata-eval 70.2%

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

       4. div-sub 70.2%

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

       5. metadata-eval 70.2%

          \[\leadsto 0.5 \cdot \left(x - \left(\frac{\color{blue}{0}}{-x} - \frac{y \cdot y}{-x}\right) \cdot z\right) \]

       6. add-sqr-sqrt 70.2%

          \[\leadsto 0.5 \cdot \left(x - \left(\frac{0}{-x} - \frac{\color{blue}{\sqrt{y \cdot y} \cdot \sqrt{y \cdot y}}}{-x}\right) \cdot z\right) \]

       7. sqrt-unprod 64.5%

          \[\leadsto 0.5 \cdot \left(x - \left(\frac{0}{-x} - \frac{\color{blue}{\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}}}{-x}\right) \cdot z\right) \]

       8. sqr-neg 64.5%

          \[\leadsto 0.5 \cdot \left(x - \left(\frac{0}{-x} - \frac{\sqrt{\color{blue}{\left(-y \cdot y\right) \cdot \left(-y \cdot y\right)}}}{-x}\right) \cdot z\right) \]

       9. sqrt-unprod 21.7%

          \[\leadsto 0.5 \cdot \left(x - \left(\frac{0}{-x} - \frac{\color{blue}{\sqrt{-y \cdot y} \cdot \sqrt{-y \cdot y}}}{-x}\right) \cdot z\right) \]

       10. add-sqr-sqrt 78.3%

           \[\leadsto 0.5 \cdot \left(x - \left(\frac{0}{-x} - \frac{\color{blue}{-y \cdot y}}{-x}\right) \cdot z\right) \]

       11. frac-2neg 78.3%

           \[\leadsto 0.5 \cdot \left(x - \left(\frac{0}{-x} - \color{blue}{\frac{y \cdot y}{x}}\right) \cdot z\right) \]

       12. associate-/l* 85.4%

           \[\leadsto 0.5 \cdot \left(x - \left(\frac{0}{-x} - \color{blue}{\frac{y}{\frac{x}{y}}}\right) \cdot z\right) \]

   11. Applied egg-rr 85.4%

       \[\leadsto 0.5 \cdot \left(x - \color{blue}{\left(\frac{0}{-x} - \frac{y}{\frac{x}{y}}\right)} \cdot z\right) \]
   12. Step-by-step derivation

       1. div0 85.4%

          \[\leadsto 0.5 \cdot \left(x - \left(\color{blue}{0} - \frac{y}{\frac{x}{y}}\right) \cdot z\right) \]

       2. neg-sub0 85.4%

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

       3. associate-/l* 78.3%

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

       4. distribute-neg-frac 78.3%

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

       5. neg-mul-1 78.3%

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

       6. associate-*l/ 78.3%

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

       7. metadata-eval 78.3%

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

       8. associate-/r* 78.3%

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

       9. neg-mul-1 78.3%

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

       10. *-commutative 78.3%

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

       11. associate-*l* 85.4%

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

       12. associate-*r/ 85.4%

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

       13. *-rgt-identity 85.4%

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

   13. Simplified 85.4%

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

   if -1.39999999999999998e-150 < x < 3.4999999999999999e-124 or 7.7999999999999993e-92 < x < 3.5e12 or 8.49999999999999954e28 < x < 2.05000000000000021e90

   1. Initial program 99.5%

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

      1. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.3%

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

   if 3.4999999999999999e-124 < x < 7.7999999999999993e-92 or 2.05000000000000021e90 < x

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 31.1%

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

      2. div-inv 31.1%

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

      3. *-commutative 31.1%

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

      4. *-commutative 31.1%

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

      5. swap-sqr 27.6%

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

      6. add-sqr-sqrt 27.6%

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

   5. Applied egg-rr 27.6%

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

   6. Taylor expanded in x around inf 30.3%

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

   7. Taylor expanded in x around 0 71.8%

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

      1. +-commutative 71.8%

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

      2. mul-1-neg 71.8%

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

      3. unsub-neg 71.8%

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

      4. unpow2 71.8%

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

      5. associate-/l* 75.7%

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

      6. associate-/r/ 76.5%

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

   9. Simplified 76.5%

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

   10. Taylor expanded in y around 0 76.5%

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

       1. unpow2 76.5%

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

       2. associate-*r/ 83.3%

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

   12. Simplified 83.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-150}:\\ \;\;\;\;0.5 \cdot \left(x - z \cdot \left(y \cdot \frac{y}{-x}\right)\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-124}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-92}:\\ \;\;\;\;0.5 \cdot \left(x - z \cdot \left(y \cdot \frac{y}{x}\right)\right)\\ \mathbf{elif}\;x \leq 3500000000000:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+28}:\\ \;\;\;\;0.5 \cdot \left(x - z \cdot \left(y \cdot \frac{y}{-x}\right)\right)\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+90}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x - z \cdot \left(y \cdot \frac{y}{x}\right)\right)\\ \end{array} \]

Alternative 3: 53.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-202}:\\ \;\;\;\;0.5 \cdot \left(x - z \cdot \left(y \cdot \frac{y}{-x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x - z \cdot \left(y \cdot \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.2e-202)
   (* 0.5 (- x (* z (* y (/ y (- x))))))
   (* 0.5 (- x (* z (* y (/ y x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.2e-202) {
		tmp = 0.5 * (x - (z * (y * (y / -x))));
	} else {
		tmp = 0.5 * (x - (z * (y * (y / 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 <= (-6.2d-202)) then
        tmp = 0.5d0 * (x - (z * (y * (y / -x))))
    else
        tmp = 0.5d0 * (x - (z * (y * (y / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.2e-202) {
		tmp = 0.5 * (x - (z * (y * (y / -x))));
	} else {
		tmp = 0.5 * (x - (z * (y * (y / x))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.2e-202:
		tmp = 0.5 * (x - (z * (y * (y / -x))))
	else:
		tmp = 0.5 * (x - (z * (y * (y / x))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.2e-202)
		tmp = Float64(0.5 * Float64(x - Float64(z * Float64(y * Float64(y / Float64(-x))))));
	else
		tmp = Float64(0.5 * Float64(x - Float64(z * Float64(y * Float64(y / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6.2e-202)
		tmp = 0.5 * (x - (z * (y * (y / -x))));
	else
		tmp = 0.5 * (x - (z * (y * (y / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.2e-202], N[(0.5 * N[(x - N[(z * N[(y * N[(y / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x - N[(z * N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.2e-202

   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. flip-+ 50.7%

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

      2. div-inv 50.6%

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

      3. *-commutative 50.6%

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

      4. *-commutative 50.6%

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

      5. swap-sqr 47.2%

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

      6. add-sqr-sqrt 47.2%

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

   5. Applied egg-rr 47.2%

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

   6. Taylor expanded in x around inf 37.1%

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

   7. Taylor expanded in x around 0 64.0%

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

      1. +-commutative 64.0%

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

      2. mul-1-neg 64.0%

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

      3. unsub-neg 64.0%

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

      4. unpow2 64.0%

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

      5. associate-/l* 64.6%

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

      6. associate-/r/ 65.2%

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

   9. Simplified 65.2%

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

       1. frac-2neg 65.2%

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

       2. neg-sub0 65.2%

          \[\leadsto 0.5 \cdot \left(x - \frac{\color{blue}{0 - y \cdot y}}{-x} \cdot z\right) \]

       3. metadata-eval 65.2%

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

       4. div-sub 65.2%

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

       5. metadata-eval 65.2%

          \[\leadsto 0.5 \cdot \left(x - \left(\frac{\color{blue}{0}}{-x} - \frac{y \cdot y}{-x}\right) \cdot z\right) \]

       6. add-sqr-sqrt 65.2%

          \[\leadsto 0.5 \cdot \left(x - \left(\frac{0}{-x} - \frac{\color{blue}{\sqrt{y \cdot y} \cdot \sqrt{y \cdot y}}}{-x}\right) \cdot z\right) \]

       7. sqrt-unprod 59.6%

          \[\leadsto 0.5 \cdot \left(x - \left(\frac{0}{-x} - \frac{\color{blue}{\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}}}{-x}\right) \cdot z\right) \]

       8. sqr-neg 59.6%

          \[\leadsto 0.5 \cdot \left(x - \left(\frac{0}{-x} - \frac{\sqrt{\color{blue}{\left(-y \cdot y\right) \cdot \left(-y \cdot y\right)}}}{-x}\right) \cdot z\right) \]

       9. sqrt-unprod 21.5%

          \[\leadsto 0.5 \cdot \left(x - \left(\frac{0}{-x} - \frac{\color{blue}{\sqrt{-y \cdot y} \cdot \sqrt{-y \cdot y}}}{-x}\right) \cdot z\right) \]

       10. add-sqr-sqrt 72.2%

           \[\leadsto 0.5 \cdot \left(x - \left(\frac{0}{-x} - \frac{\color{blue}{-y \cdot y}}{-x}\right) \cdot z\right) \]

       11. frac-2neg 72.2%

           \[\leadsto 0.5 \cdot \left(x - \left(\frac{0}{-x} - \color{blue}{\frac{y \cdot y}{x}}\right) \cdot z\right) \]

       12. associate-/l* 79.3%

           \[\leadsto 0.5 \cdot \left(x - \left(\frac{0}{-x} - \color{blue}{\frac{y}{\frac{x}{y}}}\right) \cdot z\right) \]

   11. Applied egg-rr 79.3%

       \[\leadsto 0.5 \cdot \left(x - \color{blue}{\left(\frac{0}{-x} - \frac{y}{\frac{x}{y}}\right)} \cdot z\right) \]
   12. Step-by-step derivation

       1. div0 79.3%

          \[\leadsto 0.5 \cdot \left(x - \left(\color{blue}{0} - \frac{y}{\frac{x}{y}}\right) \cdot z\right) \]

       2. neg-sub0 79.3%

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

       3. associate-/l* 72.2%

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

       4. distribute-neg-frac 72.2%

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

       5. neg-mul-1 72.2%

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

       6. associate-*l/ 72.2%

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

       7. metadata-eval 72.2%

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

       8. associate-/r* 72.2%

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

       9. neg-mul-1 72.2%

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

       10. *-commutative 72.2%

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

       11. associate-*l* 79.3%

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

       12. associate-*r/ 79.3%

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

       13. *-rgt-identity 79.3%

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

   13. Simplified 79.3%

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

   if -6.2e-202 < x

   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. flip-+ 47.9%

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

      2. div-inv 47.8%

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

      3. *-commutative 47.8%

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

      4. *-commutative 47.8%

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

      5. swap-sqr 38.7%

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

      6. add-sqr-sqrt 38.7%

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

   5. Applied egg-rr 38.7%

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

   6. Taylor expanded in x around inf 22.8%

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

   7. Taylor expanded in x around 0 41.0%

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

      1. +-commutative 41.0%

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

      2. mul-1-neg 41.0%

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

      3. unsub-neg 41.0%

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

      4. unpow2 41.0%

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

      5. associate-/l* 42.3%

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

      6. associate-/r/ 42.9%

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

   9. Simplified 42.9%

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

   10. Taylor expanded in y around 0 42.9%

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

       1. unpow2 42.9%

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

       2. associate-*r/ 45.5%

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

   12. Simplified 45.5%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-202}:\\ \;\;\;\;0.5 \cdot \left(x - z \cdot \left(y \cdot \frac{y}{-x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x - z \cdot \left(y \cdot \frac{y}{x}\right)\right)\\ \end{array} \]

Alternative 4: 53.2% accurate, 9.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 0.5 * (x - (z * (y * (y / 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 - (z * (y * (y / x))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(x - N[(z * N[(y * N[(y / x), $MachinePrecision]), $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. flip-+ 48.9%

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

   2. div-inv 48.8%

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

   3. *-commutative 48.8%

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

   4. *-commutative 48.8%

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

   5. swap-sqr 41.7%

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

   6. add-sqr-sqrt 41.7%

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

5. Applied egg-rr 41.7%

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

6. Taylor expanded in x around inf 27.9%

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

7. Taylor expanded in x around 0 49.2%

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

   1. +-commutative 49.2%

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

   2. mul-1-neg 49.2%

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

   3. unsub-neg 49.2%

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

   4. unpow2 49.2%

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

   5. associate-/l* 50.2%

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

   6. associate-/r/ 50.8%

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

9. Simplified 50.8%

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

10. Taylor expanded in y around 0 50.8%

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

    1. unpow2 50.8%

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

    2. associate-*r/ 55.2%

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

12. Simplified 55.2%

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

13. Final simplification 55.2%

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

Alternative 5: 50.7% 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 52.6%

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

5. Final simplification 52.6%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023182 
(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)))))